arXiv:math.AT/0412552 v1 31 Dec 2004

On the Foundation of Algebraic Topology

G. Fors

8/12-2004

Abstract

In this paper we add “a real unit w.r.t. topological join” to the classi-

cal category of topological spaces and continuous maps, turning it into a
convenient category for mathematical physics, since this addition provides
the category with a true monoidal structure w.r.t. topological join. It also
sharpens definitions concerning manifold boundaries by unifying the classi-
cal relative and the reduced homology functors into one single functor that
allows a K¨

unneth Theorem for joins. The term “the real 0” (w.r.t. ordinal

addition) was introduced by Mac Lane in [27] p. 178 as he pointed out that
“the topologist’s ∆” does not contain this, for Topological Quantum Field
Theories, vital object, cf. [24] p. 187-8.

2000 Mathematics Subject Classification. Primary 55Nxx, 55N10; Secondary 57P05, 57Rxx
key words and phrases. Algebraic Topology, Augmental Homology, join, manifolds
E-mail: goranf@math.su.se

Contents

1 Introduction

1.1

Relating Combinatorics to Commutative Algebra: . . . . . .

1.2

Looking Elsewhere for “Simplicial Complexes” . . . . . . .

1.3

Relating Combinatorics to Logics: . . . . . . . . . . . . . .

1.4

The K¨

unneth Formula for Joins: . . . . . . . . . . . . . . .

1.5

Relating General Topology to Combinatorics: . . . . . . . .

2 Augmental Homology Theory

2.1

Definition of Underlying Categories and Notations . . . . .

2.2

Realizations of Simplicial Complexes

: . . . . . . . . . . . .

2.3

Topological Properties of Realizations: . . . . . . . . . . . .

2.4

Simplicial Augmental Homology Theory . . . . . . . . . . .

2.5

Singular Augmental Homology Theory . . . . . . . . . . . .

3 Augmental Homology Modules for Products and Joins

3.1

Background to the Product- and Joindefinitions . . . . . . .

3.2

Definitions of Products and Joins . . . . . . . . . . . . . . .

3.3

Augmental Homology for Products and Joins . . . . . . . .

3.4

Local Augmental Homology Groups for Products and Joins

3.5

Singular Homology Manifolds under Products and Joins . .

4 Relating General Topology to Combinatorics

4.1

Realizations with respect to Simplicial Products and Joins .

4.2

Connectedness and Local Homology with respect to Simplicial Products and Joins 33

5 Relating Combinatorics to Commutative Algebra

5.1

Definition of Stanley-Reisner rings . . . . . . . . . . . . . .

5.2

Buchsbaum & (2-)Cohen-Macaulay Complexes are Weak Manifolds 38

5.3

Stanley-Reisner Rings for Simplicial Products are Segre Products 44

5.4

Gorenstein Complexes without Cone Points are Homology Spheres 45

6 Simplicial Manifolds

6.1

Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.2

Auxiliaries . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.3

Products and Joins of Simplicial Manifolds . . . . . . . . .

6.4

Simplicial Homology

Manifolds and Their Boundaries . . .

7 Appendices

7.1

The 3×3-lemma (also called “The 9-lemma”) . . . . . . . .

7.2

Simplicial Calculus . . . . . . . . . . . . . . . . . . . . . . .

8 Addendum:

8.1

The Importance of Simplicial Complexes . . . . . . . . . . .

8.2

−

Homology Groups for Joins of Arbitrary Simplicial Complexes 64

Introduction

1.1

Relating Combinatorics to Commutative Algebra:

Studying Gr¨abe’s paper, [16] 4.2 p. 171, this author noticed a boundary
formula for joins of simplicial manifolds, the singular homology analog of
which were nonexisting. This insufficiency of algebraic topology was traced
back to the fact that algebraic topologists still use the classical category of
simplicial complexes and simplicial maps - abandoned in the 70th by the
combinatorialists for the following:

Definition. 1 An (augmented abstract) Simplicial Complex Σ on a vertex
set V

is a collection of finite subsets σ, the simplices, of V

satisfying:

(a)

If v ∈ V

, then {v} ∈ Σ.

(b)

If σ ∈ Σ and τ ⊂ σ then τ ∈ Σ.

The Simplicial Join “Σ

∗ Σ

”, of two disjoint simplicial complexes Σ

and

is defined through:

∗ Σ

:= {σ

∪ σ

| σ

∈ Σ

(i = 1, 2)}.

Definition 1 has three type-theoretical levels - a complexe is a set of

simplices, which are finite sets of vertices. ∅ plays a dual role; it’s both a
(−1)-dimensional simplex and a (−∞)-dimensional complex. See p. 13 for
the definition of “dimension”.

Definition. 2. ([40] p. 108-9) A classical (abstract) Simplicial Complex Σ
on a vertex set V

is a collection of non-empty finite subsets σ, the simplices,

of V

satisfying:

(a)

If v ∈ V

, then {v} ∈ Σ.

(b)

If σ ∈ Σ and τ ⊂ σ then τ ∈ Σ.

The classical Simplicial Join “Σ

∗ Σ

”, of two disjoint classical simplicial

complexes Σ

and Σ

is defined through:

∗ Σ

:= Σ

∪ Σ

∪ {σ

∪ σ

| σ

∈ Σ

(i = 1, 2)}.

The difference between Def. 1 and Def. 2, is basically that the former

allows the empty set ∅ as a simplex, implying that the simplicial complex
induced by any vertex v, called a (combinatorial) 0-ball, is the set • :=
{∅, {v}} resp. {{v}}. Similarly, the (combinatorial) 0-sphere is the set
•• := {∅, {v}, {w}} resp. {{v}, {w}}.

The lethal combination of the complex-part of Def. 2 and the join-part

of Def. 1 is quite common, even in refereed papers. In this environment the
join-operation results in a simplicial complex (in any sense) only if one of
the join-factors is empty.

The above change of foundation took place the minute combinatorial-

ists began to use (“clashed into” - cf. [28]) commutative algebra to solve
combinatorial problems. The only homology-apparatus invented by the
combinatorialists to handle their ingenious category modification were a
jargon like -“We’ll use reduced homology with e

−1

({∅}) = Z”, which we’ll

now use as an eye-opener.

Let K be the classical category of simplical complexes and simplicial maps

resulting from Def. 2, and let K

be the category of simplices resulting

from Def. 1. Define E

: K → K

to be the functor adjoining ∅, as a

simplex, to each classical simplicial complex. E

has an inverse E : K

→

K deleting ∅. This idea is older than category theory itself. It was suggested
by S. Eilenberg & S. MacLane in [11], i.e. in their Group Extensions and
Homology, Ann. of Math. (4) 43 (1942), 757-831, where they in page
820, when exploring the 0-dimensional homology of a simplicial complex K
using the incidence numbers [ : ], wrote:

An alternative procedure would be to consider K “augmented”
by a single (−1)-cell σ

−1

such that [σ

: σ

−1

] = 1 for all σ

Any link of a simplicial homology manifold is either a simplical homology

ball or a sphere. Its boundary is the set of simplices having ball-links, i.e.

Bd Σ := {σ ∈ Σ| e

−#σ

(Lk

σ) = 0},

with

σ := {τ ∈ Σ|[σ ∩ τ = ∅] ∧ [σ ∪ τ ∈ Σ]}.

Since, Lk

∅ = Σ (“the missing link”), Bd P

= {∅} 6= ∅, for the real projec-

tive plane P

. For any point •, Bd(•) = Bd({∅, {v}}) = {∅} =“the join-unit”,

i.e. the boundary of a 0-ball is the −1- sphere. • is the only finite orientable

manifold having {∅} as its boundary. For the 0-sphere Bd(••) = ∅ - the join-
zero w.r.t. Def. 1. The classical setting gives;

Bd({{v}}) = Bd({{v}, {w}}) = ∅ = “the join-unit” w.r.t. to Def. 2,

implying that the boundary formula below, for the join of two homology
manifolds, can’t hold classically but, as will be shown, always holds if al-
gebraic topology is based on Def. 1. Moreover,

E(BdΣ) = Bd(E(Σ)),

i.e., what remains after deleting ∅ is exactly the classical boundary - always!
{∅} will serve as the (abstract simplicial) (−1)-standard simplex.

We’ll see that the join-operation in the category of simplicial complexes

and simplicial sets in their augmented form, will turn into (graded) tensor
product when subjected to either our homology functor

−

H or the Stanley-

Reisner-ring functor.

We’ll give a few more simple examples showing how our non-classical

approach simplifies algebraic topology through:

Bd(M

∗ M

) = ((BdM

) ∗ M

) ∪ (M

∗ (BdM

)).

• • = •∗• is a figurative way of saying that the simplicial complex generated
by a 1-simplex equals the join of two 0-dimensional simplicial complexes
each generated by a 0-simplex, so:

Bd(• •) = Bd(• ∗ •) = ({∅} ∗ •) ∪ (• ∗ {∅}) = • ∪ • =: • • .

Set

• •

• • := (••) ∗ (••) = (the simplicial) 1-sphere. So:

Bd(• •

• •) = Bd((••) ∗ (••)) = (∅ ∗ (••)) ∪ ((••) ∗ ∅) = ∅.

It’s also trivial to calculate the following three low-dimensional join

boundaries of;

1. M ∗ • = the cone of the M¨obius band M,

2. P

∗ • = the cone of the real projective plane P

3. S

∗ • = “the cone of the 2-sphere S

” = B

, the solid 3-dimensional

ball, usually called the 3-ball,

regarding the factors as homology manifolds using the prime field Z

with

three elements as coefficient module for the homology groups.

1. Bd

(M ∗ •) = ((Bd

M) ∗ •) ∪ (M ∗ Bd

•) = ((S

) ∗ •) ∪ (M ∗ Bd

•) =

∪ (M ∗ {∅} = B

∪ M = P

:= the real projective plane. B

is the

2-dimensional disk or “2-ball”.

2. Bd

∗•) = ((Bd

)∗•)∪(P

∗Bd

•) = ({∅}∗•)∪(P

∗{∅}) = •∪P

3. Bd

∗ •) = ((Bd

) ∗ •) ∪ (S

∗ Bd

•) = (∅ ∗ •) ∪ (S

∗ {∅}) = S

∗ • is a homology

3-manifold with boundary • ∪ P

if p 6= 2, by

[17] p. 36 and Th. 12 p. 54, while M ∗ • fails to be one due to the cone-
point. However, M ∗ • is a quasi-3-manifold, see p. 48, which through
{σ ∈ M ∗ •| e

−#σ

(Lk

M ∗•

σ) = 0} have the real projective plane P

as a well-

defined boundary as given above. The topological realisation, cf. p. 15,
of P

∗ • is not a topological manifold, due to its cone-point. This is

true also classically. Topological n-manifolds can only have n − 1-, −1-
or −∞-dimensional boundaries as homology manifolds, due to their local
orientability. The global non-orientability of M and P

becomes local non-

orientability at the cone-point in M ∗ • resp. P

∗ •.

1.2

Looking Elsewhere for “Simplicial Complexes”

Fritsch & Piccinini gives the next definition and example in Ch. 3 in [15] p.
110: [15] is an excellent book but, with respect to our intensions, it exhibits
an important (and common) compatibility problem between its Chapter 3
and its Chapter 4, where the “topologist’s ∆”

∆

is used.

Definition 3 (Equivalent to our Definition 1.) A Simplicial
Complex is a set Σ of finite sets closed under the formation of
subsets, i.e., any subset of a member of Σ is also a member of
Σ; more formally:

σ ∈ Σ ∧ τ ⊂ σ =⇒ τ ∈ Σ.

Example. 1. ([15] p. 110) Let {U

: λ ∈ Λ} be a family of

arbitrary sets; then the set K(Λ) of all finite subsets of Λ such
that

λ∈x

6= ∅ is a simplicial complex. Note that in general

the vertex set of this simplicial complex K(Λ) is not the index

set Λ itself, but only its subsets consisting of the indices λ with
U

6= ∅. Now, if Z is a space and {U

: λ ∈ Λ} is a covering of

Z (see Section A.3), then the simplicial complex K(Λ) obtained
in this way is called the nerve of the covering {U

: λ ∈ Λ}.

Note.

∈∅

= Universe 6= ∅ ⇒ x = ∅ ∈ K(Λ) (cf. [23], Kelley: General

Topology, p. 256). So, we note that any nerv K(Λ) 6= ∅, as just defined,
contains the simplex ∅, in full accordance with the definition of a simplicial
complex in definition 3 above.

By mainly just well-ordering the vertices in each simplex of a simpli-

cial complex we posses a standard procedure, cf. [15] p. 152 Ex. 4, to
associate the above defined “simplicial complex” in Def. 1 with an “aug-
mented simplicial set” in the functor category of Augmented Simplicial
Sets, cf. [27] p. 178-9, where an object is a contravariant functor from
the Simplicial Category to the Category of Sets. Equivalently, the com-
plexes resulting from Def. 2 is associated with those in the category of
Simplicial Sets. The natural transformations constitute the morphisms.
Formally, both these categories allows the notation as the functor cate-
gory (Category of sets)

(Simplicial Category)

. Obviously, the Simplicial Category

is a somewhat controversial concept among algebraic topologists. Indeed,
Fritsch & Piccinini are quite explicit about the definition of their Simplicial
Category, which they call the finite ordinals and we quote from [15] p. 220-1:

-“A short but systematic treatment of the category of finite or-
dinals can be found in MacLane (1971) under the name “the
simplicial category”. Our category of finite ordinals is not ex-
actly the same as MacLane’s; however, it is isomorphic to the
subcategory obtained from MacLane’s category by removing its
initial object”.

The last sentence pinpoints the disguised “initial simplex”-deletion de-

scribed in the following quotation from Mac Lane [27] p. 178 (=2:nd ed. of
“Mac Lane (1971)”):

-“By

∆

we denote the full subcategory of

∆

with objects all

the positive ordinals {1, 2, 3, . . . } (omit only zero). Topologists
use this category, call it

∆

, and rewrite its objects (using the

geometric dimension) as {0, 1, 2, . . . }. Here we stick to our

∆

which contains the real 0, an object which is necessary if all
face and degeneracy operations are to be expressed as in (3), in
terms of binary product µ and unit η.”

This mentioned Eq. (3) is part of Mac Lane’s description in [27] p. 175ff,

of the Simplicial Category with objects being the generic ordered abstrac-
tions of the simplices in Definition 1 p. 3 above, and with the empty set
∅, denoted 0 by Mac Lane, as an initial object which induces a unit oper-
ation η in a “universal monoid”-structure imposed by “ordinal addition”.

The latter being the equivalent of the union “σ

∪σ

” involved in the join-

definition in Definition 1 above. In particular, the join-operation has a much
more central role in modern mathematical physics than yet recognized.

This was hinted at in [24] p. 188 where J. Kock describe the classical

use of the “topologist ∆”, there named “topologist’s delta” and denoted
△, within category theory, in the following words:

In this way, the topologist’s delta is a sort of bridge between cat-
egory theory and topology. In these contexts the empty ordinal
(empty simplex) is not used, but in our context it is important
to keep it, because without it we could not have a monoidal
structure.

Next definition is equivalent to our Def. 2. and is given in p. 2 in Andr´e

Joyal & Myles Tierney; An Introduction to Simplicial Homotopy Theory;
Aug. 5 1999, found on www at the Hopf Topological Archive;

Definition 4 A Simplicial Complex K is a collection of non-
empty, finite subsets (called simplices) of a given set V (of ver-
tices) such that any non-empty subset of a simplex is a simplex.

The pure “Def. 2”-based, i.e. classical, Algebraic Topology demands the
classical combination of the relative homology functor and the ad-hoc in-
vented reduced homology functor, the latter of which is completely side-
stepped in the classical definition of the boundary of a homology manifold.
External adjunction of a −1-dimensional subspace but else following the
classical introduction of relative simplicial homology groups will produce a
unified homology functor,

−

H, resulting in clearer definitions and the final

elimination of the classical reduced functor, cf. p. 18. Other interesting
aspects of the development of mathematics are found in [26] and [28].

1.3

Relating Combinatorics to Logics:

The “standard procedure” mentioned in the last subsection, can’t handle
our new “Def. 1”-generated simplicial complex {∅}, since the simplicial
complex ∅ occupies the possibility of collapsing ∆ into the empty set. We
therefore disjointly add to each classical set S a final element ℘, regarding
S

℘

:= S ⊔{℘} as a category of its subsets w.r.t. partial injections, dumping

extras on ℘, i.e., Mor

℘

, B

℘

) 6= ∅ iff A

℘

⊃ B

℘

. Finally we add ∅. Now, a

0-dimensional simplicial complex equals its set of vertices, as in the classical
setting.

Viewing a set as a discrete topological space we add to each topological

space X, using topological sum, an external element ℘, resulting in X

℘

X + {℘}. Finally we add the universal initial object ∅. This is actually
a familiar routine from Homotopy Theory providing all free spaces with
a common base point, cf. [46]; G.W. Whitehead: Elements of Homotopy
Theory, GTM 61 Springer 1978 p. 103. Working with X

℘

instead of X

makes it possible, e.g., to deduce the K¨

unneth Formula for joins.

1.4

The K¨

unneth Formula for Joins:

That the join-operation is att the hart of algebraic topology, becomes ap-
parent for instance through Milnor’s construction of the universal principal
fiber bundle in [30] where he also formulate the non-relative K¨

unneth for-

mula for joins as;

∗ Y

)

∼

+j=q

⊕

(

) ⊗

)

+j=q−1

⊕ Tor

(

), e

)

i.e. the “X

= ∅”-case in our Th. 4 p. 26. Milnor’s results, apparently, in-

spired G.W. Whitehead to introduce the Augmental Total Chain Complex

S(◦) and Augmental Homology, e

⋆

(◦), in [45]. e

S(◦) comprise an in-built

dimension shift, which in our version becomes an application of the sus-
pension operator on the underlying singular chain. G.W. Whitehead gives
the empty space, ∅, the status of a (-1)-dimensional standard simplex but,
in his pair space theory he never took into account that the join-unit ∅ then
would have the identity map, Id

∅

, as a generator for its (-1)-dimensional sin-

gular augmental chain group, which, correctly interpreted, actually makes
his pair space theory identical to the ordinary relative homology functor.
G.W. Whitehead states thate

S(X∗Y ) and e

S(X)⊗e

S(Y ) are chain equivalent,

≈

, cf. our Theorem 3 p. 25 and then in a footnote he points out, referring

to [30] p. 431 Lemma 2.1, that;

This fact does not seem to be stated explicitly in the litera-
ture but is not difficult to deduce from Milnor’s proof of the
“K¨

unneth theorem” for the homology groups of the join.

On the chain level it’s indeed “not difficult” to see what’s needed to achieve

S(X∗Y) ≈ e

S(X) ⊗ e

S(Y), since the right hand side is wellknown, but then to

actually do it for the classical category of topological spaces and within the
frames of the Eilenberg-Steenrod formalism is, unfortunately, impossible,
since the need for a (−1)-dimensional standard simplex is indisputable and
the initial object ∅ just won’t do. This is observed in [29] p. 108, leading
up to a refutation of the above approach, considering that the classical
(as well as Whitehead’s) join-unit ∅ already plays a definite role in the
Eilenberg-Steenrod formalism, cf. [12] p. 3-4, and that the “convention”
H

(·) = H

(·, ∅; Z), cp. [12] pp. 3 + 273, is more than a mere “convension”

in that it connects the single space and the pair space theories.

For “Def. 1”-simplicial complexes, G.W. Whitehead’s “chain equivalent”-

claim above (=the non-relative Eilenberg-Zilber theorem for joins), is easily
proven. Our Th. 3 p. 25, answers G.W. Whitehead’s almost 60 years old
quest above for a proof.

Our K¨

unneth Theorem for joins i.e. Th. 4 p. 26, holds for arbitrary

topological spaces and the generalization, i.e. Theorem 7 p. 31, of the
above boundary formula, holds also for non-triangulable homology man-
ifolds, which we’ve restricted to be locally compact Hausdorff but weak
Hausdorff k-spaces, cf. [15] p. 243ff, would do since we only need the T

separation axiom (⇔points are closed).

Restricting to topological k-spaces, [15] p. 157ff remains true also after

our category modifications, i.e. Geometric realization preserves finite limits
and all colimits. The simplicial as well as the topological join-operation are
(modulo realization, equivalent) cases of colimits, as being a simplicial resp.
a topological attachment.

As pointed out in p. 8, the ”Def. 1”-induced (simplicial) join is, through

the “ordinal addition”, rooted directly at the fundament of mathematical
logics, as is Commutative Algebra, as we learn from Note iii p. 37 that the
Stanley-Reisner (St-Re) ring of a join of two simplicial complexes is the
tensor product of their respective St-Re rings. The join operation has been
unfairly ignored in the literature, except, of course for [4], [10] and [14].

1.5

Relating General Topology to Combinatorics:

Contemporary Combinatorics as well as Fritsch & Piccinini [15] Ch. 3,
through Def. 1/Def. 3 above, gives ∅ and {∅} as simplicial complexes, which
isn’t compatible with the “Def. 2/Def. 4”-based [15] Ch. 4 (pp. 132-222)
or any other introduction of Simplicial Sets except Mac Lane’s category of
Augmented Simplicial Sets in [27] p. 178-9.

A simplicial set is usually perceived through its target as a dimension-

indexed collection of its non-degenerate simplices, i.e. as a simplicial com-
plex, which itself usually is visualized through its realization. The mor-
phisms in the category of simplicial sets, i.e. the natural transformations,
now materializes as realisations of simplicial maps. Every mathematician
has a clear picture of what is ment by a “realisation of a simplicial com-
plex”. Here we find a plausible reason to why e.g. the strange ad hoc
splitting of the homology functor into a relative and a reduced part hasn’t
been rectified. The category of classical topological spaces and continu-
ous functions

isn’t compatible with any reasonable “realization func-

tor”, since ∅ is a (−∞)-dimensional zero element while {∅} is the (-1)-
dimensional unit element w.r.t. the simplicial join operation, where “com-
patible” should imply that their realizations should hold the same positions
w.r.t. topological join.

