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

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Note that L(1,q) is the sphere S

, but this case has to be treated in a slightly

special way if one wants a nice cell decomposition. One has to add another vertex

at -e

Lens spaces were ﬁrst deﬁned by Tietze ([Tiet08]). We list some of their proper-

ties below:

(1) They are closed compact three-dimensional manifolds.

(2) They can be triangulated.

(3) L(2,1) is homeomorphic to P

(4) The homology groups of L(p,q) are

(5) c(L(p,q)) = 0.

(6) L(p,q) is homeomorphic to L(p,q¢) if and only if

(7) L(p,q) and L(p,q¢) have the same homotopy type if and only if qq¢ or -qq¢ is a

quadratic residue modulo p.

Properties (1)–(3) are easy to prove. Figure 7.14(b) should make clear what one

needs to do for (2). For properties (4) and (5) use the cell decomposition of L(p,q)

induced by the cell decomposition of D

described above to compute its homology

groups. Property (6) was proved by Reidemeister ([Reid35]). Property (7) was proved

by Whitehead ([Whit41]). More details about lens spaces can also be found in [SeiT80]

and [HilW60].

Although it is obvious that homotopy equivalence is a weaker relation than home-

omorphism (consider a disk and a point or, more generally, any deformation retract),

it is not so obvious with respect to some types of spaces like manifolds without bound-

ary. This makes the next example and lens spaces all the more interesting.

7.2.4.7. Example. A consequence of properties (6) and (7) is that the 3-manifolds

L(7,1) and L(7,2) have the same homotopy type but are not homeomorphic.

7.2.5 Incidence Matrices

This section discusses the so-called incidence matrices. These matrices played an

important role in the history of combinatorial topology. An excellent detailed account

of these matrices can be found in [Cair68]. Computers can easily use them to compute

homology groups.

qq p¢∫±

()

±1

mod .

HLpq

()()

7.2 Homology Theory 399

Let K be a simplicial complex of dimension n and, as before, let n

= n

(K) be the

number of q-simplices in K. For each q, assume that we have chosen an orientation

for all the q-simplices in K and let

be the set of these oriented q-simplices of K. The set S

will be a basis for the free

abelian group C

(K). Deﬁne integers e

by the equation

and note that

Deﬁnition. The integer e

is called the incidence number of [s

] and [s

j+1

]. The qth

incidence matrix E

, 0 £ q < dim K, of K is deﬁned to be the (n

¥ n

q+1

)-matrix

whose rows and columns are indexed by the elements of S

and S

q+1

, respectively.

7.2.5.1. Example. Suppose that K =·v

Ò and that we have chosen the S

follows:

The incidence matrices of K are then given by

E and E

012

01 12 02

10 1

110

011

[]

[][][]

vvv

vv vv vv

012

01 12 02

012

{}

[][][]

{}

[]

{}

vvv

vv vv vv

vvv

,, ,

()

[][]

[]

++ +

ee e

ss s

MMOM

q+1

qq qq+1

if is a face of

otherwise

∂e

[]

[][]

[]

{}

ss s

, ,...,

400 7 Algebraic Topology

Next, we deﬁne the incidence matrices with respect to arbitrary bases

for the free abelian groups C

(K). Since we have bases, there are unique integers h

such that

Deﬁnition. The qth incidence matrix of K with respect to the bases {c

q+1

} and {c

} is

deﬁned to be the (n

¥ n

q+1

)-matrix

Let c be an arbitrary (q + 1)-chain. If we express c with respect to the basis {c

q+1

then

for some unique integers a

and

This shows that the boundary homomorphisms ∂

of K, and hence the homology

groups of K, are completely determined once the incidence matrices are known with

respect to some bases. Our goal in the remainder of this section is to show how the

homology groups of K can be computed from knowledge of the basic incidence matri-

ces E

of K alone.

7.2.5.2. Lemma. Choose a basis for each group C

(K). The matrix product of any

two successive incidence matrices with respect to any such choice of bases is the zero

matrix. Using the notation above, this means that for all q

Proof. This is a straightforward consequence of Lemma 7.2.1.3(2).

