Agoston M.K. Computer Graphics and Geometric Modelling: Mathematics

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7.2.2.3. Lemma. j

is a well-deﬁned homomorphism.

Proof. First of all, by Lemma 7.2.2.2(1), the deﬁnition makes sense, since j

(z) Œ

(L). To show that j

is well deﬁned, let a Œ H

(K) and assume that a = [z] = [z¢], z,

z¢ŒZ

(K). Then z - z¢ belongs to B

(K). Therefore,

by Lemma 7.2.2.2(2), that is, [j

(z)] = [j

(z¢)]. This proves that j

is well deﬁned.

Next, let [z

] = z

+ B

(K) be elements of H

(K). Then

Thus, j

is a homomorphism and Lemma 7.2.2.3 is proved.

Deﬁnition. The maps j

are called the homomorphisms on homology induced by the

chain map j. In particular, if f : K Æ L is a simplicial map, we shall let

denote the map on the homology group induced by the chain map f

Consider the simplicial complex K =∂ ·v

Ò. The next two examples compute

for two simplicial maps f:K Æ K.

7.2.2.4. Example. To compute f

when f is the constant map deﬁned by f(v

) = v

Solution. The given f induces the constant map ΩfΩ:ΩKΩÆΩKΩ, ΩfΩ(x) = v

. We know

from Example 7.2.1.5 that

and

so that we only have to worry about what happens in dimensions 0 and 1. The map

(K) Æ C

(K) is obviously the zero map by deﬁnition, and so f

(K) Æ H

(K)

H K for q

()

=>01,

HK HK

()

ª Z

fHKHL

qq q*

()

[]

()

()()

()

()()

()

()()

()

()()

[]

()

qqq

qq q

qq q q

zz zzBK

zz L

zzBL

zBL zBL

12 12

[[]

()

jj j

qq q q

zz zzBL

()

¢¢

[]

()

[]

()

qq q

zzzZK,.

7.2 Homology Theory 379

is also the zero map. Next, note that the group H

(K) is generated by the element

+ B

(K) and

This implies that f

is the identity map.

7.2.2.5. Example. To compute f

when f is deﬁned by f(v

) = v

, f(v

) = v

, and f(v

)

= v

Solution. It follows from an argument similar to the one in the previous example

that f

= 0 if q > 1 and f

is the identity map. Only f

is different this time. Recall

from Example 7.2.1.5 that

is a generator of H

(K). Since

it follows that f

is the negative of the identity map.

The next lemma lists some basic properties of the maps f

and f

that are easy

to prove.

7.2.2.6. Lemma. Let f:K Æ L and g:L Æ M be simplicial maps between simplicial

complexes. Then

(1) (g

= g

(K) Æ C

(M).

(2) (g

= g

(K) Æ H

(M).

(3) If K = L and f = 1

, then f

and f

are also the identity homomorphisms.

Proof. This is Exercise 7.2.2.1.

Now simplicial complexes and maps are basically only tools for studying topo-

logical spaces and continuous maps. We shall show next how continuous maps induce

homomorphisms on homology groups.

Deﬁnition. Let K and L be simplicial complexes and suppose that f:ΩKΩÆΩLΩ is a

continuous map. A simplicial approximation to f is a simplicial map j:K Æ L with the

following property: If x ŒΩKΩ and if f(x) Œ s for some simplex s Œ L, then ΩjΩ(x) Œ s.

The next lemma summarizes two important properties of simplicial

approximations.

fa f BK

ff ff ff BK

()

[]

()

()()

[]

()()

[]

()()

[]

()

[]

()

1 1 01 12 20 1

01 12 20 1

02 21 10 1

vv vv vv

aBK=

[]

()

vv vv vv

01 12 20 1

fBKf BK BK

()()

()

00 0 00 0 0 0

vvv

380 7 Algebraic Topology

7.2.2.7. Lemma. Let f : ΩKΩÆΩLΩ be a continuous map and suppose that j:K Æ L

is a simplicial approximation to f.

(1) The map ΩjΩ:ΩKΩÆΩLΩ is homotopic to f.

(2) If f = ΩyΩ, where y:K Æ L is a simplicial map, then y=j.

Proof. To prove (1), deﬁne a homotopy h:ΩKΩ¥[0,1] ÆΩLΩ between f and ΩjΩ

for x ŒΩKΩ and t Œ [0,1]. That h(x,t) actually lies in ΩLΩ follows from the fact that

ΩjΩ(x) and f(x) lie in a simplex of L, which means that the segment [ΩjΩ(x),f(x)] is

contained in ΩLΩ because simplices are convex.