Since classical Combinatorics and classical General Topology cannot sep-

arate the “join zero” from the “join unit” or the boundary of the 0-ball from
that of the 0-sphere, any theory constructed thereon - classical Algebraic
Topology in particular - will need ad-hoc definitions/reasoning. The use of
a relative and a separate reduced homology functor instead of one single
homology functor, is only one example.

A moment of reflexion on “realization functor”-candidates | · |, reveals

the need for a real topological join unit, {℘} = |{∅}| 6= |∅| = ∅.

We’re obviously dealing with a non-classical situation, which we must

not try to squeeze into a classical framework.

As for E

in p. 4 let F

℘

: D → D

℘

be the functor adjoining to each classical

topological space a new non-final element denoted ℘. Put X

℘

:= X + {℘},

resembling Maunder’s modified K

-object in [29] p. 317. F

℘

has an inverse

F : D

℘

−→ D deleting ℘.

{℘} becomes our new (−1)-dimensional singular standard simplex and

join unit.

Prop. 1 p. 27, partially quoted below, reveals a connection between the

combinatorial and the topological local structures and is in itself a strong
motivation for introducing a topological (-1)-object {℘}, imposing the fol-
lowing definition (from p. 15) of a “point-setminus” “\

”, in D

℘

. (We’re

using the convension x ↔ {x, ℘}, as apposed to the classical x ↔ {x}.)

We can’t risk dropping out of our new category so for X

℘

6= ∅; X

℘

x :=

∅ if x = ℘ and F

℘

else. (“

”:= classical “setminus” and “∅

” is ∅

regarded as a simplex, and not as a simplicial complex.)

“The contrastar of σ ∈ Σ” = cost

σ := {τ ∈ Σ| τ 6⊇ σ}, implying that:

cost

∅

= ∅ and cost

σ = Σ iff σ 6∈ Σ.

Proposition. (p. 27) Let G be any module over a commutative ring A
with unit. With α ∈ Intσ and α = α

iff σ = ∅

the following module

isomorphisms are all induced by chain equivalences, cf. [33] p. 279 Th. 46.2
quoted here in p. 25.

−

(Lk

σ; G)

∼

−

(Σ, cost

σ; G)

∼

−

(|Σ|, |cost

σ|; G)

∼

−

(|Σ|, |Σ| \

α; G).

Astro-Physical inspiration: Today astronomers know that any galaxy, in-

cluding the Milky Way, possesses a black hole, active or passive, here inter-
preted as {℘}, and so, eliminating it, would dispose off the whole galaxy,
hinting at the above “X

℘

℘ := ∅”. Well, then again, the last proposition

leaves us with no other option, since Lk

∅

= Σ by definition.

These new elements {∅} resp. {℘} induces, through axiomatically in-

vented functors, “natural” tensor product units into the various algebras
used within Topological Quantum Field Theories, as well as through our
simplicial and singular chain functors, pp. 18-20, and the Stanley-Reisner
ring functor defined in p. 37.

This paper features augmented simplicial and singular (co)homology

functors that unifies the classical relative and reduced counterparts. In this
process an augmental algebraic topology emerge that has the augmented
pair-category of topological spaces and continuous functions as source cat-
egory. The result is a synchronization of general and algebraic topology to
that of the tensor categories within commutative algebra that is completely
compatible with the equivalent synchronization of combinatorics that took
place already in the seventies. These augmental structures are also seen
to fit like a hand in a glove when it comes to topological quantum field
theories.

Augmental Homology Theory

2.1

Definition of Underlying Categories and Notations

§2.1 is ment to lay down a firm formal foundation for our extension to

an overall augmented environment.

The typical morphisms in the classical category K of simplicial complexes

with vertices in W are the simplicial maps as defined in [40] p. 109, implying
in particular that;
Mor

(∅,Σ) = {∅} = {0

∅,Σ

Mor

(Σ, ∅) = ∅, if Σ 6= ∅ and Mor

(∅, ∅) =

{∅} = {0

∅,∅

} = {id

∅

} where 0

Σ,Σ′

= ∅ = the empty function from Σ to Σ

′

So; 0

Σ,Σ′

∈ Mor

(Σ, Σ

′

) ⇐⇒ Σ = ∅.

If in a category ϕ

∈ Mor(R

, S

), i = 1, 2, we put;

⊔ ϕ

: R

⊔ R

−→ S

⊔ S

: r 7→

(r) if r ∈ R

where ⊔ := “disjoint union”.

Definition. (of the objects in K

) An (abstract) simplicial complex Σ on

a vertex set V

is a collection (empty or non-empty) of finite (Empty or

non-empty) subsets σ of V

satisfying;

(a) If v ∈ V

, then {v} ∈ Σ.

(b) If σ ∈ Σ and τ ⊂ σ then τ ∈ Σ.

So, {∅} is allowed as an object in K

We will write “concept

” or

“concept

℘

” when we want to stress that a concept is to be related to our

modified categories.

If #σ := card(σ) = q

1 then dim σ := q and σ is said to be a q-face

or a

q-simplex

of Σ

and dim Σ

:=sup{dim(σ )|σ ∈ Σ

}. Writing ∅

when using

∅ as a simplex, we get dim(∅) = −∞ and dim({∅

}) = dim(∅

) = −1.

Note. Any simplicial complex

6= ∅ in K

includes {∅

} as a subcomplex.

A typical object

in K

is Σ ⊔ {∅

} or ∅ where Σ ∈ K and ψ is a morphism

in K

if;

(a) ψ = ϕ ⊔ id

{∅}

for some ϕ ∈ Mor

(Σ, Σ

′

)

(b) ψ = 0

∅,Σ

In particular, Mor

(Σ

, {∅}) = ∅ if and only if Σ

6= {∅}, ∅.

A functor E: K

−→K:

Set E(Σ

) = Σ

\ {∅} ∈ Obj(K) and given a morphism ψ: Σ

→ Σ

′

define;

E(ψ) =

(

if ψ fulfills (a) above and

∅,E(Σ

)

if ψ fulfills (b) above.

A functor E

: K −→ K

Set E

(Σ)=Σ ⊔ {∅}∈ Obj(K

) and given ϕ : Σ → Σ

′

, we put ψ := ϕ ⊔ id

{∅}

which gives; EE

=id

; imE

= Obj(K

) \ {∅} and E

E = id

except for

E(∅) = {∅}.

Similarly, let C be the category of topological spaces and continuous

maps.

Consider the category D

℘

with objects: ∅ together with X

℘

X + {℘}, for all X ∈ Obj(C), i.e. the set X

℘

:= X ⊔ {℘} with the weak

topology, τ

X℘

, w.r.t. X and {℘}.

℘

∈ Mor

℘

, Y

℘

) if;

℘











a) f +id

{℘}

(:= f ⊔ id

{℘}

) with f ∈ Mor

(X, Y ) and

f is on X to Y, i.e the domain of f is the whole of X

and X

℘

= X+{℘}, Y

℘

= Y +{℘} or

b) 0

∅,Y

℘

(= ∅ = the empty function from ∅ to Y

℘

There are functors F

℘

(

C −→ D

℘

X 7→ X+{℘}

, F :











℘

−→ C

X+{℘} 7→ X

∅ 7→ ∅

resembling E

resp. E .

Note. The “F

℘

-lift topologies”,

:= F

℘

(τ

) ∪ {∅} = {O

= O ⊔ {℘} | O ∈ τ

} ∪ {∅}

and

:= τ

∪ {X

} = {O

\ (N ⊔ {℘}) ; N closed in X} ∪ {X

}

would also give D

℘

due to the domain restriction in a, making D

℘

a link

between the two constructions of partial maps treated in [4] pp. 184-6.
Also compare [12] Ch. X §6 p. 269ff.

No extra morphisms

℘

has been allowed into D

℘

) in the sense that the

morphisms

℘

are all targets under F

℘

) except 0

∅,Y

℘

defined through item

b, re-establishing ∅ as the unique initial object.

The underlying principle for our definitions is that a concept in C (K)

is carried over to D

℘

) by F

℘

) with addition of definitions of the

concept

for cases that isn’t a proper image under F

℘

). The definitions

of the product/join operations “×

”, “ ∗

”, “

∧

∗” in page 21 and “\

” below,

certainly follows this principle.

Definition.

℘

(

∅ if X

℘

= ∅

℘

F(X

℘

)

F(X

℘

)

if X

℘

6= ∅ in D

o℘

for X

℘2

⊂ X

℘1

where “/” is the classical “quotient” except that

F(X

℘

)

∅

:= F(X

℘1

), cp. [4]

p. 102.

Definition. Let “+” denote the classical “topological sum”, then;

℘1

℘2

(

℘1

if X

℘2

= ∅ (or vice versa)

℘

(F(X

℘1

) + F(X

℘2

)) if X

℘1

6= ∅ 6= X

℘2

To avoid dropping out of the category D

℘

we introduce a setminus “\

”

in D

℘

, giving Prop. 1 p. 15 a compact form. “\

” might be non-associative

if {℘} is involved.

Definition.

℘

′

℘

(

∅

if X

℘

= ∅, X

℘

⊂

′

℘

or X

′

℘

= {℘}

℘

F(X

℘

)

F(X

′

℘

)

else. (“

”:= classical “setminus”.)

2.2

Realizations of Simplicial Complexes

∅ 6= {∅

} and both are non-final, which under any useful definition of the

realization of a simplicial complex implies that |∅| 6= |{∅

}| and demands

the addition of a non-final object {℘} = |{∅

}| into the classical category of

topological spaces as join-unit and (-1)-dimensional standard simplex. If

℘

6= ∅, {℘} then (X

℘

, τ

X℘

) is a non-connected space. We therefore define

℘

6= ∅, {℘} to have a certain topological “property

℘

” if F(X

℘

) has the

“property”, e.g. X

℘

is connected

℘

iff X is connected.

We will use Spanier’s definition of the function space realization |Σ

| as

given in [40] p. 110, unaltered, except for the “

”’s and the underlined

addition where ℘ := α

and α

(v) ≡ 0 ∀ v ∈ V

“

We now define a covariant functor from the category of simpli-

cial complexes

and simplicial maps

to the category of topolog-

ical spaces

and continuous maps

. Given a nonempty simplicial

complex

, let |Σ

| be the set of all functions α from the set

of vertices of Σ

to I := [0, 1] such that;

(a) For any α, {v ∈ V

Σo

|α(v) 6= 0} is a simplex

of Σ

(in particular, α(v) 6= 0 for only a finite set of vertices).

(b) For any α 6= α

∈V

Σo

α(v) = 1.

If Σ

= ∅ , we define |Σ

| = ∅.

”

The barycentric coordinates

defines a metric

d(α, β) =

[α(v) − β(v)]

∈

Σo

on |Σ

| inducing the topological space |Σ

with the metric topology.

We will equip |Σ

| with another topology and for this purpose we define

the closed geometrical simplex

|σ

| of σ

∈ Σ

i.e.:

|σ

| := {α ∈ |Σ

| | [α(v) 6= 0] =⇒ [v ∈ σ

]}.

Definition. For Σ

6= ∅, |Σ

| is topologized through |Σ

| := |E(Σ

)| + {α

which is equivalent to give |Σ

| the weak topology w.r.t. the |σ

|’s, naturally

imbedded in R

+{℘} and we define Σ

to be connected if |Σ

| is, i.e. if

F(|Σ

|) ≃ |E(Σ

)| is.

Proposition.

|Σ

| is always homotopy equivalent to |Σ

([15] pp. 115, 226.).

and

|Σ

| is homeomorphic to |Σ

iff Σ

is locally finite ([40] p. 119 Th. 8.).

2.3

Topological Properties of Realizations:

[8] p. 352 ff., [12] p. 54 and [22] p. 171 emphasize the importance of tri-
angulable spaces, i.e. spaces homeomorphic to the realization

(p. 15) of

a simplicial complex

. Non-triangulable topological (i.e. 0-differentiable)

manifolds have been constructed, cp. [35] but since all continuously n-
differentiable (n > 0) manifolds are triangulable, cf. [32] p. 103 Th. 10.6,
essentially all spaces within mathematical physics are triangulable, in par-
ticular, they are CW-complexes

. A topological space has the homotopy

type of the realization (p. 15) of a simplicial complex iff it has the type of
a CW-complex, which it has iff it has the type of an ANR, cf. [15] p. 226
Th. 5.2.1. Our CW-complexes

will have a (−1)-cell {℘}, as the “relative

CW-complexes” defined in [15] p. 26, but their topology is such that they
belong to the category D

℘

, as defined in p. 14. {℘} is called an “ideal cell”

in [7] p. 122. CW-complexes are compactly generated, perfectly normal
spaces, cf. [15] pp. 22, 112, 242, that are locally contractible in a strong
sense and (hereditarily) paracompact, cf. [15] pp. 28-29 Th. 1.3.2

Th.

1.3.5 (Ex. 1 p. 33).
Notations. We have used w.r.t.:=with respect to, and τ

:=the topology of

X. We’ll also use; PID:= Principal Ideal Domain and “an R-PID-module
G”:= G is a module over a PID R. l.h.s.(r.h.s.):= left (right) hand side,
iff := if and only if, cp.:= compare (cf.:=cp.!), L

−

HS := Long

−

Homology

Sequence, M-Vs := Mayer-Vietoris sequence and w.l.o.g.:= without loss of
generality.

Let, here in Ch. 2, ∆ = {∆

,∂} be the classical singular chain complex.

“≃” denotes “homeomorphism” or “chain isomorphism”.

Note. In Definition 1 p. 1 we had {℘} = {∅} and in the realization defini-
tion we had {℘} = {α

}. ∅ is unquestionable the universal initial object.

Our {℘} could be regarded as a local initial object and ℘ as a member iden-
tifier, since every object with ℘ as its unique “−1-dimensional” element,
has therethrough a “natural” membership-tag. Note also that d(α

, α) ≡ 1

for any α, i.e., α is an isolated point.

2.4

Simplicial Augmental Homology Theory

−

H denotes the simplicial as well as the singular augmental (co)homology

functor

Choose ordered q-simplices to generate C

(Σ

; G), where the coefficient

module G is a unital (↔ 1

◦g = g) module over any commutative ring A

with unit. Now, with 0 as the additive unit-element;

-C

(∅; G) ≡ 0 in all dimensions.

-C

({∅

};G) ≡ 0 in all dimensions except for C

−1

({∅

};G) ∼

= G.

-C

(Σ

; G) ≡ e

C(E(Σ

); G) ≡ the classical “{∅

}”-augmented chain.

We use ordered simplices instead of oriented since the former is more nat-

ural in relation to our definition p. 31, of the Ordered Simplicial Cartesian
Product.

By just hanging on to the “{∅

}-augmented chains”, also when defin-

ing relative chains

, we get the Relative Simplicial Augmental Homology

Functor for K

-pairs, denoted

−

∗

- fulfilling; (H

∗

) denotes the classical

(reduced) homology functor.)

−

(Σ

, Σ

; G) =











(E(Σ

), E(Σ

); G) if Σ

6= ∅

(E(Σ

), G)

if Σ

6= {∅

}, ∅, and Σ

= ∅

∼

= G if i = −1
= 0

if i 6= −1

when Σ

= {∅

} and Σ

= ∅

f or all i when Σ

= Σ

= ∅.

2.5

Singular Augmental Homology Theory

The |σ

℘

|, defined in p. 16, imbedded in R

+{℘} generates a satisfying set

of “standard simplices

℘

” and “singular simplices

℘

”. This implies in partic-

ular that the “p-standard simplices

℘

”, denoted ∆

℘p

, are defined by ∆

℘p

∆

+ {℘} where ∆

denotes the usual p-dimensional standard simplex and

+ is the topological sum, i.e. ∆

℘p

:= ∆

⊔{℘} with the weak topology w.r.t.

∆

and {℘}. Now, and most important: ∆

℘

(−1)

:= {℘}.

Let T

denote an arbitrary classical singular p-simplex (p ≥ 0). The “sin-

gular p-simplex

℘

”, denoted σ

℘p

, now stands for a function of the following

kind:

℘p

: ∆

℘p

= ∆

+{℘} −→ X+{℘} where σ

℘p

(℘) = ℘ and

℘p

|∆p

= T

for some classical p - dimensional singular simplex T

∀ p ≥ 0.

In particular;

℘

(−1)

: {℘} −→ X

℘

= X+{℘} : ℘ 7→ ℘.

The boundary function ∂

℘

is defined by ∂

℘p

(σ

℘p

) := F

℘

(∂

)) if p > 0

where ∂

is the classical singular boundary function, and ∂

℘0

(σ

℘

) ≡ σ

℘

(−1)

for every singular 0-simplex

℘

. Let ∆

℘

= {∆

℘p

, ∂

℘

} denote the singular

augmental chain complex

℘

. Observation; |Σ

| 6= ∅ =⇒ |Σ

| = F

(|E(Σ

)|) ∈

By the strong analogy to classical homology, we omit the proof of the

next lemma.

Lemma. (Analogously for co

−

Homology.)

−

℘1

, X

℘2

; G) =











(F(X

℘1

), F(X

℘2

); G) if X

℘2

6= ∅

(F(X

℘1

); G)

if X

℘1

6= {℘}, ∅ and X

℘2

= ∅

∼

= G if i = −1
= 0

if i 6= −1

when X

℘1

= {℘} and X

℘2

= ∅

f or all i when X

℘1

= X

℘2

= ∅.

Note.

i. ∆

, X

; G) ∼

= ∆

℘i

℘

), F

℘

); G) always. So;

−

H-

℘

), F

℘

); G) ≡ 0.

ii. ∆

℘

℘1

, X

℘2

) ≃ ∆(F(X

℘1

), F(X

℘2

)) except iff X

℘1

6= X

℘2

= ∅ when the

only non-isomorphisms occurs for ∆

℘

(-1)

℘1

, ∅) ∼

= Z

≇

∼

= ∆

(-1)

(F(X

℘1

), ∅)

when, if X

℘1

6= ∅;

−

℘1

, ∅) ⊕ Z ∼

= H

(F(X

℘1

), ∅).

Remember that: Mor

℘

-pairs

((X

℘

, X

′

℘

), (X

℘

, ∅)) 6= ∅ iff X

′

℘

= ∅.

iii. C

(Σ

, Σ

; G) ≈ ∆

℘

(|Σ

|, |Σ

|; G) connects the simplicial and sin-

gular functor

. (“≈” stands for “chain equivalence”.)

iv.

−

℘+

℘

, {℘}; G) =

−

℘

, {℘}; G) ⊕

−

℘

, {℘}; G) but

−

℘+

℘

, ∅; G) =

−

℘

, ∅; G) ⊕

−

℘

, {℘}; G) =

−

℘

, {℘}; G) ⊕

−

℘

, ∅; G).

Definition. The p:th Singular Augmental Homology Group of X

℘

w.r.t.

G =

−

℘

; G) :=

−

℘

, ∅; G).

The Coefficient Group

−

−1

({℘}, ∅; G).

Using F

℘

), we “lift” the concepts of homotopy, excision and point

in C (K) into D

℘

-concepts (K

-concepts) homotopy

℘

, excision

℘

and point

℘

(=: •), respectively.

So; f

, g

∈ D

℘

are homotopic

℘

if and only if f

= g

= 0

∅,Y

or there are

homotopic maps f

, g

∈ C such that f

= f

+ Id

{℘}

, g

= g

+ Id

{℘}

An inclusion

, i

) : (X

, A

) −→ (X

, A

), U

6= {℘}, is an

excision

℘

if and only if there is an excision (i, i

) : (X

U, A

U ) −→ (X, A)

such that i

= i + Id

{℘}

and i

= i

+ Id

{℘}

{P, ℘} ∈ D

℘

is a point

℘

iff

{P} + {℘} = F

℘

({P}) and {P} ∈ C is a point.

So, {℘} is not a point

℘

Conclusions: (

−

H, ∂

℘

), abbreviated

−

H, is a homology theory on the h-category

of pairs from D

℘

), c.f. [12] p. 117, i.e.

−

H fulfills the h-category analogues,

given in [12] §§8-9 pp. 114-118, of the seven Eilenberg-Steenrod axioms from
[12] §3 pp. 10-13. The necessary verifications are either equivalent to the
classical or completely trivial. E.g. the dimension axiom is fulfilled since
{℘} is not a point

℘

Since the exactness of the relative Mayer-Vietoris sequence of a proper

triad, follows from the axioms, cf. [12] p. 43 and, paying proper attention
to Note iv above, we’ll use this without further motivation.

H(X) =

−

H(F

℘

(X), ∅) explains all the ad-hoc reasoning surrounding the

H-functor.

Augmental Homology Modules for Products and
Joins

3.1

Background to the Product- and Joindefinitions

What is usually called simply “the product topology” w.r.t. products with
infinitely many factors is actually the Tychonoff product topology which
became the dominant product topology when the Tychonoff product the-
orem for compact spaces was introduced. J.L. Kelley (1950) (J. Lo´s &
C. Ryll-Nardzewski (1954)) has proven the Tychonoff product theorem (∼
for Hausdorff spaces) to be equivalent to the axiom of choice (the Boolean
prime ideal theorem (i.e. the dual of the ultra filter theorem)). Compare [3].

3.2

Definitions of Products and Joins

∇

below denotes the classical :







topological product × from [40] p. 4 or
topological join ∗ from [45] p. 128 or
topological join

∧

∗ from [30].

Recall that; X

∧

∗ ∅ = X ∗ ∅ = X = ∅ ∗ X = ∅

∧

∗X classically.

Definition.