()()

=¥

()

-matrix of zeros.

∂∂ h

qjq

cac ac

qqq

===

()

ÂÂÂ

111

cac

()

∂h

()

ccc c

{}

, ,...,

7.2 Homology Theory 401

Lemma 7.2.5.2 and some purely algebraic manipulations of matrices lead to

7.2.5.3. Theorem. It is possible to choose bases for all the groups C

(K) simul-

taneously with respect to which the qth incidence matrix has the normalized

form

(7.3)

where

(1) the d

’s are positive integers,

(2) d

i+1

divides d

, and

(3) n

-g

≥g

q-1

Outline of Proof. Choose a basis

for each group C

(K) and assume that N

is the qth incidence matrix with respect

to these bases. We shall transform the matrices N

into the form (7.3) by appropri-

ate changes to the bases and so we need to know how changes to bases affect the

matrices. First of all, note that changing the basis {c

} clearly affects both N

and

q-1

Claim 1. The matrices N

have the following properties:

(a) Replacing

corresponds to changing all signs in the ith column of N

q-1

and ith row of N

(b) Replacing

ccccbycccc

,..., ,..., ,..., ,..., ,..., ,...,

{}{}

cccbyc cc

,..., ,..., ,..., ,...,

{}

ccc c

{}

, ,...,

rows

n rows

qq q

00 0

000

000 0

columns columns

674846744 844

402 7 Algebraic Topology

corresponds to interchanging the ith and jth columns in N

q-1

and the ith and jth rows

in N

for some integers k and i π j corresponds to replacing the ith column of N

q-1

the ith column plus k times the jth column and replacing the jth row of N

by the jth

row minus k times the ith row.

The proof of parts (a) and (b) of Claim 1 is easy and left to the reader. Part (c) is

readily deduced from the identities

and

Claim 1 is proved.

Now consider the 0th incidence matrix E

= (e

) of K. Note how each column has

only two nonzero entries, namely, one +1 and one -1.

Claim 2. Any integer matrix P = (p

) having the property that each nonzero

column has precisely two nonzero entries, one of which is +1 and the other is -1, can

be transformed into a normalized matrix as deﬁned by Theorem 7.2.5.3 via a sequence

of matrix operations of the type described in Claim 1 above.

Claim 2 is proved by induction on the number of rows (or columns) in P.

First, by interchanging rows, we may assume that p

=+1. The next step is to

zero out the ﬁrst row past the ﬁrst entry by a sequence of column operations.

Speciﬁcally, if p

=±1 for some j > 1, then replace the jth column of P by (jth column

- p

(1st column)). The new matrix P¢=(p¢

) will have p¢

= 0 and either the

jth column is zero or there are again just two nonzero entries, one +1 and the other

-1. By a sequence of such operations we arrive at a matrix P≤=(p≤

) such that

p≤

= 1, p≤

= 0, for j > 1, and P≤ also satisﬁes the same hypotheses as the original

matrix P. Now p≤

will equal -1 for some i > 1. Subtracting the 1st row from the

ith row of P≤, will give us a matrix Q = (q

), such that q

= q

= 0 for i, j > 1. The

inductive hypothesis would then apply to the matrix (q

)

2£i,2£j

and ﬁnish the proof

of Claim 2.

In order to prove our theorem we shall prove the following assertions for

k > 0:

∂hhhhh

ttij

ccckckc c

q q

==π

()

==+

()

ÂÂ

11,,

∂∂∂

ckc c kc+

()

cccbycckcc

,..., ,..., ,..., ,...,

{}

7.2 Homology Theory 403

Assertion A

: It is possible to choose bases for the groups C

(K) so that with

respect to these bases the incidence matrices are

(All other incidence matrices are uninteresting since they are zero.)

Note the following important property of the matrices N

in assertion A

that

follows easily from the fact that, by Lemma 7.2.5.2, the product N

k-1

is the zero

matrix.

Claim 3. The last g

k-1

rows of the matrix N

in assertion A

are zero.