To prove (2), let v be a vertex of K. Then w = f(v) is a vertex of L. Since a vertex

is also a 0-simplex, the deﬁnition of a simplicial approximation implies that j(v) = w.

This proves the lemma.

Part (2) of Lemma 7.2.2.7 means that the only simplicial approximation to a sim-

plicial map is the map itself. An arbitrary continuous map does not have a unique

simplicial approximation, however.

If K is a simplicial complex, deﬁne a new simplicial complex, denoted by sd(K),

as follows:

(1) The vertices of sd(K) are the barycenters b(s) of the simplices s in K.

(2) The q-simplices of sd(K), q > 0, are all the q-simplices of the form b(s

)b(s

)

···b(s

), where the s

are distinct simplices of K and s

Ɱ s

Ɱ ...Ɱ s

It is easy to show that sd(K) is a simplicial complex (Exercise 7.2.2.3) that is a sub-

division of K. Clearly, Ωsd(K)Ω=ΩKΩ.

Deﬁnition. The simplicial complex sd(K) is called the (ﬁrst) barycentric subdivision

of K. The nth barycentric subdivision of K, denoted by sd

(K), is deﬁned inductively

Figure 7.7 shows a simplex and its barycentric subdivision.

7.2.2.8. Theorem. (The Simplicial Approximation Theorem) Let K and L be sim-

plicial complexes and suppose that f :ΩKΩÆΩLΩ is a continuous map. Then there is

an integer N ≥ 0 such that for each n ≥ N, f admits a simplicial approximation j:

(K) Æ L.

Proof. See [AgoM76].

sd K K

sd K sd sd K for n

()

≥

htt tfxx x,

()

()()

j 1

7.2 Homology Theory 381

Associated to barycentric subdivisions are natural homomorphisms

that correspond to sending an oriented simplex [s] to the sum of the oriented sim-

plices into which the barycentric subdivision divides s. For example,

See Figure 7.7. More precisely, deﬁne the maps sd

inductively on the oriented

simplices as follows:

(1) If v is a vertex of K, then sd

(v) = v.

(2) Assume 0 < q < dim K and sd

#q-1

has been deﬁned. If [s] is an oriented q-

simplex of K, then

(We are using the expression w[v

...v

] to denote the oriented simplex

[wv

...v

] and let this operation distribute over sums.)

If q < 0 or dim K < q, then we deﬁne sd

to be the zero map.

7.2.2.9. Lemma. The maps sd

are well-deﬁned homomorphisms. Furthermore,

∂

= sd

#q-1

∂

, so that sd

= (..., sd

#-1

,sd

, . . .) is a chain map that induces

homomorphisms

Proof. This is an easy exercise. See [AgoM76].

We can extend our deﬁnitions and deﬁne homomorphisms

sd C K C sd K

()

sd H K H sd K

qq q

()

()()

sd b sd

qqq##

.ss s

[]

()

[]

()()

-1

∂

sd b b b b b b

bb b b

#2 012 012 0 01 012 01 1 012 1 12

012 12 2 012 2 02

vvv vvv v vv vvv vv v vvv v vv

vvv vv v vvv v vv

[]

()

()()

[]

()()

[]

()()

[]

()()

[]

()()

[]]

()()

[]

bbvvv vv v

012 0 2 0

sd C K C sd K

qq q#

()

()()

382 7 Algebraic Topology

b(v

)

b(v

)

b(v

)

b(v

)

sd (K)

Figure 7.7. A barycentric subdivision.

inductively by

The maps sd

induce homomorphisms (actually isomorphisms)

We are now ready to show how continuous maps induce homomorphisms on

homology groups. Let K and L be simplicial complexes and let

be a continuous map. The Simplicial Approximation Theorem implies that there is an

n ≥ 0, such that f admits a simplicial approximation

Deﬁnition. The homomorphism

deﬁned by

is called the homomorphism induced on the qth homology group by the continuous

map f.

7.2.2.10. Lemma.

(1) f

is a well-deﬁned homomorphism.

(2) If K = L and f = 1

, then f

is the identity homomorphism.

(3) If M is a simplicial complex and g : ΩLΩÆΩMΩ is a continuous map, then

= g

Proof. See [AgoM76].

7.2.2.11. Theorem. Let K and L be simplicial complexes and suppose that f,

g:ΩKΩÆΩLΩ are continuous maps that are homotopic. Then f

= g

(K) Æ H

(L)

for all q.