℘1

∇

℘2

∅

if X

℘1

= ∅ or X

℘2

= ∅

℘

(F(X

℘1

)

∇

F(X

℘2

)) ifX

℘1

6= ∅ 6= X

℘2

From now on we’ll delete the ℘/

-indices. So, e.g. ×, ∗,

∧

∗ now means

, ∗

∧

∗

respectively, while “X connected” means “F(X) connected”.

Equivalent Join Definition. Put ∅ ⊔

X = X⊔

∅ := ∅. If X 6= ∅; {℘}⊔

X =

{℘},X⊔

{℘}= X. For X,Y 6=∅, ℘ let X⊔

Y denote the set X×Y ×(0, 1] pasted

to the set X by ϕ

: X × Y × {1} −→ X; (x, y, 1) 7→ x , i.e. the quotient

set of X × Y × (0, 1]

⊔ X, under the equivalence relation (x, y , 1) ∼ x

and let p

: X × Y × (0, 1]

⊔ X −→ X ⊔

Y be the quotient function. For

X, Y = ∅ or {℘} let X⊔

Y := Y ⊔

X and else the set X × Y × [0, 1) pasted

to the set Y by the function ϕ

: X × Y × {0} −→ Y ; (x, y, 0) 7→ y , and

let p

: X × Y × [0, 1)

⊔ Y −→ X ⊔

Y be the quotient function. Put

X◦Y := X ⊔

∪ X ⊔

(x, y, t) ∈ X ×Y × [0, 1] specifies the point (x, y, t) ∈ X⊔

Y ∩ X⊔

Y, one-

to-one, if 0 < t < 1 and the equivalence class containing x if t = 1 (y if
t = 0), which we denote (x, 1) ((y , 0)). This allows “coordinate functions”
ξ : X◦Y → [0, 1], η

: X⊔

Y → X, η

: X⊔

Y →Y extendable to X◦Y through

(y, 0) :≡ x

∈ X resp. η

(x, 1) :≡ y

∈ Y and a projection p : X ⊔ X × Y ×

[0, 1]

⊔ Y → X◦Y.

Let X

∧

∗Y denote X◦Y equipped with the smallest topology making ξ, η

, η

continuous and X∗Y, X◦Y with the quotient topology w.r.t. p, i.e. the
largest topology making p continuous (⇒ τ

∧

∗

⊂

∗Y

Pair-definitions. (X

, X

)

∇

⊖

, Y

) := (X

∇

, (X

∇

) ⊖ (X

∇

)),

where ⊖ stand for “∪” or “∩” and if either X

or Y

is not closed (open),

∗Y

, (X

∗Y

)⊖(X

∗Y

)) has to be interpreted as (X

∗Y

, (X

∗Y

)⊖(X

∗Y

))

i.e. (X

◦ Y

) ⊖ (X

◦ Y

) with the subspace topology in the 2:nd component.

Analogously for simplicial complexes with “×” (“∗”) from [12] p. 67 Def.
8.8 ( [40] p. 109 Ex. 7.)

Note.

i. (X ∗ Y )

≥0.5

is homeomorphic to the mapping cylinder w.r.t.

the coordinate map q

: X ×Y → X.

ii. X

∧

∗Y

is a subspace of X

∧

∗Y

by [4] 5.7.3 p. 163. X

∗Y

is a subspace

of X

∗Y

if X

are closed (open). cf. [9] p. 122 Th. 2.1(1).

iii. (X

◦ Y

) ∩ (X

◦ Y

) = X

◦ Y

and with

∇

∪

for short in pair

operations; (X

, {℘}) × (Y

, Y

) = (X

, ∅) × (Y

, Y

) if Y

6= ∅.

iv.

∧

∗ and ∗ are both commutative but, while

∧

∗ is associative by [4] p. 161,

∗ isn’t in general, cf. p. 31.

v. “×

∩

” is (still, cf. [8] p. 15,) the categorical product on pairs from D

℘

3.3

Augmental Homology for Products and Joins

Through Lemma

Note ii p. 19 we convert the classical K¨

unneth formula

cf. [40] p. 235, mimicking what Milnor did, partially (≡ line 1), at the end
of his proof of [30] p. 431 Lemma 2.1.

The ability of a full and clear understanding of Milnor’s proof is definitely

sufficient prerequisites for our next six pages.

The “new” object {℘} gives the classical K¨

unneth formula (≡4:th line)

additional strength but much of the classical beauty is lost - a loss which
is regained in the join version i.e. in Theorem 4 p. 26

Theorem 1. For {X

×Y

, X

×Y

} excisive, q ≥ 0, R a PID, and assuming

Tor

(G, G

′

) = 0 then;

−

((X

, X

) × (Y

, Y

); G ⊗

′

)

∼











[

−

; G)⊗

−

; G

′

)]

⊕(

−

; G)⊗

′

)⊕(G⊗

−

; G

′

))⊕T

if C

[

−

; G) ⊗

−

, Y

; G

′

)]

⊕ (G⊗

−

, Y

; G

′

)) ⊕ T

if C

[

−

, X

; G) ⊗

−

; G

′

)]

⊕ (

−

, X

; G) ⊗

′

) ⊕ T

if C

[

−

, X

; G) ⊗

−

, Y

; G

′

)]

⊕ T

if C

(1)

The torsion terms, i.e. the T-terms, splits as those ahead of them, resp.,

e.g.

= [Tor

−

; G),

−

; G

′

)

]

−1

⊕

⊕ Tor

−

q−1

; G), G

′

⊕ Tor

−

q−1

; G

′

)

and

= [Tor

(

−

, X

; G),

−

, Y

; G

′

)

]

-1











:= “X

×Y

6= ∅, {℘} and X

= ∅ = Y

”,

:= “X

×Y

6= ∅, {℘} and X

= ∅ 6= Y

”,

:= “X

×Y

6= ∅, {℘} and X

6= ∅ = Y

”,

:= “X

×Y

= ∅,{℘} or X

6= ∅ 6= Y

”.

and [...]

is still, i.e. as in

[40] p. 235 Th. 10, to be interpreted as

...

i+j=q & i,j≥0

Lemma. For a relative homeomorphism f: (X, A) → (Y, B) (⇔ f : X → Y
continuous and f : X \ A → Y \ B a homeomorphism), let F : N × I → N
be a (strong) [neighborhood] deformation retraction of N onto A. If B and
f (N ) are closed in N

′

:= f (N \ A) ∪ B, then B is a (strong) [neighborhood]

deformation retract of N

′

through;

′

: N

′

× I → N

′

;

(y,t) 7→ y

if y ∈ B , t ∈ I

(y,t) 7→f◦F (f

(y)

, t) if y ∈ f (N ) \ B = f (N \ A), t ∈ I.

Proof. F

′

is continuous as being so when restricted to the closed subspaces

f (N )×I resp. B×I, where N

′

×I = (f (N )×I)∪(B×I), by [4] p. 34; 2.5.12.

Note. X (Y ) is a strong deformation retract of the mapping cylinder, w.r.t.
product projection, i.e. of (X∗Y )

t>s

((X∗Y )

1−s

). Equivalently for “

∧

∗”-join,

by the Lemma. So;

r: X

∧

∗Y → X ∗ Y ;

(

(x, y, t) 7→ x if t ≥ 0.9, y if t ≤ 0.1

(x, y,

0.5
0.4

(t − 0.5) + 0.5) else,

is a homotopy inverse of the identity, i.e.,

∧

∗ Y and X ∗ Y are homotopy equivalent.

Theorem 2. (Analogously for

∧

∗ by the Note, mutatis mutandis.) If (X

, X

) 6=

({℘}, ∅) 6= (Y

, Y

) and G is an A-module, then;

−

((X

, X

)×(Y

, Y

); G)

∼

−

((X

)∗ (Y

); G)⊕

−

((X

)∗ (Y

)

≥0.5

+ (X

)∗ (Y

)

≤0.5

; G)=

∼











−

∗ Y

; G) ⊕

−

; G) ⊕

−

; G)

−

((X

, ∅) ∗ (Y

, Y

); G) ⊕

−

, Y

; G)

−

((X

, X

) ∗ (Y

, ∅); G) ⊕

−

, X

; G) if

−

((X

, X

) ∗ (Y

, Y

); G)

(2)

where











:= “X

×Y

6= ∅, {℘} and X

= ∅ = Y

”,

:= “X

×Y

6= ∅, {℘} and X

= ∅ 6= Y

”,

:= “X

×Y

6= ∅, {℘} and X

6= ∅ = Y

”,

:= “X

×Y

= ∅,{℘} or X

6= ∅ 6= Y

”.

Proof. Split X ∗ Y at t = 0.5 then; X (Y ) is a strong deformation re-
tract of (X ∗ Y )

≥0.5

((X ∗ Y )

≤0.5

), i.e. the mapping cylinder w.r.t. prod-

uct projection.

The relative M-Vs w.r.t. the excisive couple of pairs

{((X

) ∗ (Y

))

≥0.5

, ((X

) ∗ (Y

))

≤0.5

} splits since the inclusion of their

topological sum into (X

) ∗ (Y

) is pair null-homotopic, cf. [33] p. 141

Ex. 6c, and [21] p. 32 Prop. 1.6.8. Since the 1:st (2:nd) pair is acyclic if
Y

) 6= ∅ we get Theorem 2. Equivalently for

∧

∗ by the Note.

Milnor finished his proof of [30] Lemma 2.1 p. 431 by simply comparing

the r.h.s. of the C

-case in Eq. 1 with that of Eq. 2. Since we are aiming at

the stronger result of “natural chain equivalence” in Theorem 3 this isn’t
enough and so, we’ll need the following three auxiliary results to prove
our next two theorems. We hereby avoid explicit use of “proof by acyclic
models”.

(“≈” stands for “chain equivalence”.)

(5.7.4 ([4] p. 164.)). (E

:= {e, ℘} = • denotes a point, i.e. a 0-disc.)

There is a homeomorphism:

ν : X

∧

∗Y

∧

∗E

−→ (X

∧

∗E

) × (Y

∧

∗E

)

which restricts to a homeomorphism:

∧

∗Y → ((X

∧

∗E

) × Y ) ∪ (X × (Y

∧

∗E

)).

Corollary. (from [21] p. 210.)

If φ: C ≈ E with inverse ψ and φ

′

: C

′

≈ E

′

with inverse ψ

′

, then

φ ⊗ φ

′

: C ⊗ C

′

≈ E ⊗ E

′

with inverse ψ ⊗ ψ

′

The ⊗-operation in the module-category in the last corollary can be

substituted for any monoid-inducing operation in any category, cf. [24]
Ex. 2 p. 168.

Theorem (46.2). (from [33] p. 279) For free chain complexes C, D van-
ishing below a certain dimension and if a chain map λ : C → D induces
homology isomorphisms in all dimensions, then λ is a chain equivalence.

Theorem 3. (The relative Eilenberg-Zilber theorem for joins.) For an exci-
sive couple {X

∧

∗Y

, X

∧

∗Y} from the category of ordered pairs ((X, X

), (Y, Y

))

of topological pairs

℘

, s(∆

℘

(X, X

) ⊗ ∆

℘

(Y, Y

)) is naturally chain equivalent to

∆

℘

((X, X

)

∧

∗ (Y, Y

)). (“s” stands for suspension i.e. the suspended chain

equals the original except that dimension i in the original is dimension i

in the suspended chain.)

Proof. The second isomorphism is the key and is induced by the pair home-
omorphism in [4] 5.7.4 p. 164. For the 2:nd last isomorphism we use [21]
p. 210 Corollary 5.7.9 and that L

−

HS-homomorphisms are “chain map”-

induced. Note that the second component in the third module is an excisive
union.

−

∧

∗ Y)

∼

−

∧

∗ Y

∧

∗ {v, ℘}, X

∧

∗ Y)

∼

−

((X

∧

∗{u,℘})×(Y

∧

∗{v, ℘}),

(

∧

∗ {u,℘})×Y

)

∪

(

X×(Y

∧

∗ {v, ℘})

)

−

((X

∧

∗{u, ℘}, X)×(Y

∧

∗{v, ℘}, Y ))

∼

Motivation: The underlying chains on the l.h.s. and r.h.s. are,

by Note ii p. 19 isomorphic to their classical counterparts on
which we use the classical Eilenberg-Zilber Theorem.

∼

−

(∆

℘

∧

∗ {u, ℘}, X)⊗

∆

℘

∧

∗ {v, ℘}, Y ))

∼

(s∆

℘

(X)⊗

s∆

℘

(Y))

∼

(s[∆

℘

(X)⊗

∆

℘

(Y)]).

So, (X

∧

∗ Y) is naturally chain equivalent to s[∆

℘

(X)⊗

∆

℘

(Y)] by [33] p. 279

Th. 46.2 quoted above, proving the non-relative Eilenberg-Zilber Theorem
for joins. (Compare the original ×-proof as given in [40] p. 232 Theorem
6.)

⊲

Substituting, in the original ×-proof given in [40] p. 234, “

∧

∗ ”, “∆

℘

resp.

s∆

℘

”, “Theorem 3, 1:st part” for “×”, “∆”, “Theorem 6” resp. will do

since;

s ∆

℘

) ⊗ ∆

℘

)

(

s ∆

℘

) ⊗ ∆

℘

)

+ s ∆

℘

) ⊗ ∆

℘

)

= s

∆

℘

) ⊗ ∆

℘

(

∆

℘

) ⊗ ∆

℘

)

+ ∆

℘

) ⊗ ∆

℘

)

= s

(

∆

℘

)/∆

℘

)

⊗

(

∆

℘

)/∆

℘

)

[40] Cor. 4 p. 231 now gives Theorem 4, since;

−

⋆

(◦) ∼

= s

−

⋆

(◦) ∼

−

⋆

(s(◦))

and ∆

℘

((X, X

) ∗ (Y, Y

)) ≈ ∆

℘

((X, X

)

∧

∗ (Y, Y

)) by Th. 2 and [33] p. 279 Th.

46.2.

∗Y

, X

∗Y

} is excisive iff {X

×Y

, X

×Y

} is, which is seen through a

M-Vs-stuffed 9-Lemma, cf. p. 59ff, and Theorem 2 (line four).

Theorem 4. ( The K¨

unneth Formula for Toplogical Joins; cp. [40] p.

235.) If {X

∧

∗ Y

, X

∧

∗ Y

} is an excisive couple in X

∧

∗Y

, R a PID, G, G

′

R-modules and Tor

(G, G

′

) = 0, then the functorial sequences below are

(non-naturally) split exact;

0 −→

+j=q

[

−

, X

; G) ⊗

−

, Y

; G

′

)] −→

−→

−

((X

, X

)

∧

∗ (Y

, Y

); G ⊗

′

) −→

(3)

−→

+j=q−1

Tor

−

, X

; G),

−

, Y

; G

′

)

−→ 0

Analogously for the ∗-join. [40] p. 247 Th. 11 gives the co

−

Homology-

analog.

Putting (X

, X

) = ({℘}, ∅) in Theorem 4, our next theorem immediately

follows.

Theorem 5. [The Universal Coefficient Theorem for (co)

−

Homology]

−

, Y

; G)

∼

[40] p. 214

∼

−

, Y

; R ⊗

∼

(

−

, Y

; R) ⊗

)

⊕ Tor

−

−1

, Y

; R), G

for any R-PID module G.

If all

−

∗

, Y

; R) are of finite type or G is finitely generated, then;

−

, Y

; G)

∼

−

, Y

; R ⊗

∼

(

−

, Y

; R) ⊗

)

⊕ Tor

−

, Y

; R), G

3.4

Local Augmental Homology Groups for Products and
Joins

Proposition 1 motivates in itself the introduction of a topological (−1)-
object, which imposed the definition p. 15 of a “setminus”, “\

”, in D

℘

revealing the shortcomings of the classical boundary definitions w.r.t. man-
ifolds, cf. p. 4 and p. 30. Somewhat specialized, Prop. 1 below is found in
[16] p. 162 and partially also in [34] p. 116 Lemma 3.3. “X \ x” usually
stands for “X \ {x}” and we’ll write x for {x, ℘} as a notational convention.
Recall the definition of α

, p. 16, and that dim Lk

σ = dimΣ−

σ.

The contrastar of σ ∈ Σ = cost

σ := {τ ∈ Σ| τ 6⊇ σ}. cost

∅

= ∅ and

cost

σ = Σ iff σ 6∈ Σ.

Proposition. 1. Let G be any module over a commutative ring A with
unit. With α ∈ Intσ and α = α

iff σ = ∅

the following module isomorphisms

are all induced by chain equivalences, cf. [33] p. 279 Th. 46.2 quoted here
in p. 25.

−

(Lk

σ; G)

∼

−

(Σ, cost

σ; G)

∼

−

(|Σ|, |cost

σ|; G)

∼

−

(|Σ|, |Σ|\

α; G),

−

(Lk

σ; G)

∼

−

(Σ, cost

σ; G)

∼

−

(|Σ|, |cost

σ|; G)

∼

−

(|Σ|, |Σ|\

α; G).

Proof. (Cf. definitions p. 34-5.) The “ \

”-definition p. 15 and [33] Th.

46.2 p. 279

pp. 194-199 Lemma 35.1-35.2

Lemma 63.1 p. 374 gives the

two ending isomorphisms since |cost

σ| is a deformation retract of |Σ|\

α,

while already on the chain level; C

⋆

(Σ, cost

σ) = C

⋆

(st

(σ), ˙σ ∗ Lk

σ) =

⋆

(σ ∗ Lk

σ, ˙σ ∗ Lk

σ) ≃ C

⋆

#σ

(Lk

σ).

Lemma.. If x (y) is a closed point in X (Y ), then {X×(Y \

y), (X \

x)×Y },

and {X∗ (Y \

y), (X \

x) ∗ Y } are both excisive pairs.

Proof. [40] p. 188 Th. 3, since X ×(Y \

) X ∗(Y \

)

is open in (X ×

(Y \

)) ∪ ((X \

) ×Y ) (X ∗ (Y \

)) ∪ ((X \

) ∗ Y )

, which proves the

excisivity.

Theorem 6. For x∈X (y ∈ Y ) closed and (t

x ∗ y, t

) := {(x, y, t) | 0 < t

≤

t ≤ t

< 1};

−

∧

∗ Y, X

∧

∗ Y \

(x, y, t); G)

∼

−

∧

∗ Y, X

∧

∗ Y \

x ∗ y, t

); G)

∼

−

(X×Y, X×Y \

(x, y); G)

∼

A simple

calculation

∼

−

((X, X\

x)×(Y, Y \

y); G)

∼

Th. 2 p. 23

line four

∼

−

((X, X\

∧

∗ (Y, Y \

y); G).

ii.

−

∧

∗ Y, X

∧

∗ Y \

(y, 0); G)

∼

−

((X, ∅)

∧

∗ (Y, Y \

y ); G)

and equivalently for the (x, 1)-points.

All isomorphisms are induced by chain equivalences, cf. [33] p. 279 Th.

46.2 quoted here in p. 25. Analogously for “ ∗ ” substituted for “

∧

∗” and for

−

Homology.

Proof. i.

A := X ⊔

Y \

{(x

, y

, t) | t

≤ t < 1}

B := X ⊔

Y \

{(x

, y

, t) | 0 < t ≤ t

}

=⇒

A ∪ B = X

∧

∗ Y \

]

x ∗ y, t

)

A ∩ B = X×Y ×(0, 1)\

}×{y

}×(0, 1)

with x

× y

× (0, 1) := {x

} × {y

} × {t | t ∈ (0, 1)} ∪ {℘} and (x

, y

, t

) :=

{(x

, y

, t

), ℘}.

Now, using the null-homotopy in the relative M-Vs w.r.t. {(X⊔

Y, A),

(X⊔

Y, B)} and the resulting splitting of it and the involved pair deforma-

tion retractions as in the proof of Th. 2 we get;

−

∧

∗ Y, X

∧

∗ Y \

]

∗ y

, t

))

∼

−

(X×Y ×(0, 1), X×Y ×(0, 1) \

}×

}×(0, 1) )

∼

Motivation: The underlying pair on the r.h.s.

is a pair deformation retract of that on the l.h.s.

∼

−

(X×Y ×{t

, ℘}, X×Y ×{t

, ℘} \

) )

∼

−

(X×Y, X×Y \

)) =

−

(X×Y, (X×(Y \

)) ∪ ((X\

)×Y ) ) =

−

((X, X\

)×(Y, Y \

) ). ⊲

ii.

A := X ⊔

B := X ⊔

Y \

X×{y

}×[0, 1)

=⇒

A ∪ B = X

∧

∗ Y \

, 0)

A ∩ B = X×(Y \

)×(0, 1)

where (x

, y

, t) ∈ X × {y

} × [0, 1) is independent of x

and (x

, y

, t

) :=

{(x

, y

, t

), ℘}.

Now use Th. 1 p. 22 line 2 and that the r.h.s. is a pair deformation retract

of the l.h.s.; (X×Y ×(0, 1), X×(Y \

)×(0, 1)) e (X×Y , X ×(Y \

)) =

(X, ∅)×(Y, Y \

Definition.

Hip

X = {Homologically

instabile points} := {x ∈ X

−

(X,

X \

) = 0 ∀ i ∈ Z}. So; Hip

∇

) ⊃ (X

∇

Hip

) ∪ ((Hip

)

∇

3.5

Singular Homology Manifolds under Products and Joins

Definition. . ∅ is a weak homology

−

∞-manifold. Else, a T

-space (⇔ all

points are closed) X ∈ D

℘

is a weak homology

n-manifold (n-whm

) if for

some A-module R;

−

(X, X \

x; G) = 0 if i 6= n for all ℘ 6= x ∈ X,

(4.i)

−

(X, X \

x; G) ∼

= G ⊕ R for some ℘ 6= x ∈ X if X 6= {℘}. (4.ii

′

)

(

′

)

An n-whm

X is joinable (n-jwhm

) if (4.i) holds also for x = ℘.

An n-jwhm

X is a weak homology n-sphere

(n-whsp) if

−

−1

(X \

x; G) =

0 ∀ x∈X.

Definition. (of technical nature.) X is acyclic

−

(X, ∅; G) = 0 for all

i ∈ Z. So, {℘} (= |{∅

}|) isn’t acyclic

. X is weakly direct

−

(X; G) ∼

G ⊕ P for some i and some A-module P. X is locally weakly direct

−

(X, X \

x; G) ∼

= G⊕Q for some i, some A-module Q and some ℘ 6= x ∈ X.

An n-whm

X is ordinary

−

(X \

x; G) = 0, ∀ i ≥ n and ∀ x ∈ X.

Corollary. . (to Th. 6) For locally weakly direct

(⇒ X

6= ∅, {℘}) T

-spaces

, X

∗X

+ n

+ 1)-whm

⇐⇒ X

, X

both n

-jwhm

⇐⇒ X

∗X

jwhm

ii. X

× X

+ n

)-whm

⇐⇒ X

, X

both whm

iii. If n

+ n

> n

i = 1, 2, then; [X

× X

+ n

)-jwhm

] ⇐⇒ [X

, X

both

-jwhm

and acyclic

(Since by Eq. 2 p. 24; [

−

×X

; G) = 0 for i 6= n

] ⇐⇒ [X

, X

both

acyclic

] ⇐⇒ [X

× X

acyclic

]. So; X

× X

is never a whsp

iv.If

moreover X

, X

are weakly direct

then; X

, X

are both whsp

iff X

∗X

is.

Proof. Augmental

−

Homology, like classical, isn’t sensitive to base ring changes.

So, ignore A and instead use the integers Z;
(i-iii.) Use Th. 1, 4-6 and the weak directness

to transpose non-zeros from

one side to the other, using Th. 6.ii only for joins, i.e., in particular, with
ε = 0 or 1 depending on wether

∇

= × or ∗ resp., use;

(

−

∇

, X

∇

, x

); G)

∼ )

−

((X

, X

) × (X

, X

); G)

∼

Lemma p. 15
+ Eq. 1 p. 12

∼

+j=p

[

−

, X

; Z)⊗

−

, X

; G)] ⊕

⊕

+j=p−1

Tor

−

, X

; Z),

−

, X

; G)

. ⊲

iv. Use, by the Five Lemma, the chain equivalence of the second component
in the first and the last item of Th. 6.i and the M-Vs w.r.t. {(X ∗ (Y \

)), ((X \

) ∗ Y )}.

Definition. ∅ is a homology

−∞-manifold and X = •• is a homology

0-manifold. Else, a connected, locally compact Hausdorff space X ∈ D

℘

is a

(singular) homology

n-manifold (n-hm

)

if;

−

(X, X\

x; G) = 0 if i 6= n ∀ ℘ 6= x ∈ X,

(4.i)

−

(X, X \

x; G) ∼

= 0 or G ∀ ℘ 6= x ∈ Xand = G for some x ∈ X.(4.ii)

(4)

The boundary: Bd

X := {x ∈ X

−

(X, X \

x; G) = 0}. If Bd

X 6= ∅

(Bd

X= ∅), X is called a homology

n-manifold with (without) boundary.

A compact n-manifold S is orientable

−

(S, BdS; G) e

= G. An n-

manifold is orientable

if all its compact n-submanifolds are orientable

else non-orientable

. Orientability is left undefined for ∅.

An n-hm

X is joinable if (4) holds also for x = ℘. An n-hm

X6= ∅ is a

homology

n-sphere (n-hsp

) if for all x ∈ X,

−

(X, X \

x; G) = G if i = n

and 0 else. So, a triangulable n-hsp

is a compact space.

Note. 1. Triangulable manifolds 6= ∅ are ordinary by Note 1 p. 50 and
locally weakly direct

since,

−

dimΣ

(|Σ|, |Σ| \

α; G) ∼

= G with α ∈ Intσ if σ is a

maxidimensional simplex i.e. if #σ−1 =: dim σ = dim Σ, since now Lk

σ =

{∅

}. Proposition 1 p. 27 and Lemma p. 19 now gives the claim. If α ∈ Intσ,

Proposition 1 p. 27 also implies that;

−

dimΣ

(Σ, cost

σ; Z) ∼

−

dimΣ

(|Σ|, |Σ|\

α; Z)

∼

−

dimΣ

( Lk

σ; Z) is a direct sum of Z-terms.

When

∇

in Th. 7, all through, is interpreted as ×, the symbol “hm

” on the

r.h.s. of 7.1., temporarily excludes ∅, {∅

} and ••, and we assume ǫ := 0.

When

∇

, all through, is interpreted as ∗, put ǫ := 1, and let the symbol

“hm

” on the right hand side of 7.1. be limited to “any compact joinable

homology

-manifold”.

Theorem 7. For locally weakly direct

-spaces X

, X

and any A-module G:

7.1. X

∇

is a homology

+ n

+ǫ)-manifold ⇐⇒ X

is a n

-hm

, i =

1, 2.

7.2.

(• × X) = • × (Bd

X). Else; Bd

∇

) = ((Bd

)

∇

) ∪

∇

(Bd

)).

7.3. X

∇

is orientable

⇐⇒ X

, X

are both orientable

Proof. Th. 7 is trivially true for X

× • and X

∗ {℘}. Else, exactly as for the

above Corollary, adding for 7.1. that for Hausdorff-like spaces (:= all com-
pact subsets are locally compact), in particular for Hausdorff spaces, X

∗ X

is locally compact (Hausdorff) iff X

, X

both are compact (Hausdorff), cf.

[6] p. 224.

Note. 2. (X

∇

, Bd

∇

)) =

7.2 above +

pair-def. p. 21

= (X

, Bd

)

∇

, Bd

Relating General Topology to Combinatorics

4.1

Realizations with respect to Simplicial Products and
Joins

The k-iffikation k(X) of X is X with its topology enlarged to the weak

topology w.r.t. all maps with compact Hausdorff-domain. Put X ¯

× Y :=

k(X× Y ). For CW-complexes, this is a proper topology-enlargement only
if none of the two underlying complexes are locally finite and at least one is
uncountable. Let X¯

∗Y be the quotient space w.r.t. p : (X ¯

×Y ) × I → X ◦ Y

from p. 21. See [25] p. 214 for relevant distinctions. Now, simplicial ×, ∗
“commute” with realization by turning into ¯

×, ¯∗ respectively. Unlike ∗,

Def. p. 21, ¯∗ is associative for arbitrary topological spaces, cf. [43] §3.

Definition. (cf. [12] Def. 8.8 p. 67.) Given ordered simplicial complexes
∆

′

and ∆

′′

i.e. the vertex sets V

∆′

and V

∆

′′

are partially ordered so that each

simplex becomes linearely ordered resp. The Ordered Simplicial Cartesian
Product ∆

′

∆

′′

of ∆

′

and ∆

′′

(triangulates |∆

′

|¯

|∆

′′

| and) is defined through

∆′×∆

′′

:= {(v

′

, v

′′

)}=V

∆′

∆

′′

. Put w

i,j

:= (v

′

, v

′′

). Now, simplices in ∆

′

∆

′′

are

sets {w

, w

,.., w

}, with w

is,js

6= w

and v

′

≤v

′

≤..≤v

′

′′

≤

′′

≤..≤v

′′

) where v

′

, v

′

, .., v

′

′′

, v

′′

, .., v

′′

) is a sequence of vertices,

with repetitions possible, constituting a simplex in ∆

′

(∆

′′

Lemma. (cp. [12] p. 68.) η := (|p

|, |p

|), |Σ

× Σ

| → |Σ

| ¯

×|Σ

|, where

: Σ

× Σ

→ Σ

is the simplicial projection, triangulates |Σ

| ¯

×|Σ

|. If

and L

are subcomplexes of Σ

and Σ

, then η carries |L

× L

| onto

| ¯

×|L

|. Furthermore, this triangulation has the property that, for each

vertex B of Σ

, say, the correspondence x → (x, B) is a simplicial map of

into Σ

× Σ

. Similarly for joins, cp. [44] p. 99.

Proof. (×). The simplicial projections p

: Σ

× Σ

→ Σ

gives realized

continuous maps |p

|, i = 1, 2. η := (|p

|, |p

|) : |Σ

×Σ

|→ |Σ

| ¯

×|Σ

| is bijective

and continuous cf. [4] 2.5.6 p. 32

Ex. 12, 14 p. 106-7. τ

|Σ1×Σ2|

(τ

|Σ1|¯

×|Σ2|

) is

the weak topology w.r.t. the compact subspaces {|Γ

× Γ

Γi⊂Σi

({ e

= |Γ

| ×

|Γ

Γi⊂Σi

), cf. [15] p. 246 Prop. A.2.1. (|p

|, |p

|)(|Γ

× Γ

| ∩ A) = (|Γ

| × |Γ

|) ∩

(|p

|, |p

|)(A) i.e. (|p

|, |p

is continuous.

⊲

(∗). As for ×, cp. [45] (3.3) p. 59. With Σ

∗ Σ

:= {σ

∪ σ

| σ

∈ Σ

(i =

1, 2)}, and if σ = {v

′

. . .

′

′′

q+1

. . .

′′

q+r

} ∈ V

∗Σ

then put {t

′

. . .

′

q+1

′′

q+1

. . .

q+r

′′

q+r

} := α

: V

→ [0, 1]; α

(v) := t

if v ∈ σ and 0 else.

′

1≤i≤q

, t

′′

:= 1 − t

′

+1≤j≤q+r

Now; η : |Σ

∗ Σ

| ֒→

→ |Σ

| ¯∗ |Σ

| ; {t

′

. . .

′

q+1

′′

q+1

. . .

q+r

′′

q+r

} 7→

7→

(

′

{

′

. . .

′

}

′′

{

q+1

′′

q+1

. . .

q+r

′′

q+r

}

)

, where (tx, (1−t)y) := ^

(x, y, t).

Denote an ordered simplicial complex Σ when regarded as an (aug-

mented) simplicial set by ˆ

Σ, cf. p. 7, and let ˆ

× ˆ

be the (augmented)

semi-simplicial product of ˆ

and ˆ

, while ˆ|ˆ

Σˆ| is the Milnor realization

of ˆ

Σ. Now, [15] p. 160 Prop. 4.3.15 + p. 165 Ex. 1+2 gives;

|Σ

| ¯

×|Σ

| ≃ ˆ|ˆ

ˆ|¯×ˆ|ˆ

ˆ| ≃ˆ|ˆ

× ˆ

ˆ| ≃ |Σ

× Σ

(Equivalently: |Σ

|¯∗|Σ

| ≃ ˆ|ˆ

ˆ|¯∗ˆ|ˆ

ˆ| ≃ˆ|ˆ

ˆ∗ ˆ

ˆ| ≃ |Σ

∗Σ

| with ˆ∗ from [10] or

[14])

The Milnor realization ˆ|Ξˆ| of any simplicial set Ξ is triangulable by [15]

p. 209 Cor. 4.6.12. E.g; the augmental singular complex ∆

℘

(X) w.r.t. any

topological space X, is a simplicial set and, cf. [31] p. 362 Th. 4, the map
j : ˆ|∆

℘

(X)ˆ| → X is a weak homotopy equivalence i.e. induces isomorphisms

in homotopy groups, and j is a true homotopy equivalence if X is of CW-
type, cf. [15] pp. 76-77, 170, 189ff, 221-2.

[15] pp. 303-4 describes a category theoretical generalization of Milnor’s

original realization procedure in [31].

4.2

Connectedness and Local Homology with respect to Sim-
plicial Products and Joins

So; η : (|Σ

∇

|, |Σ

∇

| \

(

]

, α

)})−→

≃ (|Σ

| ¯

∇

|Σ

|, |Σ

| ¯

∇

|Σ

| \

(α

, α

)) is a

homeomorphism if η(

]

, α

) = (α

, α

) and it’s easily seen that k(|Σ| ∗ |∆|)

= |Σ ∗ ∆| = k(|Σ|

∧

∗|∆|), since these spaces have the same topology on their

compact subsets.

Moreover, if ∆

′

⊂ Σ

′

, ∆

′′

⊂ Σ

′′

then |∆

′

|¯∗|∆

′′

| is a subspace of |Σ

′

|¯∗|Σ

′′

dim(Σ × ∆) = dim Σ + dim ∆ and dim(Σ ∗ ∆) = dim Σ + dim ∆ + 1.
If α

∈ Intσ

⊂ |Σ

|, (

, α

) := η

−1

(α

, α

) ∈ Intσ ⊂ |Σ

× Σ

| and c

dim σ

+ dim σ

− dim σ then; c

≥ 0 and [c

= 0 iff σ is a maximal simplex

in ¯

× ¯

⊂ Σ

× Σ

Corollary. (to Th. 6) Let G, G

′

be arbitrary modules over a PID R such

that Tor

(G, G

′

) = 0, then, for any ∅

6= σ ∈ Σ

× Σ

with η Int(σ)

⊂

Int(σ

) × Int(σ

);

−

i+c

(Lk

Σ1×Σ2

σ; G ⊗

′

)

∼

+q=i

p,q

≥−1

[

−

(Lk

; G) ⊗

−

(Lk

); G

′

)] ⊕

⊕

+q=i−1

p,q

≥−1

Tor

−

(Lk

; G),

−

(Lk

; G

′

)

∼

−

i+1

(Lk

Σ1∗Σ2

(σ

∪ σ

); G ⊗

′

So, if ∅

6= σ and c

= 0 then

−

(Lk

Σ1×Σ2

σ; G)

∼

−

(Lk

Σ1∗Σ2

(σ

∪

); G)

and

−

(Lk

Σ1×Σ2

σ; G ⊗ G

′

)

∼

−

(Lk

Σ1

; G) ⊗

−

−1

(Lk

Σ2

; G

′

) ⊕

−

−1

(Lk

Σ1

; G) ⊗

−

(Lk

Σ2

; G

′

Proof. Note that σ 6= ∅

⇒ σ

6= ∅

, j = 1, 2. The isomorphisms of the under-

lined modules are, by Proposition 1 p. 27, Theorem 6i p. 27 in simplicial dis-
guise, and holds even without the PID-assumption. Prop. 1 p. 61, and The-
orem 4 p. 26, gives the second isomorpism even for σ

= ∅

and/or σ

= ∅

The above module homomorphisms concerns only simplicial homology,

so, the proof of them should also be possible to give purely in terms of
simplicial homology. This is however a rather cumbersome task, mainly do
to the fact that Σ

× Σ

isn’t a subcomplex of Σ

∗ Σ

Lemma 1 and 2 below are related to the defining properties for quasi-

manifolds resp. pseudomanifolds, cf. p. 47. We write out all seven items
mainly just to be able to see the details once and for all.

Read

∇

in Lemma 1 as “×” or all through as “∗” when it’s trivially true if

any Σ

= {∅

} and else for ∗, Σ

are assumed to be connected or 0-dimensional.

“codimσ ≥ 2” (⇒ dim Lk

σ ≥ 1) means that a maximal simplex, τ say,

containing σ always fulfills dim τ ≥ dim σ +2. G

, G

are A-modules. For

definitions of Intσ and ¯

σ see p. 60.

Lemma. 1.

If dim Σ

≥ 0 and v

:= dim σ

(i = , 1, 2) then D

-D are all equivalent;

)

−

(Lk

(Σ

∇Σ

)

σ; G

⊗ G

) = 0 for ∅

6= σ ∈ Σ

∇

whenever codimσ ≥ 2.

′

)

−

(Lk

; G

) = 0 for ∅

6= σ

∈ Σ

whenever codimσ

≥ 2 (i = 1, 2).

)

−

v+1

(Σ

∇

, cost

Σ1∇Σ2

σ; G

⊗ G

) = 0

for ∅

6= σ ∈ Σ

∇

, if codimσ ≥ 2.

′

)

−

(Σ

, cost

Σi

; G

) = 0 for ∅

6= σ

∈ Σ

whenever codimσ

≥ 2 (i = 1, 2).

)

−

v+1

(|Σ

∇

|, |Σ

∇

| \

α; G

⊗ G

) = 0

for all α

6= α ∈ Int(σ) if codim σ≥ 2.

′

)

−

(|Σ

|, |Σ

| \

; G

) = 0

for α

6= α

∈ Int(σ

), if codimσ

≥ 2 (i = 1, 2).

−

v+1

( |Σ

∇

|Σ

|, |Σ

∇

|Σ

| \

(α

, α

); G

⊗ G

)= 0

for all α

6= (

, α

) ∈ Int(σ) ⊂ |Σ

∇

| if codimσ ≥ 2,

where η : |Σ

∇

| −→

≃ |Σ

|¯

∇

|Σ

| and η(

, α

) = (α

, α

), (k-iffikations never

effect the homology modules.)

Proof. By the homogenity of the interior of |¯

× ¯

| we only need to deal

with simplices σ fulfilling c

= 0. By Prop. 1 p. 27 and the top of this

page, all non-primed resp. primed items are equivalent among themselves.
The above connectedness conditions aren’t coefficient sensitive, so suppose
G

:= k, a field. D

⇔ D

′

by the above Corollary. For joins of finite

complexes, this is done explicitly in [16] p. 172.

Any Σ is representable as Σ =

σm

∈Σ

, where σ

denotes the simplicial

complex generated by the maximal simplex σ

Definition. 1. Two maximal faces σ, τ ∈ Σ are strongly connected if they
can be connected by a finite sequence σ = δ

, .., δ

= τ of maximal

faces with #(δ

∩ δ

) = max

0≤j≤q

#δ

−1 for consecutives. Strong connectedness

imposes an equivalence relation among the maximal faces, the equivalence
classes of which defines the maximal strongly connected components of Σ,
cp. [4] p. 419ff. Σ is said to be strongly connected if each pair of of its
maximal simplices are strongly connected. A submaximal f ace has exactly
one vertex less then some maximal face containing it.

Note. Strongly connected complexes are pure, i.e. σ ∈ Σ maximal ⇒
dimσ = dimΣ.

Lemma. 2. ([13] p. 81 gives a proof, valid for any finite-dimensional com-
plexes.)
A) If d

:= dim Σ

≥ 0 then; Σ

×Σ

is pure ⇐⇒ Σ

and Σ

are both pure.

B) If dim σ

≥ 1 for each maximal simplex σ

∈ Σ

then;

Any submaximal face in Σ

× Σ

lies in at most (exactly) two maximal

faces ⇐⇒ Any submaximal face in Σ

lies in at most (exactly) two maximal

faces of Σ

, i = 1, 2.

C) If d

> 0 then; Σ

× Σ

strongly connected ⇐⇒ Σ

, Σ

both strongly

connected.

Note. 1. Lemma 2 is true also for ∗ with exactly the same reading but now
with no other restriction than that Σ

6= ∅ and this includes in particular

item B.

Definition. 2 ∆

∆

:= {δ ∈ ∆ | δ 6∈ ∆

} is connected as a poset (partially

ordered set) w.r.t. simplex inclusion if for every pair σ, τ ∈ ∆

∆

there is a

chain σ = σ

, σ

, ..., σ

= τ where σ

∈ ∆

∆

and σ

⊆ σ

or σ

⊇ σ

Note. 2. (cf. [16] p. 162.) Given ∆⊃

∆

⊃

{∅

}. ∆

∆

is connected as a poset

iff

|∆| \

|∆

| is pathwise connected. When ∆

= {∅

}, then the notion of

connectedness as a poset is equivalent to the usual one for ∆.

|∆| is connected iff |∆

(1)

| is. ∆

(p)

:= {σ ∈ ∆ | #σ ≤ p + 1}.

Lemma. 3. ([16] p. 163) ∆

∆

is connected as a poset iff to each pair of

maximal simplices σ, τ ∈ ∆

∆

there is a chain in ∆

∆

, σ = σ

⊇ σ

⊆ σ

⊇

... ⊆ σ

= τ, where the σ

s are maximal faces and σ

2i+1

and σ

2i+2

2i+1

are situated in different components of Lk

∆

2i+1

(i = 0, 1, ..., m−1).

Lemma. 4 (A. Bj¨

orner 1995.) (Immediate from Lemma 3.) Let Σ be a

finite-dimensional simplicial complex, and assume that Lk

σ is connected

for all σ ∈ Σ, i.e. inkluding ∅

∈ Σ, such that dimLk

σ ≥ 1. Then Σ is pure

and strongly connected.

Relating Combinatorics to Commutative Alge-
bra

5.1

Definition of Stanley-Reisner rings

—∗ ∗ ∗— What we’re aiming at. —∗ ∗ ∗—

5.2. Through the Stanley-Reisner Functor below, attributes like Buchs-
baum, Cohen-Macaulay and Gorenstein on (graded) rings and modules,
became relevant also within Combinatorics as the classical definition (Def. 2
p. 3) of simplicial complexes was altered to Def. 1 p. 3. Now this extends
to General/Algebraic Topology, cf. p 38.

Th. 8 p. 40 induces an inductive characterization of simplicial homology

manifolds, i.e., including those within Mathematical Physics.
6.2.

Corollary 1 p. 52, tells us that there is no n-manifolds with an

(n − 2)-dimensional boundary. The examples in p. 55 implicitly raise the
arithmetic-geometrical question; Which boundary-dimensions are accessi-
ble w.r.t. a certain coefficient-module?
6.4. Corollary 2 p. 58, confirms Bredon’s conjecture in [2] p. 384, that
homology manifolds with Z as coefficient-module are locally orientable,

and so, these manifolds have either an empty, a (−1)-dimensional or an
(n − 1)-dimensional boundary.