Proof. See [AgoM76].

fsd

qqq

***

=j o

fHKHL

qq q*

()

j :.sd K L

()

fK L: Æ

sd H K H sd K

()

sd zero map for n

sd identity map of C K

sd sd sd for n

()

7.2 Homology Theory 383

7.2.3 Applications of Homology Theory

Before describing some applications, it is worthwhile to brieﬂy pause and summarize

what we have accomplished so far; otherwise, it is easy to lose sight of the global

picture and get lost in a sequence of lemmas and theorems. The main results can be

summarized by the following:

Fact 1. For every simplicial complex K and every integer q there is an abelian group

(K) called the qth homology group of K.

Fact 2. For every continuous map f :ΩKΩÆΩLΩbetween the underlying spaces of two

simplicial complexes K and L there are a homomorphisms

whose natural properties are best summarized by the commutative diagram

fHKHL

qq q*

()

384 7 Algebraic Topology

|K|

|L|

|M|

(K) H

(K)

(L)

(M)

|K|

)

(f ¢)

(K)

f , f ¢

f  f ¢

The top line in the diagram deals with simplicial complexes and maps and

the bottom lines deal with groups and homomorphisms.

For our purposes, Facts 1 and 2 contain essentially everything that we need to know

about homology groups and induced maps. Many of our applications will follow in a

purely formal way from Facts 1 and 2 with the geometry being irrelevant. Actual deﬁni-

tions are only needed for a few speciﬁc computations. There is one caveat though. We

would really like to have well-deﬁned homology groups and induced maps associated to

polyhedra and their continuous maps. Singular homology theory (see Section 7.6)

accomplishes that, but we shall at times pretend that we have this here also. To avoid

such pretense and restore rigor we could pick a ﬁxed triangulation for each polyhedron

and translate continuous maps between them to maps between the underlying spaces of

the simplicial complexes. This would validate our arguments but the messy details

would obscure geometric ideas. By the way, homology theory can be described axiomat-

ically by means of the so-called Eilenberg-Steenrod axioms. Facts 1 and 2 correspond to

a subset of these axioms. Statements made in Chapter 6 about algebraic topology asso-

ciating “algebraic invariants” to spaces should make a lot more sense now.

7.2.3.1. Theorem. Two simplicial complexes K and L with homotopy equivalent

underlying spaces have isomorphic homology groups.

Proof. Let f :ΩKΩÆΩLΩand g : ΩLΩÆΩKΩ be continuous maps such that g

f  1

ΩKΩ

and f

g  1

ΩLΩ

. Then Fact 2 implies that

for all q. It follows that f

is an isomorphism and the theorem is proved.

7.2.3.2. Corollary. Homotopy equivalent polyhedra have isomorphic homology

groups.

7.2.3.3. Corollary. If a polyhedron X has the homotopy type of a point, then

In particular, these are the homology groups of D

There is one consequence of Theorem 7.2.3.1 that would be disappointing to

anyone who might have hoped to use homology groups to classify topological spaces.

They are not strong enough invariants to distinguish spaces up to homeomorphism.

For example, Corollary 7.2.3.3 shows that both a single point and the disk D

have

the same homology groups but are clearly not homeomorphic. The best we could hope

for now is that they distinguish spaces up to homotopy type. Unfortunately, they fail

to do even that except in special cases. (There exist polyhedra, such as the spaces in

Example 7.2.4.7, that have isomorphic homology groups but that are not homotopy

equivalent.) Nevertheless, homology groups are strong enough to enable one to prove

many negative results, that is, if one can show that two spaces have nonisomorphic

homology groups, then it follows that the are not homeomorphic. In fact, they would

not even have the same homotopy type.

Before we state several invariance results that can be proved using homology

groups, we need to compute the homology groups for the higher-dimensional spheres.

7.2.3.4. Theorem. If n ≥ 1, then

Proof. The case n = 1 is left as an easy exercise for the reader. Assume that n ≥ 2.

Let s = v

···v

be any n-simplex. Let M =·sÒ and N =∂M be the simplicial

complexes associated to the simplex and its boundary. Since S

n-1

is homeomorphic

to ∂s =ΩNΩ, it sufﬁces to compute H

(N). The deﬁnition of the simplicial homology

groups implies that

Therefore,

B N B M for k n or k n

Z N Z M for all k

()

££- ≥

()

if n and k

if n and either k or k n

if k or n

n-1

()

≈= =

≥==-

π-

201

001

()

q=0

0, q > 0

g f l and f g l

q q HK q q HL

(

)

(

)

==oo

7.2 Homology Theory 385

and Corollary 7.2.3.3 proves Theorem 7.2.3.4 for these values of k.