—∗ ∗ ∗—

Definition. A subset s⊂W⊃V

∆

is said to be a non-simplex (w.r.t. W) of

a simplicial complex ∆, denoted s n

∈∆, if s 6∈ ∆ but ˙s = (¯

(dim s)−1

⊂ ∆ ( i.e.

the (dim s − 1)-dimensional skeleton of ¯

s, consisting of all proper subsets

of s, is a subcomplex of ∆). For a simplex δ = {v

, . . . , v

} we define m

to be the squarefree monic monomial m

:= 1

· v

·. . .· v

∈ A[W] where

A[W] is the graded polynomial algebra on the variable set W over the
commutative ring A with unit 1

. So, m

∅

= 1

Let A

∆

:= A[W]/I

∆

where I

∆

is the ideal generated by {m

| δ n

6∈ ∆}.

∆

is called the “face ring” or “Stanley-Reisner (St-Re) ring” of ∆ over

A. Frequently A = k, a field. Typical within physics and geometry is k =
C

]

, the complex numbers.

Stanley-Reisner (St-Re) ring theory is a basic tool within combinatorics,

where it supports the use of commutative algebra. The definition of m

, in

∆

(

| δ 6∈ ∆}

)

in existing literature leaves the questions ”A

{∅

}

?” and ”A

∅

= ?” open. This is rectified in Note ii, showing that:

{∅

}

∼

= A 6= 0 =“The trivial ring”= A

∅

Note. i. Let P be the set of finite subsets of the set P then; A

= A[P].

P is known as “the full simplicial complex on P” and the natural numbers
N gives N as “the infinite simplex ”. ∆ =

6∈

∆

cost

s . (See

pp. 60ff for definitions.)

ii. A

∆

∼

A[V

∆

]

({

∈A[V

∆

]

| δ

6∈∆})

, if ∆ 6= ∅, {∅

}. So, the choice of the universe

W isn’t all that critical. If ∆ = ∅, then the set of non-simplices equals {∅},
since ∅

6∈ ∅, and ∅

((dim ∅o)−1)

= {∅

}

(−2)

= ∅ ⊂ ∅, implying; A

∅

= 0 =“The

trivial ring”, since m

∅

= 1

Since ∅ ∈ ∆ for every simplicial complex ∆ 6= ∅, {v} is a non-simplex of

∆ for every v ∈ W \ V

∆

, i.e. [v 6∈ V

∆

6= ∅]⇔[{v} n

6∈ ∆ 6= ∅]. So A

{∅

}

= A

since {δ n

6∈ ∆} = W.

iii. k

∆

∗∆

∼

= k

∆

⊗ k

∆

with I

∆

∗∆

= ({m

[δ n

6∈ ∆

∨ δ n

6∈ ∆

] ∧ [δ /

∈

∆

∗∆

]}) by [13] Example 1 p. 70. ((·) :=the ideal generated by ·)

iv. If ∆

6= ∅ i = 1, 2, it’s well known that

a. I

∆

∪∆

= I

∆

∩ I

∆

= ({m = Lcm(m

, m

)

∈ ∆

, i = 1, 2})

b. I

∆

∩∆

= I

∆

+ I

∆

= ({m

δ n

∈ ∆

∨ δ n

∈ ∆

}) in A[W]

∆

∩I

∆

and I

∆

are generated by a set (no restrictions on its car-

dinality) of squarefree monomials, if both I

∆

and I

∆

are. These squarefree

monomially generated ideals form a distributive sublattice (J

◦

; ∩, +, A[W]

of the ordinary lattice structure on the set of ideals of the polynomial ring
A[W], with a counterpart, with reversed lattice order, called the squarefree
monomial rings with unit, denoted (A

◦

[W]; ∩, +, I). A[W]

is the unique

homogeneous maximal ideal and zero element. We can use the ordinary
subset structure to define a distributive lattice structure on Σ

◦

:= “The

set of non-empty simplicial complexes over W”, with {∅} as zero element
and denoted (Σ

◦

; ∪, ∩, {∅}). The Weyman/Fr¨

oberg/Schwartau Construc-

tion eliminates the above squarefree-demand, cf. [42] p. 107.

Proposition.The Stanley-Reisner Ring Assignment Functor defines a mo-
nomorphism on distributive lattices from (Σ

◦

, ∪, ∩, {∅}) to (I

◦

[W], ∩, +, I),

which is an isomorphism for finite W.

5.2

Buchsbaum&(2-)Cohen-Macaulay Complexes are Weak
Manifolds

We are now in position to give combinatorial/algebraic counterparts of

the weak homology manifolds defined in §3.4 p. 29.

Prop. 1 below and

Th. 8 p. 40 plus Th. 11 p. 51, show how simplicial homology manifolds can
be inductively constructed.

Combinatorialists call a finite simplicial complex a Buchsbaum (Bbm

)

(Cohen-Macaulay (CM

)) complex if its Stanley-Reisner Ring is a Buchs-

baum (Cohen-Macaulay) ring. We won’t use the ring theoretic definitions
of Bbm or C-M and therefore we won’t write them out. Instead we’ll use
some homology theoretical characterizations found in [41] pp. 73, 60-1 resp.
94 to deduce, through Prop. 1 p. 27, the following consistent definitions for
arbitrary modules and topological spaces.

Definition.

X is “Bbm

” (“CM

”, 2-“CM

”) if X is an n-whm

(n-jwhm

, n-whsp

A simplicial complex Σ is “ Bbm

”, “CM

” resp. 2-“CM

” if |Σ| is. In

particular, ∆ is 2-“CM

” iff ∆ is “CM

” and

−

−1

(cost

∆

δ ; G) = 0, ∀ δ ∈ ∆,

cp. [41] Prop. 3.7 p. 94.

The n in “n-Bbm” (“n-CM” resp. 2-“n-CM

”) is deleted since any inte-

rior point α of a realization of a maxi-dimensional simplex gives;

−

dim ∆

(|∆|, |∆|\

α; G) = G. σ ∈ Σ is ”maxi-dimensional” if dim σ = dim Σ and “\

” is de-

fined in p. 15.

So we’re simply renaming n-whm

, n-jwhm

and n-whsp

to “Bbm

”,

“CM

” resp. 2-“CM

”, where the quotation marks indicate that we’re not

limited to compact spaces nor to just Z or k as coefficient modules.

N.B. The definition p. 32 of ˆ|∆

℘

(X )ˆ| provides each topological space X

with a Stanley-Reisner ring, which, w.r.t. “Bbm

”-, “CM

”- and 2-“CM

”-

ness is triangulation invariant. This constitutes a consistencyconfirmation
of our {℘}-synchronization of General Topology to Logics, Combinatorics,
Commutative Algebra and Category Theory.

Proposition. 1. The following conditions are equivalent: (We assume
dim ∆ = n.)
a. ∆ is “Bbm

”,

b. (Schenzel, [18] p. 96) ∆ is pure and Lk

∆

δ is“CM

” ∀ ∅

6= δ ∈ ∆,

c. (Reisner, [5] 5.3.16.(b) p. 229) ∆ is pure and Lk

∆

v is “CM

” ∀ v ∈ V

∆

Proof. Use Prop. 1 p. 27. and Lemma 4 p. 36. and then use Eq. I p. 61.

Example. When limited to compact polytopes and a field k as coefficient
module, we add, from [41] p. 73, the following Buchsbaum-equivalence using
local cohomology;
d. (Schenzel) ∆ is Buchsbaum iff dim

–

∆

]

∆

) ≤ ∞ if 0 ≤ i <

dim k

∆

), in which case

–

∆

]

∆

) ∼

−

−1

(|∆|; k), cf. [42] p. 144 for

proof. Here, “dim” is Krull dimension, which for Stanley-Reisner Rings is
simply “1 + the simplicial dimension”.

For Γ

, Γ

finite and CM

we get the following K¨

unneth formula for ring

theoretical local cohomology; (“·

” indicates the unique homogeneous max-

imal ideal of “·”.)
1.

[[

Γ1

]]

⊗k

[[

Γ2

]]

[[[

]]]

⊗ k

[[[

]]]

) ∼

∼

k[Γ

]⊗ k[Γ

] ∼

= k[Γ

∗ Γ

cf. iii p. 37

Eq. 3 p. 26. and Corollary i p. 29

∼

+j=

[[

Γ1

]]

[[[

]]]

)⊗ H

[[

Γ2

]]

[[[

]]]

2. Put: β

(X) := inf{j | ∃x; x ∈ X ∧

−

(X, X \

x; G) 6= 0}. For a finite ∆,

(∆) is related to the concepts “depth of the ring k

∆

” and “C-M-ness

of k

∆

” through β

(∆) = depth(k

∆

) −1, in [5] Ex. 5.1.23 p. 214 and [41]

p. 142 Ex. 34. See also [34].

Proposition. 2.

a. [∆ is “CM

”] ⇐⇒ [

−

(∆,cost

∆

δ; G) = 0 ∀ δ ∈ ∆ and ∀ i ≤ n−1].

b. (cf. [41] p. 94) [∆ is 2-“CM

”] ⇐⇒ [

−

(cost

∆

δ; G) = 0 ∀ δ ∈

∆

and ∀ i ≤

n−1].

Proof. Use the L

−

HS w.r.t. (∆, cost

∆

δ), Prop. 1 p. 27 and the fact that

cost

∆

∅

= ∅ respectively cost

∆

δ= ∆ if δ 6∈ ∆ .

Put ∆

(p)

:= {δ ∈ ∆ | #δ ≤ p + 1}, ∆

:= ∆

(p)

\ ∆

(p−1)

, ∆

′

:= ∆

(n−1)

, n := dim ∆. So,

∆

(n)

= ∆.

Theorem 8.

a. ∆ is “CM

” iff ∆

′

is 2-“CM

” and

−

−1

(∆, cost

∆

δ; G) = 0 ∀ δ ∈ ∆.

b. ∆ is 2-“CM

” iff ∆

′

is 2-“CM

” and

−

−1

(cost

∆

δ; G) = 0 ∀ δ ∈

∆

Proof. Proposition 2 p. 40 together with the fact that adding or deleting n-
simplices does not effect homology groups of degree ≤ n−2. See Proposition
3.

p. 62.

Th. 8 is partially deducible, using commutative algebra, through [19] pp.

358-60.

Our next corollary was, for G = k, originally ring theoretically proven

by T. Hibi. We’ll essentially keep Hibi’s formulation, though using that
∆

:= ∆

{τ ∈ ∆ | τ ⊃ δ

for some i ∈ I} =

∈

cost

∆

Corollary. . ([18] Corollary p. 95-6) Let ∆ be a pure simplicial complex of
dimension n and {δ

}

∈

, a finite set of faces in ∆ satisfying δ

∪

∈ ∆ ∀ i 6= j.

Set, ∆

∈

cost

∆

a) If ∆ is “CM

” and dim ∆

< n, then dim ∆

= n−1 and ∆

is “CM

”.

b) If ¯

∆

is “CM

” ∀ i ∈ I and ∆

is “CM

” of dimension n, then ∆ is also

“CM

”.

Proof. a. [δ

∪

∈ ∆∀i 6= j ∈ I] ⇐⇒[cost

∆

∪

6=i

cost

∆

= ∆] =⇒

cost

∆′

∪

j<i

cost

∆′

cost

∆

∪

j<i

cost

∆

∩

∆

′

= ∆

′

=⇒ dim ∆

= n−1 =⇒ ∆

= ∆

∩

∆

′

∈

cost

∆

∩

∆

′

∈

cost

∆′

By Th. 8 we know that ∆

′

is 2-“CM

”, implying that cost

∆′

is “CM

” ∀ i ∈ I.

Induction using the M-Vs w.r.t. (cost

∆′

j<i

cost

∆′

) gives

−

(∆

; G) = 0 ∀ i <

∆

−1.

For links, use Prop. 2 a + b p. 61. E.g. Lk

∆◦

δ =Lk

∆◦∩∆′

δ = Lk

( ∩

∩

∈

cost

∆

i)∩∆

′

= [Prop. 2.I

p. 61]=

∈

cost

∆′

δ where cost

∆′

and so Lk

cost

∆′

δ is “CM

”, ∀ i ∈ I.

⊲

b. ∆ = ∆

∈

∆

. st

∆

δ ∩ ∆

Eq. II+III p. 60ff

plus that

∪δ

∈∆

= st

∆

δ ∩ cost

∆

δ = ˙δ ∗ Lk

∆

δ.

So, by Cor. i p. 29 st

∆

δ ∩ ∆

is “CM

” iff st

∆

δ is. Induction, using the M-Vs

w.r.t. (st

∆

, ∆

j<i

∆

), gives

−

(∆ ; G) = 0 ∀ i < n

∆

. End as in I.

Now, we’ll show that our 2-“CM”-ness is consistent with that of K. Ba-

clawski.

Lemma. 1.

∆ “CM

” =⇒







(a)

−

(cost

∆

δ; G) = 0 ∀ δ ∈ ∆

if i ≤ n − 2

and
(b)

−

(cost

cost

∆

; G) = 0 ∀ δ

, δ

∈ ∆ if i ≤ n − 3.

Proof. (a) Use Proposition 1 p. 15, the definition of “CM”-ness and the
L

−

HS w.r.t. (∆, cost

∆

δ), which reads;

...

1∗

−→

−

(∆,cost

∆

δ;G)

1∗

−→

−

−1

(cost

∆

δ;G)

1∗

−→

−

−1

(∆;G)

1∗

−→

−

−1

(∆,cost

∆

δ;G)

1∗

−→

−

−2

(cost

∆

δ;G)

1∗

−→...

⊲

(b) Apply Prop. 3. p. 35 a+b to the M-Vs w.r.t. (cost

∆

, cost

∆

), then use

a, i.e;

... = ...

β1∗

−→

−

(cost

∆

(δ

∪δ

);G)

δ1∗

−→

−

−1

(cost

cost

∆

δ2

;G)

α1∗

−→

−

−1

(cost

∆

;G) ⊕

−

−1

(cost

∆

;G)

β1∗

−→

β1∗

−→

−

−1

(cost

∆

(δ

∪δ

);G)

δ1∗

−→

−

−2

(cost

cost

∆

δ2

;G)

α1∗

−→

−

−2

(cost

∆

;G) ⊕

−

−2

(cost

∆

;G)

β1∗

−→ ...

Observation. To turn the implication in lemma 1 into an equivalence we
just have to add

−

(∆;G) = 0 for i ≤ n−1, giving us the equivalence in;

∆ “CM

” ⇐⇒











−

(∆; G) = 0

i ≤ n−1

ii)

−

(cost

∆

δ; G) = 0 ∀ δ∈ ∆

i ≤ n−2

iii)

−

(cost

cost

∆

; G) = 0 ∀ δ

, δ

∈ ∆i ≤ n−3

(1)

−

−1

(cost

∆

δ; G) = 0 ∀ δ ∈ ∆ allows one more step in the proof of Lemma

1b, i.e.;

∆ is “CM

”

and

−

−1

(cost

∆

δ; G) = 0 ∀ δ∈∆ ⇐⇒

⇐⇒











−

(∆; G) = 0

i ≤ n−1

ii)

−

(cost

∆

δ; G) = 0 ∀ δ ∈ ∆

i ≤ n−1

iii)

−

(cost

cost

∆

; G) = 0 ∀ δ

, δ

∈ ∆ i ≤ n−2

(2)

Note. a: Item i in Eq. 2 follows from ii and iii by the M-Vs above.
b: The l.h.s. in Eq. 2 is, by definition, equivalent to ∆ being 2-“CM”.

Since, [∆“CM

”] ⇐⇒ [

−

(∆,cost

∆

δ; G) = 0 ∀ δ ∈ ∆ and ∀ i ≤ n−1], item iii

in Eq. 1 is, by the L

−

HS w.r.t. (∆,cost

∆

δ), totally superfluous as far as the

equivalence is concerned but never the less it becomes quite useful when
substituting cost

∆

δ for every occurrence of ∆ and using that cost

cost

∆

= cost

∆

for δ

∈ cost

∆

δ, we get;

cost

∆

δ “CM

” ∀ δ∈ ∆ ⇐⇒

(

(i)

−

(cost

∆

δ; G) = 0

i ≤ n

−1 ∀ δ∈ ∆

(ii)

−

(cost

cost

∆

; G) = 0

i ≤ n

−2 ∀ δ, δ

∈ ∆. (3)

Remark 1. From the L

−

HS w.r.t. (∆, cost

∆

δ), Corollary iii p. 29 and Note

1 p. 50 we conclude that; If ∆ is 2-“CM

” then Note 1 p. 30 plus Prop. 1

p. 27 implies that

−

∆

(∆;G) 6= 0.

Remark 2. It’s always true that; n

∆

− 1 ≤ n

≤ n

∆

if τ ⊂ δ, where

:= dim cost

∆

ϕ and n

∆

:= dim ∆.

Set • • • := {∅

, {v

}, {v

, v

}}. Now,

= n

∆

and cost

∆

v pure ∀ v∈ V

∆

] ⇐⇒ [n

= n

∆

and cost

∆

δ pure ∀ ∅

6= δ∈ ∆] =⇒

=⇒ [cost

∆

δ (= ∪

∈δ

cost

∆

v ) pure ∀ ∅

6= δ ∈ ∆ 6= • • •] =⇒ [∆ is pure] =⇒

[v ∈ V

∆

is a cone point ] ⇐⇒ [n

= n

∆

−1] ⇐⇒ [v ∈ δ

∈ ∆ if dim δ

= n

∆

]

Since, n

:= dim cost

∆

ϕ = n

∆

−1 ⇐⇒ ∅

6= ϕ ⊂ δ

∈ ∆ ∀ δ

∈ ∆ with

dim δ

= n

∆

, we conclude that;

If ∆ is pure then ϕ consists of nothing but cone points, cf. p. 45.

So, [∆ pure and n

= dim∆ ∀ δ ∈ ∆] ⇐⇒ [∆ pure and has no cone points].

([n

= n

∆

−1∀ v ∈ V

∆

] ⇐⇒ V

∆

is finite and ∆ = V

∆

(:= the full complex w.r.t. V

∆

).)

Theorem 9.

The following two double-conditions are equivalent to “∆ is 2-“CM

””;







i. cost

∆

δ is “CM

”, ∀ δ ∈ ∆

and

ii. n

:= dim cost

∆

δ = dim ∆ =: n

∆

∀ ∅

6= δ ∈ ∆.







i. cost

∆

v is “CM

”, ∀ v ∈ V

∆

and
ii. {• •

•} 6= ∆ has no cone points.

Proof. With no dimension collapse in Eq. 3 it’s equivalent to Eq. 2.

Definition. ∆ \ [{v

, . . . , v

}] := {δ ∈ ∆| δ ∩ {v

, . . . , v

} = ∅}. (∆\[{v}] =

cost

∆

v.)

Permutations and partitions within {v

, . . . , v

} doesn’t effect the result,

i.e;

Lemma. 2. ∆\[{v

, . . . ,v

}] = (∆\[{v

′

, . . . ,v

′

}])\[{v

′′

, . . . ,v

′′

}] and

∆\ [{v

, . . . ,v

}] =

=1,p

cost

∆

= cost

cost v2

cost

∆

Definition. Alternative Definition. For k ∈ N, ∆ is k-“CM

” if for

every subset T ⊂ V

∆

such that #T = k − 1, we have:

i. ∆ \ [T ] is “CM

”,

ii. dim ∆ \ [T ] = dim ∆ =: n

∆

=: n.

Changing “#T = k − 1” to “#T < k” doesn’t alter the extension of

the definition. (Iterate in Th. 9b mutatis mutandis.) So, for G = k, a
field, it’s equivalent to Kenneth Baclawski’s original definition in Europ. J.
Combinatorics 3 (1982) p. 295.

5.3

Stanley-Reisner Rings for Simplicial Products are Segre
Products

Definition.. The Segre product of the graded A-algebras R

and R

, de-

noted R = σ

, R

) or R = σ(R

, R

), is defined through;

[R]

:=[R

]

⊗

]

, ∀p ∈ N.

Example. The trivial Segre product, R

, is equipped with the trivial

product, i.e. every product of elements, both of which lacks ring term,
equals 0.
2. The “canonical” Segre product, R

⊗R

, is equipped with a product

induced by extending ( linearly and distributively) the componentwise mul-
tiplication on simple homogeneous elements: If m

′

⊗ m

′′

∈

⊗R

and

′

⊗ m

′′

∈

⊗R

then (m

′

⊗ m

′′

)(m

′

⊗ m

′′

) := m

′

⊗ m

′′

∈

⊗R

α+β

3. The “canonical” generator-order sensitive Segre product, R

⊗R

, of

two graded k-standard algebras R

and R

presupposes the existence of a

uniquely defined partially ordered minimal set of generators for R

)

in [R

]

([R

]

) and is equipped with a product induced by extending

(distributively and linearly) the following operation defined on simple ho-
mogeneous elements, each of which now are presumed to be written, in
product form, as an increasing chain of the specified linearly ordered genera-
tors: If (m

⊗m

) ∈

⊗R

and (m

⊗m

) ∈

⊗R

then (m

⊗

)(m

⊗ m

) := ((m

⊗ m

) ∈

⊗R

α+β

if by “pairwise”

permutations, (m

, m

) can be made into a chain in the product

ordering, and 0 otherwise. Here, (x, y) is a pair in (m

, m

) if x

occupy the same position as y counting from left to right in m

and

respectively.

Note. 1. ([42] p. 39-40) Every Segre product of R

and R

is module-

isomorphic by definition and so, they all have the same Hilbert series. The
Hilbert series of a graded k-algebra R =

≥0

Hilb

(t) :=

i≥0

(H(R, i))t

i≥0

(dim

, where “dim” is Krull di-

mension and “H” stands for the “The Hilbert function”.
2. If R

, R

are graded algebras finitely generated (over k) by x

, . . . , x

∈

]

, y

, . . . , y

∈ [R

]

, resp., then R

⊗R

and R

⊗R

are generated by

⊗y

), . . . , (x

⊗y

), and dim R

⊗R

= dim R

⊗R

= dim R

+dim R

−1.

3. The generator-order sensitive case covers all cases above. In the theory
of Hodge Algebras and in particularly in its specialization to Algebras with
Straightening Laws (ASLs), the generator-order is the main issue, cf. [5] §
7.1 and [20] p. 123 ff.

[13] p. 72 Lemma gives a reduced (Gr¨

obner) basis, C

′

∪D, for “I”in k

∆

= k[V

∆1

×V

∆2

]/I with C

′

:= {w

λ,µ

ν,ξ

| λ < ν ∧ µ > ξ}, w

λ,µ

:= (v

, v

), v

∈

∆1

, v

∈ V

∆2

where the subindices reflect the assumed linear ordering on the

factor simplices and with p

as the projection down onto the i:th factor;

D :=

= w

λ1,µ1

· · · · · w

,µ

(

)}

∈∆

∧

(

)} ∈ ∆

∧

<···<λ

≤···≤µ

∨

(

)} ∈ ∆

∧

(

)}

∈∆

∧

≤···≤λ

<···<µ

∨

(

)}

∈∆

∧

(

)}

∈∆

∧

<···<λ

<···<µ

′

∪ D = {m

δ n

6∈ ∆

×∆

} and the identification v

⊗ v

↔ (v

, v

) gives,

see [13] Theorem 1 p. 71, the following graded k-algebra isomorphism of
degree zero; k

∆

× ∆

= k

∆

⊗ k

∆

, which, in the Hodge Algebra

terminology, is the discrete algebra with the same data as k

∆

⊗ k

∆

cf. [5] § 7.1. If the discrete algebra is “C-M” or Gorenstein (Definition p.
26), so is the original by [5] Corollary 7.1.6. Any finitely generated graded
k-algebra has a Hodge Algebra structure, see [20] p. 145.

5.4

Gorenstein Complexes without Cone Points are Homol-
ogy Spheres

Definition. 1. v ∈ V

is a cone point if v is a vertex in every maximal

simplex in Σ.

Definition. 2. (cp. [41] Prop 5.1) Let Σ be an arbitrary (finite) complex
and put; Γ := coreΣ := {σ ∈ Σ | σ contains no cone points}.

Then; ∅ isn’t Gorenstein

, and ∅ 6= Σ is Gorenstein

(Gor

) if

−

(|Γ|, |Γ| \

α; G) =

if i 6= dim Γ

G if i = dim Γ

∀ α ∈ |Γ|.

Note.. Σ Gorenstein

⇐⇒ Σ finite and |Γ| is a homology

sphere as defined

in p. 30. δ

:= {v ∈ V

| v is a cone point} ∈ Σ. Now; v is a cone point iff

v = Σ and so, st

= Σ = (coreΣ) ∗ ¯

and

coreΣ = Lk

:= {τ ∈ Σ|[δ

∩ τ = ∅] ∧ [δ

∪ τ ∈ Σ]}.

Proposition. 1. (Cf. [13] p. 77. )

∗ Σ

Gorenstein

⇐⇒ Σ

, Σ

both Gorenstein

Gorensteinness is, unlike “Bbm

”-, “CM

”- and 2-“CM

”-ness, triangu-

lationsensitive and in particular, the Gorensteinness for products is sensi-
tive to the partial orders, assumed in the definition, given to the vertex sets
of the factors. See p. 45 for {m

δ n

6∈ ∆

×∆

}. In [13] p. 80, the product

is represented in the form of matrices, one for each pair (δ

, δ

) of maximal

simplices δ

∈ ∆

, i = 1, 2. It is then easily seen that a cone point must

occupy the upper left corner in each matrix or the lower right corner in
each matrix. So a product (dim ∆

≥ 1) can never have more than two

cone points. For Gorensteinness to be preserved under product the factors
must have at least one cone point to preserve even “CM

”-ness, by Corollary

iii p. 29.

Bd(core(∆

×∆

)) = ∅ demands each ∆

to have as many cone points as

∆

×∆

. So;

Proposition. 2. (Cf. [13] p. 83ff. for proof.) Let ∆

, ∆

be two arbitrary

finite simplicial complexes with dim∆

≥ 1, (i = 1, 2) and a linear order

defined on their vertex sets V

∆

, V

∆

respectively, then;

∆

× ∆

Gor

⇐⇒ ∆

, ∆

both Gor

and

condition I or II holds,

where;

(I) ∆

, ∆

has exactly one cone point each - either both minimal or both

maximal.

(II) ∆

has exactly two cone points, one minimal and the other maximal,

i = 1, 2.

Example. Gorensteinness is character sensitive! Let Γ := coreΣ = Σ be
a 3-dimensional Gor

complex where k is the prime field Z

of character-

istic p. This implies, in particular, that Γ is a homology

3-manifold.

Put

−

(Γ; Z), then;

−

= 0,

−

= Z and

−

has no torsion by

Lemma 1.i p. 49.

Poincare’ duality and [40] p. 244 Corollary 4 gives

−

⊕ T

−

, where T◦ := the torsion-submodule of ◦ . So a Σ =

Γ with a pure torsion

−

= Z

, say, gives us an example of a presumptive

character sensitive Gorenstein complex. Examples of such orientable com-
pact combinatorial manifolds without boundary is given by the projective
space of dimension 3, P

and the lens space L(n, k) where

−

; Z) = Z

and

−

(L(n, k); Z) = Z

. So, P

∗ • L(n, k) ∗ •

is Gor

for chark 6=

2 (chark 6= n) while it is not even Buchsbaum for chark = 2 (chark = n),
cf. [33] p. 231-243 for details on P

and L(n, k). Cf. [41] Prop. 5.1 p. 65

or [13] p. 75 for Gorenstein equivalences. A Gorenstein

∆ isn’t in general

shellable, since if so, ∆ would be CM

but L(n, k) and P

isn’t. Indeed, in

1958 M.E. Rudin published [38], An Unshellable Triangulation of a Tetra-
hedron.

Other examples are given by Jeff Weeks’ computer program “SnapPea”

hosted at http://thames.northnet.org/weeks/index/SnapPea.html, e.g.

−

(Σ

fig8

(5, 1); Z) = Z

for the old tutorial example of SnapPea, here denoted

fig8

(5, 1), i.e. the Dehn surgery filling w.r.t. a figure eight complement with

diffeomorphism kernel generated by (5,1). Σ

fig8

(5, 1) ∗ • is Gor

if chark6= 5

but not even Bbm

if chark = 5.

Simplicial Manifolds

6.1

Definitions

We will make extensive use of Proposition 1 p. 27 without explicit notifi-
cation. A topological space will be called a n-pseudomanifold or a quasi-
n-manifold if it can be triangulated into a simplicial complex that is a
n-pseudomanifold resp. a quasi-n-manifold.

Definition. 1.

An n-dimensional pseudomanifold is a locally finite n-

complex Σ such that;
(α) Σ is pure, i.e. the maximal simplices in Σ are all n-dimensional.
(β) Every (n − 1)-simplex of Σ is the face of at most two n-simplices.
(γ) If s and s

′

are n-simplices in Σ, there is a finite sequence s = s

, s

, . . . s

′

of n-simplices in Σ such that s

∩ s

i+1

is an (n− 1)-simplex for 0 ≤ i < m.

The boundary, BdΣ, of an n-dimensional pseudomanifold Σ, is the sub-

complex generated by those (n − 1)-simplices which are faces of exactly one
n-simplex in Σ.

Definition. 2. Σ = •• is a quasi-0-manifold. Else, Σ is a quasi-n-manifold
if it’s an n-dimensional, locally finite complex fulfilling;

(

)

Σ is pure.

(α is redundant since it follows from γ by Lemma 4 p. 36.)

(

)

Every (n − 1)-simplex of Σ is the face of at most two n-simplices.

(

)

σ is connected i.e.

−

(Lk

σ; G) = 0 for all σ ∈ Σ, s.a. dimσ < n−1.

The boundary with respect to G, denoted Bd

Σ, of a quasi-n-manifold

Σ, is the set of simplices Bd

Σ := {σ ∈ Σ |

−

(Σ, cost

σ; G) = 0}, where G

is a unital module over a commutative ring I. (β in Def. 1-2 ⇒ • and • •
are the only 0-manifolds.)

Note. 1. Σ is (locally) finite ⇐⇒ |Σ| is (locally) compact. If X = |Σ| is a
homology

n-manifold (n-hm

) we’ll call Σ a n-hm

and then by Th. 5 p. 26;

Σ is a n-hm

for any R-PID module G.

[40] p. 207-8 + p. 277-8 treats the case R = G = Z := {the integers}, A

simplicial complex Σ is called a hm

if |Σ| is. The n in n-hm

is deleted,

since now n = dim Σ. From a purely technical point of view we really don’t
need the “locally finiteness”-assumption, as is seen from Corollary p. 29

Definition. 3. (Let “manifold” stand for pseudo-, quasi- or homology
manifold.)
A compact n-manifold, S, is orientable

−

(S, BdS;G) ˜

=G. An n-manifold

is orientable

if all its compact n-submanifolds are orientable − else, non-

orientable

. Orientability is left undefined for ∅.

Note. 2. For a classical homology

manifold X 6= • s.a. Bd

X = ∅; Bd

℘

(X) = ∅ if X is compact and orientable and Bd

℘

(X) = {℘} else. • is the

only compact orientable manifold with boundary = {℘} ({∅

}).

Definition. 4. {B

}

∈

is the set of strongly connected boundary components

of Σ if {B

}

∈

is the maximal strongly connected components of Bd

Σ, from

Definition 1 p. 20. (⇒ B

pure and if σ is a maximal simplex in B

, then;

σ = Lk

σ = {∅

}).

Note. 3. ∅, {∅

}, and 0-dimensional complexes with either one, •, or two, ••,

vertices are the only manifolds in dimensions ≤ 0, and the |1-manifolds| are
finite/infinite 1-circles and (half)lines, while [Σ is a quasi-2-manifold] ⇐⇒
[Σ is a homology

2-manifold]. Def. 1.γ is paraphrased by “Σ is strongly

connected” and ••-complexes, though strongly connected, are the only non-
connected manifolds.

Note also that S

-1

:= {∅

} is the boundary of the 0-

ball, •, the double of which is the 0-sphere, ••. Both the (−1)-sphere {∅

}

and the 0-sphere •• has, as preferred, empty boundary.

6.2

Auxiliaries

Lemma. 1. For a finite n-pseudomanifold Σ;
i. (cf. [40] p. 206 Ex. E2.)

−

(Σ, BdΣ; Z) = Z and

−

n−1

(Σ, BdΣ; Z) has no torsion, or

−

(Σ, BdΣ; Z) = 0 and the torsion submodule of

−

n−1

(Σ, BdΣ; Z) is iso-

morphic to Z

ii. Σ orientable

⇐⇒ Σ orientable

or Tor

= G.

So, any complex Σ is orientable w.r.t. Z

iii.

−

(Σ, BdΣ;Z) = Z (Z

) when Σ is (non-)orientable.

Proof. By conditions α, β in Def. 1 p. 47, a possible relative n-cycle in
C

(Σ, BdΣ; Z

) must include all oriented n-simplices all of which with co-

efficients of one and the same value. When the boundary function is ap-
plied to such a possible relative n-cycle the result is an (n − 1)-chain that
includes all oriented (n − 1)-simplices, not supported by the boundary, all
of which with coefficients 0 or

-2c ∈ Z

. So,

−

(Σ, BdΣ; Z

) = Z

− always,

and

−

(Σ, BdΣ; Z) = Z (0) ⇐⇒

−

(Σ, BdΣ; Z

) = Z

(0) if m 6= 2. The

Universal Coefficient Theorem (= Th.5 p. 26) now gives;

−

(Σ, BdΣ; Z

)

∼

−

(Σ, BdΣ; Z ⊗

)

∼

−

(Σ, BdΣ; Z) ⊗

⊕ Tor

−

−1

(Σ, BdΣ; Z), Z

)

and,

−

(Σ, BdΣ; Z

)

∼

−

(Σ, BdΣ; Z ⊗

)

∼

−

(Σ, BdΣ; Z) ⊗

⊕ Tor

−

−1

(Σ, BdΣ; Z), Z

where the last homology module in each formula, by [40] p. 225 Cor. 11, can
be substituted by its torsion submodule. Since Σ is finite,

−

−1

(Σ, BdΣ; Z) =

⊕C

⊕...⊕Cs by The Structure Theorem for Finitely Generated Modules

over PIDs, cf. [40] p. 9. Now, a simple check, using [40] p. 221 Example
4, gives i, which gives iii by [40] p. 244 Corollary. Theorem 5 p. 26 and i
implies ii.

Proposition 1 p. 39 together with Proposition 1 p. 27 gives the next

Lemma.

Lemma. 2.i. Σ is a n-hm

=⇒

⇐=

Σ is a quasi-n-manifold

=⇒

⇐=

Σ is an n-

pseudomanifold.
ii. Σ is a n-hm

iff it’s a“Bbm

”pseudomanifold and

−

(

σ;G

)

= 0 or G∀σ 6=

∅

Th. 10 was given for finite quasi-manifolds in [16] p. 166. Def. 1 p. 35.

makes perfect sense even for non-simplicial posets like Γ

∆ (Def. 2 p. 36).

which allow us to say that Γ

∆ is or is not strongly connected (as a poset)

depending on whether Γ

∆ fulfills Def. 1 p. 35 or not. Now, for quasi-

manifolds Γ

∆ connected as a poset (Def. 2 p. 36) is equivalent to Γ

∆

strongly connected which is a simple consequence of Lemma 3 p. 36 and
the definition of quasi-manifolds, cf. [16] p. 165 Lemma 4. Σ

cost

σ is

connected as a poset for any simplicial complex Σ and any σ 6= ∅

. Σ

BdΣ

is strongly connected for any pseudomanifold Σ. See p. 40 for the definition
of ∆

Theorem 10. If G is a module over a commutative ring A with unit, Σ
a finite n-pseudomanifold, and ∆ ⊂ Γ⊂

Σ and dim Γ = dim Σ, then; in the

relative L

−

HS w.r.t. (Σ, Γ, ∆),

−

(Σ, ∆; G) →

−

(Σ, Γ; G) is injective if

∆ is strongly connected.

Proof. Each strongly connected n-component Γ

of Γ is an n-pseudomanifold

with (BdΓ

)

−1

6= ∅ since Γ⊂

Σ.

(BdΓ

)

−1

⊂ (∆

)

:= (Γ

∩ ∆)

⊂

(Γ

)

iff

−

(Γ

, ∆

; G) 6= 0, i.e. iff every n-sequence from γ ∈ (Γ

∆

)

to σ ∈ (Σ

Γ)

connects via ∆

−1

. Now, Σ

∆ can’t be strongly connected. The relative

−

HS w.r.t. (Σ, Γ, ∆) gives our claim.

Γ = cost

τ and ∆ = cost

∅

= ∅ resp. cost

τ gives b in the next Corollary.

Corollary. 1. If Σ is a finite n-pseudomanifold and σ ⊂

τ ∈ Σ, then;

−

(Σ, cost

σ; G) −→

−

(Σ, cost

τ ; G) is injective if Σ

cost

σ strongly

connected.
b.

−

(|Σ|\

α; G)

∼

−

(cost

τ ; G) =

−

(cost

τ, cost

τ ; G) = 0, if α ∈ Int(τ )

for any τ ∈ Σ. (Cp. the proof of Proposition 1 p. 27.)

Note. 1. a. above, implies that the boundary of any manifold is a sub-
complex and

b that any simplicial manifold is “ordinary”, as defined

in p. 29 and that Bd

Σ 6= ∅ ⇐⇒

−

(Σ; G) = 0, since

−

(Σ; G) =

−

(Σ, cost

δ; G) = 0 iff δ ∈ Bd

Σ by the LHS.

Corollary. 2.
i. If Σ is a finite n-pseudomanifold with #I ≥ 2 then;

−

(Σ, ∪

∪

6=i

;G) = 0

and

−

(Σ, B

;G) = 0, (with B

:= B

from Def. 4 p. 48).

ii. Both

−

(Σ;Z) and

−

(Σ, BdΣ;Z) equals 0 or Z.

Proof. i. ∪

∪

6=i

⊂

cost

σ for some B

-maxidimensional σ ∈ B

and vice versa.⊲

ii.[dimτ = n] ⇒ [[

−

(Σ, cost

τ ; G) = G]∧[BdΣ ⊂ cost

τ ]] andΣ\BdΣ strongly

connected. Theorem 10 and the L

−

HS gives the injections

−

(Σ;G) ֒→

−

(Σ, BdΣ;G) ֒→ G.

Note. 2. Σ

∗ Σ

6⊂ Σ

locally finite ⇐⇒ Σ

, Σ

both finite, and;

Σ Gorenstein =⇒ Σ finite.

Theorem 11. i.a (cp. [16] p. 168, [17] p. 32.) Σ is a quasi-manifold iff
Σ = •• or Σ is connected and Lk

σ is a finite quasi-manifold for all ∅ 6=

σ ∈ Σ.
i.b. Σ is a homology

n-manifold iff Σ = •• or

−

(Σ;G) = 0 and Lk

σ is a

finite “CM

”-homology

(n − #σ)-manifold ∀ ∅

6= σ ∈ Σ.

ii. Σ is a quasi-manifold =⇒ Bd

(Lk

σ) = Lk

σ if σ ∈ Bd

Σ and Bd

(Lk

σ) ≡

∅ else.
iii. Σ is a quasi-manifold ⇐⇒ Lk

σ is a pseudomanifold ∀ σ ∈ Σ, including

σ = ∅

Proof. i. A simple check confirms all our claims for dim Σ ≤ 1, cf. Note 3
p. 48.

So, assume dim Σ ≥ 2 and note that σ ∈ Bd

Σ iff Bd

(Lk

σ) 6= ∅.

i.a. (⇐) That Lk

σ, with dim Lk

σ = 0, is a quasi-0-manifold implies def-

inition condition 2β p. 48 and since the other “links” are all connected
condition 2γ follows.

⊲

(⇒) Definition condition 2β p. 48 implies that 0-dimensional links are • or
•• while Eq. I p. 34 gives the necessary connectedness of ‘links of links’, cp.
Lemma 4 p. 36.⊲
i.b. Lemma2. ii above plus Proposition 1 p. 39 and Eq. I 61.

⊲

ii. Pureness is a local property, i.e. Σ pure =⇒ Lk

σ pure. Put n := dim Σ.

Now;
ǫ ∈ Bd

(Lk

σ) ⇔ 0 =

−

(Lk

ε; G) =

Eq. I 61

ε∈

−

(σ∪ε)

(Lk

(σ ∪ ε); G) and ε ∈

σ. So;

ε ∈ Bd

(Lk

σ) ⇔ [σ ∪ ε ∈ Bd

Σ and ε ∈ Lk

σ] ⇔ [σ ∪ ε ∈ Bd

Σ and σ ∩ ε = ∅] ⇔

[ε ∈ Lk

σ].

iii. (⇒) Lemma 2. i p. 50 and i.a above. (⇐) All links are connected,
except ••.

Note. 3. For a finite n-manifold Σ and a n-submanifold ∆, put U :=
|Σ| \ |∆|, implying that |Bd

∆| ∪ U is the polytope of a subcomplex, Γ,

of Σ i.e. |Γ| = |Bd

∆| ∪ U, and Bd

Σ ⊂ Γ, cp. [33] p. 427-429. Consis-

tency of Definition 3 p. 48 follows by excision in simplicial

−

Homology since;

−

(Σ, Bd

Σ; G) ֒→

−

(Σ, Γ; G) ∼

−

(Σ \ U, Γ \ U; G) =

−

(∆, Bd

∆; G). E.g.,

σ = ¯

σ ∗ Lk

σ, is an orientable quasi-manifold if Σ

is, as is Lk

σ by The-

orem 11 and Theorem 12 below. Moreover, δ ⊂ σ ⇒ st

δ non-orientable if

σ is.

Corollary. 1. For any quasi-n-manifold Σ except infinite 1-circles;

dim B

≥n−2 =⇒ dim B

=n−1.

Proof. Check n ≤ 1 and then assume that n ≥ 2. If dim σ = dim B

n − 2 and σ ∈ B

then; Lk

σ = Lk

σ = {∅

Now, by Th. 11ii Lk

σ =

(Lk

σ) =

σ is, by Th. 11i, a finite quasi-

1-manifold i.e. (a circle or) a line.

= (∅ or) • •. Contradiction!

Denote Σ by Σ

, Σ

and Σ

when it’s assumed to be a pseudo -, quasi-

resp. a homology manifold. Note also that; σ ∈ Bd

⇐⇒ Bd

(Lk

σ)

σ 6= ∅by Th. 11.ii p. 51

Corollary. 2.i. If Σ is finite quasi-n-manifold with Bd

= ∪

∪

∈I

and −1 ≤

dim B

< dim Σ−1 for some i ∈ I then,

−

(Σ, Bd

Σ; G) = 0. In particular, Σ

is non-orientable

ii. If Bd

= ∪

∪

∈I

with dim B

:= dim Bd

, then B

is a pseudomanifold.

iii.