To compute H

n-1

(N), note that

and so

We shall show that S is in fact a generator of Z

n-1

(N). If

then

Let s < t. The coefﬁcient of the oriented (n - 2)-simplex [v

...vˆ

...v

] is

Since this coefﬁcient has to vanish, it is easy to check that z = a

S. It follows that

n-1

(N) = ZS. But B

n-1

(N) = 0 and Z

n-1

(N) has no elements of ﬁnite order, so that

and the theorem is proved.

7.2.3.5. Theorem.

(1) The spheres S

and S

have the same homotopy type only when n = m. In

particular, S

is homeomorphic to S

if and only if n = m.

(2) The Euclidean space R

is homeomorphic to R

only when n = m.

Proof. Part (1) follows from Theorem 7.2.3.4 and Corollary 7.2.3.2. To prove part

(2), we use the stereographic projection

:.Se R-Æ

HNZN

nn--

()

aa.

1101

()

= ◊◊◊ ◊◊◊

[]

()

◊◊◊ ◊◊◊ ◊◊◊

[]

()

◊◊◊ ◊◊◊ ◊◊◊

[]

=+=

ÂÂ

∂∂

nin in

jin

ijn

vv v v

vvvvv vvvvv

ˆˆ ˆˆ

ËË

za ZN

iinn

= ◊◊◊ ◊◊◊

[]

()

vv v v

01 1

S= ◊◊◊

[]

()

∂

nnn

ZNvv v

01 1

∂∂

nn n-

()

◊◊◊

[]

()

101

0o vv v ,

H N H M for k n

()

π-,,1

386 7 Algebraic Topology

Suppose that h:R

Æ R

is a homeomorphism. Deﬁne

The map H will be a homeomorphism and therefore n = m by part (1). The theorem

is proved.

The next three theorems are less trivial.

7.2.3.6. Theorem. (Invariance of Dimension) If K and L are simplicial complexes

with ΩKΩªΩLΩ, then dim K = dim L.

Proof. See [AgoM76].

Deﬁnition. The dimension of a polyhedron is deﬁned to be the dimension of any

simplicial complex that triangulates it.

Theorem 7.2.3.6 shows the dimension of a polyhedron is a well-deﬁned topologi-

cal invariant.

7.2.3.7. Theorem. (Invariance of Boundary) If K and L are simplicial complexes

and h:ΩKΩÆΩLΩ is a homeomorphism, then h(Ω∂KΩ) =Ω∂LΩ.

Proof. See [AgoM76].

Theorem 7.2.3.7 makes it possible to deﬁne the boundary of a polyhedron.

Deﬁnition. Let X be a polyhedron. Deﬁne the boundary of X, denoted by ∂X, by

where (K,j) is any triangulation of X.

7.2.3.8. Theorem (Invariance of Domain) If U and V are homeomorphic subsets of

and if U is open in R

, then so is V.

Proof. See [AgoM76].

Returning to our deﬁnition of topological manifolds in Section 5.3, we are ﬁnally

able to prove the claimed invariance of two aspects of the deﬁnition.

7.2.3.9. Corollary. The dimension of a topological manifold and its boundary are

well deﬁned.

∂j∂X =

()

H p h p if and

mn n

xxxe

()

oo ,,

: SSÆ

7.2 Homology Theory 387

Proof. The corollary is an easy consequence of Theorems 7.2.3.5 and 7.2.3.8.

Next, we return to the Euler characteristic as deﬁned in Chapter 6. We are now

in a position to put this invariant in a more general context. What we had in Chapter

6 was a combinatorial concept deﬁned for surfaces that was easy to compute by some

simple counting and yet was claimed to be a topological invariant. We can now deﬁne

that topological invariant in a rigorous manner.

Deﬁnition. If K is a simplicial complex, let n

(K) denote the number of q-simplices

in K and deﬁne the Euler-Poincaré characteristic of K, c(K), by

What makes c(K) a topological invariant is the fact that it is related to the Betti

numbers b

(K) of K.

7.2.3.10. Theorem. (The Euler-Poincaré Formula) Let K be a simplicial complex.

Then

Proof. By deﬁnition, the boundary map ∂

(K) Æ B

q-1

(K) is onto and has kernel

(K) and the natural projection Z

(K) Æ H

(K) is onto and has kernel B

(K). There-

fore, Theorem B.5.8 implies that

These identities and the fact that rank (C

(K)) = n

(K) gives us that

c KnK

rank C K

rank B K rank H K rank B K

rank H K rank B K

qqq

()

() ()

() ()()

()()

[]

() ()()

() (

dim

))()

()()

[]

() ()()

rank B K

rank H K

dim

rank C K rank B K rank Z K and

rank Z K rank H K rank B K

qq q

qqq

()()

-1

cbKK

()

() ()

dim

c KnK

()

() ()

dim

388 7 Algebraic Topology