−

(Σ, BdΣ

; G

′

−

(Σ, Bd

; G

′

−

(Σ, Bd

; G

′

), i = 0, 1 even

if G 6= G

′

Orientability is independent of G

′

, as long as Tor

′

6= G

′

(Lemma

1. ii p. 49). Moreover, Bd

= Bd

always, while (BdΣ

)

= (Bd

)

and

(BdΣ

)

= (Bd

)

except for infinite 1-circles in which case (BdΣ

)

∅ 6= {∅

} = (Bd

)

iv. For any finite orientable

quasi-n-manifold Σ, each boundary component

:= B

" {∅

} in Def. 4 p. 48, is an orientable (n − 1)-pseudomanifold

without boundary.

v. Tor

,G) = 0 =⇒ Bd

= Bd

vi. BdΣ

= Bd

⊆ Bd

with equality if Bd

Σ = ∅ or dimB

n−1 ∀j ∈ I, except if Σ is infinite and Bd

= {∅

} 6= ∅ = BdΣ

(by Lemma 1

i+ii p. 49 plus Th. 5 p. 26 since ∅

∈ Bd

6= ∅ if Σ

is infinite. If Σ

Gorenstein

then; Bd

= BdΣ

Proof. i. dim B

< dim Σ−2 from Cor. 1 above, gives the 2:nd equality and

Cor. 1.a p. 50 gives the injection-arrow in;

−

(Σ, Bd

Σ; G) =

−

(Σ, B

∪ ( ∪

∪

6=i

); G) =

−

(Σ, ∪

∪

6=i

; G) ֒→

−

(Σ, cost

σ; G) = 0

if σ ∈ Bd

∪

6=i

⊲

ii. The claim is true if dim Σ ≤ 1 and assume it’s true for dimensions
≤ n − 1. α and γ are true by definition of B

so only β remains. If σ ∈ B

with dim σ = dim B

− 1, then σ /

∈ ∪

∪

6=i

and so, Bd

(Lk

σ)

= Lk

σ = Lk

from which we conclude, by the induction assumption, that the r.h.s. is a
0-pseudomanifold i.e. • or • • .

⊲

iii. Note 1.

p. 50, plus Corollary 1 and the fact that; [σ∈ Σ

∩ BdΣ

iff

σ = •]. ⊲

iv. i and ii implies that only orientability and Bd(B

)

= ∅ remains.

Bd(B

)

6= ∅ iff

−

n−1

((B

)

; G) = 0 by Note 1 p. 50, and so, Bd

(Lk

σ)

6= ∅

iff ∃B

∋ σ by iii. From Def. 4 p. 48 we get, dim B

= dim B

= n−1, s, t ∈ I

=⇒ dim B

∩ B

≤ n − 3 if s 6= t, which, since (BdΣ

)

−i

= (Bd

)

−i

i = 1, 2,

gives;
. . . −→

−

(Bd

Σ, ∪

∪

6=i

′

)

}

= 0 for dimensional reasons.

−→

−

(Σ, ∪

∪

6=i

; G

′

)

}

= 0 by Cor. 2 p. 51

−→

−

(Σ, Bd

Σ; G

′

)

}

=G by assumption if G′=G

֒→

−

(Bd

Σ, ∪

∪

6=i

; G

′

) =

−

; G) for dimensional reasons.

In the above truncated relative LHS w.r.t. (Σ, Bd

Σ, ∪

∪

6=i

), the choice

of L is irrelevant by iii. Lemma 1.ii p. 49 gives the orientability

. If G

′

= Z

in the LHS, the injection gives our claim. Otherwise Tor

,G) = G and

we are done.

⊲

6∋ σ ∈ Bd

⇐⇒

−

((Lk

)

;G) 6= 0 =

−

((Lk

)

;Z) ⇐⇒

⇐⇒ Bd

(Lk

σ)

= ∅ 6= Bd

(Lk

σ)

. (Bd(Lk

δ)

)

(n−

#δ

)−i

= (Bd

(Lk

δ)

)

(n−

#δ

)−i

i = 1, 2, by

iii

=⇒ n −

σ − 3 ≥ dim Bd

(Lk

σ)

=⇒ 0 6=

−

((Lk

)

;G)

Th.

−

((Lk

)

;Z)

}

= 0 by assumption.

⊗G ⊕ Tor

(

−

((Lk

)

;Z), G) =

dimensional

reasons.

= Tor

(

−

((Lk

)

, Bd

(Lk

σ)

;Z), G) = 0. Contradiction! - since

Tor

,G) = 0 and since the torsion module of

−

((Lk

)

, Bd

(Lk

σ)

;Z) is

either 0 or homomorphic to Z

by Lemma 1.i p. 49, giving the contradiction

since only the torsion sub-modules matters in the torsion product by [40]
Corollary 11 p. 225. ⊲

vi.

iv and that Bd

⊆ Bd

, by Corollary 2 p. 51 plus Theorem 5

p. 26.

6.3

Products and Joins of Simplicial Manifolds

Let in the next theorem, when

∇

, all through, is interpreted as ×, the word

“manifold(s)” in 12.1 temporarily excludes ∅, {∅

} and ••.

When

∇

, all through, is interpreted as ∗ let the word “manifold(s)” in

12.1 stand for finite“pseudo-manifold(s)” (“quasi-manifold(s)”, cf. [16] 4.2
pp. 171-2). We conclude, w.r.t. joins, that Th. 12 is trivial if Σ

or Σ

= {∅

}

and else, Σ

, Σ

must be finite since otherwise, theire join isn’t locally finite.

ǫ = 0 (1) if

∇

= × (∗).

Theorem 12. If G is an A-module, A commutative with unit, and V

Σi

6= ∅

then;

12.1 Σ

∇

is a (n

+ n

+ ǫ)-manifold ⇐⇒ Σ

is a n

-manifold.

12.2 Bd(•×Σ) = •×(BdΣ). Else; Bd(Σ

∇

) = ((BdΣ

)

∇

)∪(Σ

∇

(BdΣ

)).

12.3 If any side of 12.1 holds; Σ

∇

is orientable

⇐⇒ Σ

, Σ

are both

orientable

Proof. (12.1) [Pseudomanifolds.] Lemma 2 p. 35. ⊲

[Quasi-manifolds.] Lemma 1+2 p. 34-35.⊲

Prop. 1 p. 27 and Th. 7 p. 31 gives the rest, but we still add a combinatorial
proof.

(12.2) [Quasi-manifolds; ×] Put n := dim Σ

×Σ

= dim Σ

+dim Σ

+ n

. The invariance of local

−

Homology within Intσ

× Intσ

implies,

through Prop. 1 p. 27, that w.l.o.g. we’ll only need to study simplices with
c

= 0 (Def. p. 33). Put v := dim σ = dim σ

+ dim σ

=: v

+ v

. Remembering

that now, all links are quasi-manifolds, “σ ∈ Bd

×Σ

⇐⇒ σ

∈ Bd

” follows from Corollary p. 33 with G

′

:= Z which, after deletion

of a trivial torsion term through Note 1 p. 30, gives;
♦

−

(Lk

Σ1×Σ2

σ; G)

∼

−

(Lk

Σ1

; Z) ⊗

−

(Lk

Σ2

; G) ⊕ Tor

−

(Lk

Σ1

; Z),

−

(Lk

Σ2

; G)

. ⊲

[Pseudomanifolds; ×] Reason as for Quasi-manifolds but restrict to sub-
maximal simplices only, the links of which are 0-pseudomanifolds.

⊲

[Pseudo- & Quasi-manifolds; ∗] Use Corollary p. 33 as for products, cf. [16]
4.2 pp. 171-2.

⊲

(12.3)

(Σ

∇

, Bd

(Σ

∇

)) = [Th. 12.2] = (Σ

∇

, Σ

∇

∪ Bd

∇

) =

= [

According to the

pair-definition p. 21

] = (Σ

, Bd

)

∇

(Σ

, Bd

). By Cor. 2.iv p. 53 we

can w.l.o.g. confine our study to pseudomanifolds, and choose the co-
efficient module to be, say, a field k (chark 6= 2) or Z. Since any finite
maxi-dimensional submanifold, i.e. a submanifold of maximal dimension,
in Σ

×Σ

(Σ

∗ Σ

, cp. [45] (3.3) p. 59) can be embedded in the product

(join) of two finite maxi-dimensional submanifolds, and vice versa, we con-
fine, w.l.o.g., our attention to finite maxi-dimensional submanifolds S

, S

of Σ

resp. Σ

. Now, use Eq. 1 p. 22 (Eq. 3 p. 26) and Note 3 p. 52.

Example.

1. For a triangulation Γ of a two-dimensional cylinder Bd

two circles. Bd

• = {∅

}. By Th. 12 Bd

(Γ∗•)

= Γ∪({two circles}∗•).

So, Bd

(Γ∗•)

, R

- realizable as a pinched torus, is a 2-pseudomanifold

but not a quasi-manifold.

2. Cut, twist and glue the cuts, to turn the cylinder Γ in Ex. 1 into a

M¨obius band M. “The boundary (w.r.t. Z) of the cone of M” =
Bd

(M ∗ •)

= (M ∗ {∅

}) ∪ ({a circle} ∗ •)= M ∪ {a 2-disk} which

is a well-known representation of the real projective plane

, i.e. a

homology

2-manifold with boundary {∅

} 6= ∅ if p 6= 2, cf. [17] p. 36.

:= The prime-number field modulo p.

= P

= M ∪

{a 2-disk} confirms the obvious - that the n-sphere

is the unit element w.r.t. the connected sum of two n-manifolds, cf.
[33] p. 38ff + Ex. 3 p.

366. Leaving the S

-hole empty turns P

into M. “ ∪

” denotes “union through identification of boundaries”.

(M ∗ S

)

= (M ∗ ∅) ∪ (S

∗ S

)= S

. So, Bd

∗ M

)

= (S

∗

)

∪

∗ S

)

, a quasi-4-manifold. Is Bd

∗M

)

orientable or

is Bd

(Bd

∗ M

)

={∅

3. Let

(

) be a triangulation of the projective plane (projective space

with dim P

= 4) implying Bd

= Bd

= {∅

}. So, by Th. 12, Bd

∗

)

= P

∪P

, P 6= 2 (Bd

∗P

)

= ∅ if P = 2), and dim Bd

∗P

)

= 4

while dim(P

∗ P

)

= 7, cp. Corollary 1 p. 52. Bd

∗ ••)

= •• and

∗ •)

= P

∪ • .

4. E

:= the m-unit ball. With n := p + q, p, q ≥ 0, S

= BdE

≃

Bd(E

∗ E

) ≃ E

∗ S

−1

∪ S

−1

∗ E

≃ Bd(E

×E

) ≃ E

×S

−1

∪ S

×E

by Th. 12 and Lemma p. 32. Cp. [29] p. 198 Ex. 16 on surgery.
S

≃ S

∗ S

and E

≃ E

∗ S

also hold.

See also [33] p. 376 for some non-intuitive manifold examples. Also
[42] pp. 123-131 gives insights on different aspects of different kinds
of simplicial manifolds.

Proposition. If Σ

is finite and −1 ≤ dim B

< dim Σ − 1 then Lk

δ is

non-orientable

for all δ ∈ B

. (Note that Cor. 2.i p. 52 is the special case

∅

= Σ.)

Proof. If σ ∈ Bd

∪

6=i

, then dim Bd

(Lk

σ)

≤ n−

σ − 3 in st

σ, cf. Note 3

p. 52.

6.4

Simplicial Homology

Manifolds and Their Boundaries

In this section we’ll work mainly with finite simplicial complexes and

though we’re still working with arbitrary coefficient modules we’ll delete
those annoying quotation marks surrounding “CM

”. The coefficient mod-

ule plays, through the St-R ring functor, a more delicate role in commuta-
tive ring theory than it does here in our

−

Homology theory, so when it isn’t

a Cohen-Macaulay ring we can not be sure that a CM complex gives rise
to a CM St-R ring.

Lemma. 1. i. Σ is a homology

manifold iff [[Σ = •• or Σ is connected

and Lk

v is a finite CM

pseudo manifold for all vertices v ∈ V

. ] and [

= Bd

or else; [[Bd

= Bd

= ∅ 6= {∅

} = Bd

] or [ ∃ {∅

}

⊂ Σ

and Tor

,G) = G]]]].

ii. For a CM

homology

manifold ∆, Bd

∆

= Bd

∆

for any module G.

So, for any homology

manifold Σ, Bd

Σ = ∅, {∅

} or dim B

= (n−1) for

boundary components.

Proof. i. Prop. 1 p. 39 Lemma 2.ii p. 50 and the proof of Corollary 2.v
p. 53.
ii. Use Theorem 5 p. 26 and i along with G = Z

in Corollary 2.iii p. 52.

Lemma. 2. For a finite Σ

, δ

−

(Σ, Bd

Σ; G) →

−

−1

(Σ, B

∩ ( ∪

∪

6=i

); G)

in the relative M-Vs

w.r.t. {(Σ, B

), (Σ, ∪

∪

6=i

)} is injective if #I ≥ 2 in

Definition 4 p. 48.

So, [

−

−1

(Σ, {∅}; G) = 0] =⇒ [

−

(Σ, Bd

Σ; G) = 0 or Bd

Σ is strongly con-

nected ], e.g., if Σ 6= •, •• is a CM

quasi-n-manifold.

If Σ is a finite CM

quasi-n-manifold then Bd

Σ = ∅ or it is strongly

connected and dim(Bd

Σ) = n

− 1 (by Lemma 1).

Proof. Use the relative M-Vs

w.r.t. {(Σ, B

), (Σ, ∪

∪

6=i

)}, dim(B

∩ ∪

∪

6=i

) ≤

− 3 and Corollary 2.i p. 51.

Theorem 13. i. (Cf. [41] p. 70 Th. 7.3.) If Σ is a finite orientable

homology

n-manifold with boundary then, Bd

Σ is an orientable

homology

(n − 1)-manifold without boundary.

ii. Moreover; Bd

Σ is Gorenstein

Proof. i. Induction over the dimension, using Theorem 11.i-ii, once the
connectedness of the boundary is established through Lemma 2, while
orientability

resp. Bd

(Bd

Σ) = ∅ follows from Corollary 2.iii-iv p. 52.

⊲
ii. Σ ∗ (••) is a finite orientable

homology

(n + 1)-manifold with

boundary by Th. 12 + Cor. i p. 29.

(Σ ∗ ••) = [

Th. 12.2

or Th. 7.2

] = Σ ∗

(••)

∪ (Bd

Σ) ∗ (••)

= Σ ∗ ∅ ∪ (Bd

Σ) ∗ (••) = Bd

∗ (••) where the

l.h.s. is an orientable

homology

n-manifold without boundary by the first

part. So, Bd

Σ is an orientable

homology

(n − 1)-manifold without

boundary by Th. 12 i.e. it’s Gorenstein

Note. 1. ∅ 6= ∆ is a 2-CM

⇔ ∆ = core∆ is Gorenstein

⇔ ∆ is a

homology

sphere.

Corollary. 1. (Cp. [29] p. 190.) If Σ is a finite orientable

homology

manifold with boundary, so is 2Σ except that Bd

(2Σ) = ∅. 2Σ =“the double

of Σ” := Σ ∪

Σ where

Σ is a disjoint copy of Σ and “ ∪

” is “the union through

identification of the boundary vertices”. If Σ is CM

then 2Σ is 2-CM

Proof. Use [20] p. 57 (23.6) Lemma, i.e., apply the non-relative augmental
M-Vs to the pair (Lk

v, Lk

··

v) using Prop. 2.a p. 61 and then to (Σ,

Σ) for

the CM

case.

Theorem 14. If Σ is a finite CM

-homology

n-manifold, then Σ is orientable

Proof. Σ finite CM

⇐⇒ Σ CM

for all prime fields Z

of characteris-

tic P, by (M.A. Reisner, 1976) induction over dim Σ, Theorem 5 p. 26
and the Structure Theorem for Finitely Generated Modules over PIDs.
So, Σ is a finite CM

homology

n-manifold for any prime number P by

Lemma 1.i, since Bd

Σ = Bd

Σ by Lemma 1.ii above. In particular, Bd

is Gorenstein

by Lemma 1.ii p. 49 and Theorem 13 above. If Bd

Σ = ∅ then

Σ is orientable

. Now, if Bd

Σ 6= ∅ then dim Bd

Σ = n−1 by Cor. 2.iii+iv

p. 52. and, in particular, Bd

Σ = Bd

Σ is a quasi-(n−1)-manifold.

(Bd

Σ) = ∅ since Bd

Σ is Gorenstein

and so, dim Bd

(Bd

Σ) ≤

n − 4 by Cor. 1 p. 52. So if Bd

(Bd

Σ) 6= ∅ then, by Cor. 2.ii p. 52.

Σ is nonorientable

i.e.

−

−1

(Bd

Σ, Bd

(Bd

Σ); Z) =

−

−1

(Bd

Σ; Z) = 0

and the torsion submodule of

−

−2

(Bd

Σ, Bd

(Bd

Σ); Z) =

For dimen-

sional reasons.

−

−2

(Bd

Σ; Z) is isomorphic to Z

by Lemma 1.i p. 49. In particular

−

−2

(Bd

Σ; Z) ⊗ Z

6= 0 implying, by Th. 5 p. 26, that

−

−2

(Bd

Σ; Z

) =

−

−2

(Bd

Σ; Z

) =

−

−2

(Bd

Σ; Z)⊗Z

⊕Tor

−

−3

(Bd

Σ; Z), Z

6= 0 contradicting

the Gorenstein

-ness of Bd

Σ.

Corollary. 2. Each simplicial hm

is locally orientable (by Prop. 1 p. 39).

Note. 2. Corollary 2 confirms G.E. Bredon’s conjecture stated in [2] Re-
mark p. 384 just after the definition of Borel-Moore cohomology manifolds
with boundary, the reading of which is aiming at Poincar´e duality and
therefore only allows BdX = ∅ or dim BdX = n − 1, cp. our Ex. 3 p. 56.
Weak homology manifolds over PIDs are defined in [2] p. 329 and again we
would like to draw the attention to their connection to Buchsbaum rings,
cf. our p. 38, and that, for polytopes, ”join” becomes ”tensor product”
under the St-R ring functor as mentioned in our Note iii p. 37.

Appendices

7.1

The 3×3-lemma (also called “The 9-lemma”)

In this appendix we’ll make use of the relative Mayer-Vietories sequence
(M-Vs) to form a 9-Lemma-grid which clarifies some relations between
Products and Joins. In particular that {X

∧

∗Y

, X

∧

∗Y

} is excisive iff {X

, X

×Y

} is.

Ex.

6.16.

(3×3 Lemma) (J.J. Rotman: An Intr. to Homological Algebra p. 175-6.)

Consider the commutative diagram of modules (on the right): If the columns are exact and
if the bottom two (or the top two) rows are exact, then the top row (or the bottom row) is
exact. (Hint: Either use the Snake lemma or proceed as follows: first show that αα

′

= 0, then

regard each row as a complex and the diagram as a short exact sequence of complexes, and
apply theorem 6.3.)
(Cf. Ex. 5., S. Mac Lane: Categories for the Working Mathematician, c

1988, p. 208.)

A 3×3 diagram is one of the form (to the right) (bordered by zeros).

(a) Give a direct proof of the 3×3 lemma: If a 3×3 diagram is commutative and all three
columns and the last (first) two rows are short exact sequences, then so is the first (last) row.
(b) Show that this lemma also follows from the ker-coker sequence.
(c) Prove the middle 3×3 lemma: If a 3×3 diagram is commutative, and all three columns
and the first and third rows are short exact sequences, then so is the middle row.
Ex.

16. (P.J. Hilton & S. Wylie: Homology Theory p. 227.) Let (the diagram on the

right) be a commutative diagram in which each row and each column is an exact sequence of
differential groups (Def. p. 99). Then there are defined (see 5.5.1 (p. 196)) homomorphism
ν

∗

: H(C

) → H(C

), ν

∗

: H(C

) → H(A

), ν

∗

: H(C

) → H(A

), ν

∗

: H(A

) → H(A

). Prove

that ν

∗

= −ν

∗

. (Reversed notation.)

Ex. 2. (A. Dold: Lectures on Algebraic Topology p. 53.)...if (the diagram on the right is) a
commutative diagram of chain maps with exact rows and columns then the homology sequence
of these rows and columns constitute a 2-dimensional lattice of group homomorphisms. It is
commutative except for the (∂

∗

-∂

∗

)-squares which anticommute.



0−→A

−→

′

−→

α A

−→ 0



0−→B

−→ B

−→B

−→ 0



0−→C

−→ C

−→C

−→ 0



Put ˆ

CX:=X

∧

∗ {(x

, 1)} where x

∈ X

and ˆ

CY := Y

∧

∗ {(y

, 0)} where y

∈ Y

, with (x

, 0) :=

, y

, 1) and (y

, 1) :=

, y

, 0).

Put











(

:= ˆ

×Y

:= ˆ

×Y

(

:= X

× ˆ

:= X

× ˆ

then











∩ B ∩ C ∩ D =

= (A ∩ D) ∪ (B ∩ C)

= X

×Y

(A ∩ B) ∪ (C ∩ D) =
= (A ∪ C) ∩ (B ∪ D)

= X

∧

∗

(A ∪ B) ∩ (C ∪ D) =
= (A ∩ C) ∪ (B ∩ D)

= X

×Y

∪X

×Y

(A ∪ B) ∪ (C ∪ D) =
= (A ∪ C) ∪ (B ∪ D)

= X

∧

∗

∪ X

∧

∗

(A ∪ D) ∩ (B ∪ C)

= X

∧

∗

∪ [X

×Y

∪X

×Y

]

(A∩C)∩(B∩D)=
=(A∩B)∪(C∩D)

(A ∪ D) ∩ (B ∪ C)

(A∪C)∩(B∪D)=
=(A∩B)∪(C∩D)

77oo

(A∩C)∪(B∩D)=
=(A∪B)∩(C∪D)

(A∪C)∪(B∪D)=
=(A∪B)∪(C∪D)

The last commutative square provide us with a “9-lemma”-grid as that

above, constituted by two horisontal and two vertical relative Mayer-Vietoris
sequences, using the naturality of the M-Vs exhibited in Munkres [33]
p. 186-7. The most typical use of the 9-lemma is to create relative se-
quences out of non-relative as shown in Dold [8] p. 53.

The entries consists of three levels where the upper (lower) concerns the

vertical (horisontal) M-Vs. X

, Y

are assumed to be non-empty.

↓δ

3∗

(−#)

↓δ

1∗

↓δ

2∗

...

−→

∧

∗

( ˆ

∪

×ˆ

)∩

∩

( ˆ

∪

×ˆ

)) =

∧

∗

∧

∗

∧

∗

( ˆ

∩ ˆ

)∪

∪

×ˆ

∩

×ˆ

))

∗

−

−−−

→

n(X1

( ˆ

∩

×ˆ

)∩

∩

( ˆ

∩

×ˆ

)) =

n(X1

( ˆ

∩ ˆ

)∩

∩

×ˆ

∩

×ˆ

))

−−→

, ˆ

∩ ˆ

)⊕

⊕

n(X1

×ˆ

∩

×ˆ

) =

n(Y1

)⊕

n(X1

, ˆ

∩ ˆ

)⊕

⊕

n(X1

×ˆ

∩

×ˆ

)

−

−−−

→



−j

)



−j

)



−j

)

...

−−→

(X1

∧

∗

Y1, ˆ

CX1×Y2 ∪X1×ˆ

CY2 )⊕

⊕

∧

∗

, ˆ

∪

×ˆ

∧

∗

∧

∗

)⊕

∧

∗

∧

∗

(X1

∧

∗

Y1, ˆ

CX1×Y2 ∪X1×ˆ

CY2 )⊕

⊕

∧

∗

, ˆ

∪

×ˆ

)

∗

−

→

n(X1

, ˆ

∩

×ˆ

)⊕

⊕

n(X1

, ˆ

∩

×ˆ

) =

n(X1

)⊕

n(X1

, ˆ

∩

×ˆ

)⊕

⊕

n(X1

, ˆ

∩

×ˆ

)

′
∗

−

→

, ˆ

)⊕

, ˆ

)⊕

⊕

n(X1

×ˆ

)⊕

n(X1

×ˆ

n(Y1

)⊕

n(Y1

)⊕

n(X1

)⊕

n(X1

, ˆ

)⊕

, ˆ

)⊕

⊕

n(X1

×ˆ

)⊕

n(X1

×ˆ

)

∗

(= 0)

−

−−−

→



−l

)



−l

)



−l

)

...

−−→

∆

℘

( ˆ

∧

∗

)

/[∆

℘

(ˆ

∪

× ˆ

∆

℘

(ˆ

∪

× ˆ

]

n(X1

∧

∗

∧

∗

∪

∧

∗

)

= H

∆

℘

∧

∗

)

/[∆

℘

( ˆ

∪ ˆ

∆

℘

× ˆ

∪

× ˆ

)

]

∗

−

−−−

→

∆

℘

( ˆ

)

/[∆

℘

(ˆ

∩

× ˆ

∆

℘

(ˆ

∩

× ˆ

)

]

n(X1

∪

)

= --

n(X1

( ˆ

∪ ˆ

)∩

∩

× ˆ

∪

× ˆ

))

−

−−−

→

∆

℘

(

)

/[∆

℘

(

∆

℘

(

)

]

⊕

∆

℘

(

× ˆ

)

/[∆

℘

(

× ˆ

∆

℘

(

× ˆ

)

]

(=)

n(Y1

)⊕

n(X1

)

CX1×Y1 , ˆ

CX1 ×Y2 ∪ ˆ

CX2 ×Y1 )⊕

⊕

n(X1

× ˆ

∪

× ˆ

)

−−→

↓δ

3∗

(−#)

↓δ

1∗

↓δ

2∗

The groups within curly braces in the last row doesn’t play any part in

the deduction of the excisivity equivalence but fits the underlying pattern.

7.2

Simplicial Calculus

The complex

of all subsets of a simplex

σ is denoted ¯

σ, while the boundary

of σ, ˙σ, is the set of all proper subsets. So, ˙σ := {τ | τ ⊂

σ} = ¯

{σ},

¯∅

= {∅

} and ˙∅

= ∅.

“The closed star of σ w.r.t. Σ” = st

σ := {τ ∈ Σ| σ ∪ τ ∈ Σ}.

“The open (realized) star of σ w.r.t. Σ” = st

σ := {α ∈ |Σ| | [v ∈ σ] =⇒

[α(v) 6= 0]}. So, α

6∈ st

σ except for st

∅

= |Σ|. So, st

(σ) = {α∈|Σ|

α∈ T

∈ σ

{v}}.

“The closed geometrical simplex |σ| w.r.t. Σ” = “The (realized) closure of σ
w.r.t. Σ” = |σ| := {α ∈ |Σ| | [α(v) 6= 0] =⇒ [v ∈ σ]}. So, |∅

| := {α

“The interior of σ w.r.t. Σ”= hσi = Int(σ) := {α ∈ |Σ| | [v ∈ σ] ⇐⇒ [α(v) 6=
0]}. So, Int

(σ) is an open subspace of |Σ| iff σ is a maximal simplex in Σ

and Int(∅

) := {α

The barycenter ˆ

σ of σ 6= ∅

is the α ∈ Int(σ) ∈ |Σ| fulfilling v ∈ σ ⇒ α(v) =

#σ

while ˆ

∅

:= α

“The link of σ w.r.t. Σ” = Lk

σ := {τ ∈ Σ|[σ ∩ τ = ∅] ∧ [σ ∪ τ ∈ Σ]}. So,

∅

= Σ, σ ∈ Lk

τ ⇐⇒ τ ∈ Lk

σ and Lk

σ = ∅ iff σ 6∈ Σ, while Lk

τ = {∅

} iff

τ ∈ Σ is maximal.

The join Σ

∗Σ

:= {σ

∪ σ

|σ

∈ Σ

(i = 1, 2)}.

In particular Σ ∗{∅

} =

{∅

} ∗Σ = Σ.

Proposition. 1. If V

∩ V

= ∅ then,

[τ ∈ Σ

∗

] ⇐⇒ [∃! σ

∈ Σ

so that τ = σ

∪ σ

], which follows

immediately from the definition.
and

Σ1∗Σ2

(σ

∪ σ

) = (Lk

Σ1

) ∗ (Lk

Σ2

), which is proved by bracket juggling.

Using the convention Lk

v := Lk

{v}, any link is an iterated link of ver-

tices and Lk

τ = {∅

} ∗ Lk

τ = Lk

σ ∗ Lk

τ =

Prop. 1

above

= Lk

∗

(σ

∪

τ) = Lk

(σ

∪

τ ) =

(

∪

)

(

∪

)

= Lk

(σ

∪

τ ). So, τ /

∈ Lk

σ ⇒ Lk

τ = ∅ while;

τ ∈ Lk

σ ⇒ Lk

τ = Lk

(σ∪τ ) = Lk

σ ⊂ Lk

σ∩Lk

(I)

τ ∈ Lk

σ =⇒

σ = ¯

σ ∗ Lk

σ = [Eq. I] = ¯

σ ∗ Lk

τ = Lk

∅

∗ Lk

τ =

Prop. 1

above

= Lk

∗

τ = Lk

τ.

The p-skeleton is defined through; ∆

(p)

:= {δ ∈ ∆ | #δ ≤ p + 1}, implying

that ∆

(n)

= ∆, if n := dim ∆, when ∆

′

:= ∆

(n−1)

. ∆

:= ∆

(p)

∆

(p−1)

Γ ⊂ Σ is full in Σ if for all σ ∈ Σ; σ ⊂ V

=⇒ σ ∈ Γ.

Proposition. 2.

i. Lk

∆1∪∆2

δ = Lk

∆1

δ ∪ Lk

∆2

δ,

ii. Lk

∆1∩∆2

δ = Lk

∆1

δ ∩ Lk

∆2

δ,

iii ∆

∗∆

′

= (∆

′

∗∆

) ∪ (∆

∗∆

′

iv ∆ ∗ ( ∩

∈

∆

) = ∩

∈

(∆ ∗ ∆

and

∆ ∗ ( ∪

∈

∆

) = ∪

∈

(∆ ∗ ∆

(iv also holds for arbitrary topological spaces under the

∧

∗-join.)

b. ∆ pure ⇐⇒ Lk

∆

′

= Lk

∆′

δ ∀ ∅

6= δ ∈ ∆.

c. [Γ ∩ Lk

∆

δ = Lk

δ ∀ δ ∈ Γ] ⇐⇒ [Γ is full in ∆] ⇐⇒ [Γ ∩ st

∆

δ = st

∀ δ ∈ Γ].

(II)

Proof. a. Associativity and distributivity of the logical connectives. ⊲

b. (=⇒) ∆ pure =⇒ ∆

′

pure, so; Lk

∆′

δ = {τ ∈ ∆||τ ∩δ = ∅ ∧ τ ∪δ ∈ ∆

′

} = {τ ∈

∆||τ ∩δ = ∅ ∧ τ ∪δ ∈ ∆ ∧ #(τ ∪δ) ≤ n } = {τ ∈ Lk

∆

δ || #(τ ∪δ) ≤ n} = {τ ∈

∆

δ || #τ ≤ n−#δ} = {τ ∈ Lk

∆

δ || #τ ≤ dim Lk

∆

δ} = Lk

∆

′

(⇐=) If ∆ non-pure, then ∃ δ

∈ ∆

′

maximal in both ∆

′

and ∆ i.e.,

∆

′

= {∅

}

′

= ∅ 6= {∅

} = Lk

∆′

⊲

c. Lk

σ = {τ | τ ∩ σ = ∅ & τ ∪ σ ∈ Γ} =

true ∀ σ∈Γ

iff Γ full

= {τ ∈ Γ| τ ∩ σ =

∅ & τ ∪ σ ∈ ∆} = Γ ∩ Lk

∆

σ.

Γ ∩ st

∆

δ = Γ ∩ {τ ∈ Γ| τ ∪ σ ∈ ∆} ∪ {τ ∈ ∆| τ /

∈ Γ & τ ∪ σ ∈ ∆} = {τ ∈

Γ| τ ∪ σ ∈ ∆} =

True for all σ∈ Γ iff Γ

is a full subcomplex.

= {τ ∈ Γ| τ ∪ σ ∈ Γ} = st

δ.

The contrastar of σ ∈ Σ = cost

σ := {τ ∈ Σ| τ 6⊇ σ}. cost

∅

= ∅ and

cost

σ = Σ iff σ 6∈ Σ.

Proposition. 3 Without assumptions whether δ

, δ

∈ ∆ or not, the follow-

ing holds;

a. cost

∆

(δ

∪ δ

) = cost

∆

∪ cost

∆

and δ = {v

, ..., v

} ⇒ cost

∆

δ =

∈ 1, p

cost

∆

b. cost

cost

∆

= cost

∆

∩ cost

∆

= cost

cost

∆

and

∈ 1, q

cost

∆

= cost

costδ2

cost

∆

(

i. δ /

∈ ∆ ⇔

[

cost

∆

δ = cost

∆

v = ∅

]

ii. δ ∈ ∆ ⇔

[

v /

∈ δ ⇔ δ ∈ cost

∆

v ⇔ Lk

cost

∆

δ = cost

∆

v ⊃ {∅} (6=∅)

]

δ ∈ ∆ ⇔

[

v ∈ δ ⇔ δ /

∈ cost

∆

v ⇔ Lk

cost

∆

δ = ∅ 6= {∅} ⊂ Lk

∆

δ = cost

∆

]

d. δ, τ ∈ ∆ =⇒

[

δ ∈ cost

∆

τ ⇐⇒ ∅ 6= Lk

cost

∆

δ = Lk

cost

∆

(

)

δ = cost

∆

(τ

δ)

]

(

d ⇒ Lk

cost

∆

δ = Lk

∆

δ iff δ

∪

τ /

∈ ∆ or equivalently, iff τ

δ /

∈ st

∆

δ.)

e. If δ 6= ∅ then;

1. [ cost

∆

′

= cost

∆′

δ] ⇐⇒ [n

= n

∆

]

2. [cost

∆

δ = cost

∆′

δ] ⇐⇒ [n

= n

∆

− 1]

where n

:= dim cost

∆

and

∆

:= dim∆ < ∞.

(We always have; τ ⊂ δ ⇒ n

∆

−1 ≤ n

≤ n

∆

∀ ∅

6= τ, δ.)

∆

δ ∩ cost

∆

δ = cost

∆

δ = ˙δ ∗ Lk

∆

δ.

(

So, cost

∆

δ is a quasi-/homology manifold if ∆ is.)

Proof. If Γ ⊂ ∆ then cost

γ = cost

∆

γ ∩ Γ ∀ γ and cost

∆

(δ

∪ δ

) = {τ ∈

∆|| ¬[δ

∪ δ

⊂τ ]} = {τ ∈ ∆| ¬[δ

⊂τ ] ∨¬[δ

⊂τ ]}= cost

∆

∪ cost

∆

, giving a and b.

c. A “brute force”-check gives c, which is the “τ={v}”-case of d.

d. τ ∈ Lk

∆

δ =⇒{v} ∈ Lk

∆

δ ∀ {v} ∈ τ ∈ ∆ ⇐⇒ δ ∈ Lk

∆

v (⊂ cost

∆

v) ∀ v ∈ τ ∈ ∆

gives d from a, c and Proposition 2a i above.

e. cost

∆

δ = [iff n

= n

∆

− 1 ] = (cost

∆

δ)

∩

∆

′

= cost

∆′

= (cost

∆

δ)

′

= [iff n

= n

∆

] = (cost

∆

δ)

∩

∆

′

f. st

∆

δ = ¯

δ∗Lk

∆

δ = ˙δ∗Lk

∆

δ ∪ {τ ∈ ∆ || δ ⊂ τ }.

With δ, τ ∈ Σ; δ ∪ τ /

∈ Σ

⇐⇒ δ /

∈ st

τ ⇐⇒ st

δ ∩ |st

τ | = {α

} ⇐⇒ st

τ ∩ st

δ =

{α

}

and

|st

σ|

σ = | ˙σ ∗ Lk

σ| = |st

σ ∩ cost

σ|, by [33] p. 372, 62.6 and

Proposition 3.f above.

σ = {τ ∈ Σ| σ ∪ τ ∈ Σ} = ¯

σ ∗ Lk

(III)

and identifying |cost

σ| with its homeomorphic image in |Σ| through; |cost

σ| ≃

|Σ|

σ, we get st

σ = |Σ|

|cost

σ|.

——∗ ∗ ∗—— —–∗ ∗ ∗—— ——∗ ∗ ∗——

8.1

The Importance of Simplicial Com-
plexes

In [12] p. 54 Eilenberg and Steenrod wrote:

-“Although triangulable spaces appear to form a

rather narrow class, a major portion of the spaces oc-
curring in applications within topology, geometry and
analysis are of this type. Furthermore, it is shown in
Chapter X that any compact space can be expressed as
a limit of triangulable spaces in a reasonable sense. In
this sense, triangulable spaces are dense in the family
of compact spaces.”

Moreover, not only are the major portion of the spaces

occurring in applications triangulable, but the triangulable
spaces plays an even more significant theoretical role than
that, when it comes to determining the homology and homo-
topy groups of arbitrary topological spaces, or as Sze-Tsen
Hu, in his Homotopy Theory ([22] 1959) p. 171, states, after
giving a description of the Milnor realization of the Singular
Complex S(X) of an arbitrary space X:

-“The significance of this is that, in computing the

homotopy groups of a space X, we may assume without
loss of generality that X is triangulable and hence lo-
cally contractible. In fact, we may replace X by S(X).”

Recall that the Milnor realization of S(X) of any space X is

weakly homotopy equivalent to X and triangulable, as is any
Milnor realization of any semi-simplicial set, cf. [15] p. 209
Cor. 4.6.12. Since triangulable spaces are so frequently en-
countered, their singular homology is often most easily deter-
mined using a triangulation and calculate its simplicial homol-
ogy groups. But the usefulness also goes the other way: E.G.
Sklyarenko: Homology and Cohomology Theories of General
Spaces

([39] p. 132):

The singular theory is irreplaceable in problems of

homotopic topology, in the study of spaces of contin-
uous maps and in the theory of fibrations. Its impor-
tance, however, is not limited to the fact that it is ap-
plicable beyond the realm of the category of polyhedra.
The singular theory is necessary for the in depth un-
derstanding of the homology and cohomology theory
of polyhedra themselves”. . . . In particular it is used in
the problems of homotopic classification of continuous
maps of polyhedra and in the description of homology
and cohomology of cell complexes. The fact that the
simplicial homology and cohomology share the same
properties is proved by showing that they are isomor-
phic to the singular one.

8.2

−

Homology Groups for Joins of Ar-

bitrary Simplicial Complexes

As said in p. 10, it’s easily seen what Whitehead’s e

S(X ∗ Y)

need to fulfill to make e

S(X ∗ Y) ≈ e

S(X) ⊗ e

S(Y) true, since

the right hand side is well known as soon as e

S(X) and e

S(Y) are

known. But, a priori, we don’t know what e

S(X∗Y) actually

looks like. Now, Theorem 3 p. 25 gives the following complete
answer:

∆

℘

((X, X

)

∧

∗ (Y, Y

))

∼ s(∆

℘

(X, X

) ⊗ ∆

℘

(Y, Y

)).

Below, we will express

((Γ

, ∆

) ∗ (Γ

, ∆

); G

⊗G

) in

terms of

(Γ

, ∆

; G

) for arbitrary simplicial complexes. By

Prop. 1 p. 61 every simplex γ ∈ Γ

∗ Γ

splits uniquely into

∪

∈

retain their original orientation and put every vertex in Γ

“after” the vertices in Γ

. Let [γ

] ⊎ [γ

] := [γ] stand for

the chosen generator representing γ in C

(Γ

∗ Γ

) and,

extend the “⊎” distributively and linearly, C

(Γ

∗ Γ

) =

+q=v1 +v2

(Γ

) ⊎ C

(Γ

)).

The boundary function operates on generators as;

∗

[γ] = δ

dim γ

+dim γ

∗

[γ] =

= (δ

Γ1

dim γ

[γ

]) ⊎ [γ

]

+ [γ

] ⊎ (−1)

dim γ

(δ

dim γ

[γ

])

Define a morphism through its effect on the generators of

(Γ

∗Γ

)=C

(Γ

∗Γ

) by f

([γ])=f

([γ

] ⊎ [γ

]) :=

[γ

] ⊗ (−1)

[γ

], into [C

(Γ

) ⊗ C

(Γ

)]

+q=v

(Γ

) ⊗

(Γ

)), where the boundary function is given in [40] p. 228

through its effect on generators by,

Co(Γ

)⊗Co(Γ

)

[τ ] = δ

dim γ

+dim γ

Co(Γ

)⊗Co (Γ

)

[γ

] ⊗ [γ

]

= (δ

dim γ

[γ

]) ⊗

[γ

]

+ [γ

] ⊗ (−1)

dimγ

(δ

dimγ

[γ

])

. f

∗

is obviously a chain

isomorphism of degree -1, and so,

∗ Γ

and s C

(Γ

)⊗C

(Γ

)

are isomorphic as chains,

where the “s” stands for “suspension” meaning that the sus-
pended chain equals the original except that the dimension i
in the original is dimension i + 1 in the suspended chain.

The argument motivating the formula for relative simplicial

homology is equivalent to the one we used to motivate relative
singular homology case in p. 25ff.

Lemma. On the category of ordered pairs of abitrary simpli-
cial pairs

(Γ

, ∆

) and (Γ

,∆

) there is a natural chain eqiva-

lence of

(Γ

∗Γ

)/C

(Γ

∗ ∆

) ∪ (∆

∗ Γ

)

with

(Γ

)/C

(∆

)

⊗

(Γ

)/C

(∆

)

[40] p. 231 Corollary 4, with R a PID, G and G

′

R-modules

and Tor

(G, G

′

) = 0, gives;

Proposition 1.

((Γ

, ∆

) ∗ (Γ

, ∆

); G ⊗ G

′

) e

(♦)

+j=q

[

(Γ

, ∆

; G) ⊗

(Γ

, ∆

; G

′

)]⊕

⊕

+j=q−1

Tor

(Γ

, ∆

; G),

(Γ

, ∆

; G

′

)

—∗ ∗ ∗—

From ♦ we could use some category theory to prove our:

(X ∗ Y ; Z) e

+q=i

[

(X; Z) ⊗

(Y ; Z)]

⊕

+q=i−1

Tor

(X; Z),

(Y ; Z)

(♥)

Switch to realizations of our simplicial complexes and to

topological join in ♦. From [8] p. 367, Exercise 2 we learn
that singular homology is the Kan-extension of simplicial ho-
mology, i.e. a direct limit (cf. [8] p. 362, 3.4), w.r.t. to poly-
topes over the topological space in question. Also direct sums
of modules can be regarded as direct sums. Any two direct
limits commute by [36] p. 49, Theorem 2.21. Moreover, both
these direct limits has directed index sets, by [8] p. 362, 3.4
and [36] p. 44-45 respectively, and so, they both commute with
Tor

by [36] p. 223 Theorem 8.11. So, the r.h.s. of ♥ is the

direct limit of the r.h.s. of ♦ w.r.t. to underlying polytopes.

For the l.h.s. we use [40] p. 175 Th. 6 to se that we only

have to prove ♥ for compact spaces X and Y .

The simplicial projections w.r.t. to the simplicial join of

two simplicial complexes are indeed simplicial, implying that
also the join of polytopes over X resp. Y are cofinal w.r.t. to
polytopes over X ∗ Y . We conclude that also

(X ∗ Y ; Z)

is the the direct limit of the groups

(Γ

∗ Γ

; Z) w.r.t.

simplicial polytopes Γ

and Γ

over X resp. Y , i.e. ♥ follows.

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Department of Mathematics
University of Stockholm
SE-104 05 Stockholm, Sweden

Document Outline

Introduction
Augmental Homology Theory
Augmental Homology Modules for Products and Joins
Relating General Topology to Combinatorics
- Realizations with respect to Simplicial Products and Joins
- Connectedness and Local Homology with respect to Simplicial Products and Joins
Relating Combinatorics to Commutative Algebra
Simplicial Manifolds
Appendices
- The 33-lemma (also called ``The 9-lemma")
- Simplicial Calculus
Addendum:
- The Importance of Simplicial Complexes
- to4.0pt--to Homology Groups for Joins of Arbitrary Simplicial Complexes

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