We shall use induction on k to prove the assertions A

above. Assertion A

follows

from Claims 1 and 2. These claims show that E

can be transformed into the nor-

malized form N

by changing the basis of C

(K) and C

(K) appropriately. Although

the 1st incidence matrix may have changed, the ith incidence matrices for i ≥ 2 have

not. One can check that the proof actually shows that d

= 1, for 1 £ i £g

Assume inductively that assertion A

is true for some k > 0. One can show that N

can be transformed into a normalized form such as is required for Theorem 7.2.5.3

by a sequence of matrix operations of the type described in Claim 1. This fact is a

special case of a normalization theorem for matrices that is not hard but too long to

reproduce here. A proof can, for example, be found in [Cair68].

Using Claim 3 we may assume that only the ﬁrst n

-g

k-1

rows of N

will be affected

as we transform the matrix to its normalized form. Translating the changes we make

in the matrix into the corresponding changes in the basis for C

(K) and their effect

on N

k-1

, we can easily see that only the ﬁrst n

-g

k-1

columns of N

k-1

are manipu-

lated. Since these consist entirely of zeros, the matrix N

k-1

is left unchanged. Of

course, the matrix E

k+1

will certainly have changed, but we have established assertion

k+1

. By induction, assertion A

is true for all k > 0. Assertion A

proves Theorem

7.2.5.3.

A detailed version of the qth incidence matrix (7.3) of Theorem 7.2.5.3: Recall

that the rows and columns of the incidence matrices are indexed by the chains in the

chosen basis of the appropriate chain groups of K. Therefore, assume that 0 £ q £ n

and that the basis elements of C

(K) corresponding to the rows of N

have been labeled

as follows:

The ﬁrst g

basis elements are labeled as A

’s (note that g

-1

= 0),

the last g

q-1

are labeled as C

’s,

the remaining b=n

-g

q-1

basis elements, if there are any, are labeled as B

’s,

and

if the d

are the elements shown in the normalized matrix (7.3), then the integer

is deﬁned by

With this notation, matrix (7.3) can be rewritten as

i=>

{}

()

max , .01

NNNE E

kkk n

-+-0111

,..., , , ,..., .

404 7 Algebraic Topology

The detailed version of the incidence matrices makes computing the homology

groups of K easy because it tells us all we need to know about the groups C

(K) and

homomorphisms ∂

. In particular, it is easy to see that

is a basis for H

(K). Also,

o([A

]) =d

, with the d

’s being the torsion coefﬁcients of H

(K),

o([B

]) =•, and

rk H

(K) =b

Finally, incidence matrices can be deﬁned for CW complexes using their cells. All

the information about homology groups that one could deduce in the simplicial case

remain valid. This greatly simpliﬁes computations because the dimensions of these

matrices will be much smaller.

7.2.6 The Mod 2 Homology Groups

A more precise name for the homology groups of a simplicial complex K as deﬁned

in Section 7.2.1 is to call them the homology groups “with coefﬁcients in Z.” The

reason is that chain group C

(K) consisted of formal integer linear combinations of

oriented q-simplices. It is easy to generalize this.

Let G be an arbitrary abelian group and let S

again denote the set of oriented q-

simplices of K. If 0 £ q £ dim K, deﬁne the group C

(K;G) of q-chains with coefﬁcients

in G by

AABB

qqqq

[]

{}

,..., , ,...,

AABBCCC C

q qqqqqq q

q q

qqqqq

q q

++++++

◊◊◊ ◊◊◊ ◊◊◊ ◊◊◊

00 0

000

000 0

gbrrg

000 0

7.2 Homology Theory 405

Clearly, C

(K) is just C

(K;Z). The elements of C

(K;G) can be thought of as formal

sums

(7.4)

where g

Œ G, motivating the terminology. Elements are added by collecting the coef-

ﬁcients of the a and then adding them using the group addition. Pretty much every-

thing we did earlier carries over without any problem if we simply replace Z by G.

There is a boundary homomorphism

and we can deﬁne the subgroup Z

(K;G) of q-cycles and the subgroup B

(K;G) of q-

boundaries with coefﬁcients in G. Again, B

(K;G) Ã Z

(K;G). Finally,

Deﬁnition. The group

is called the qth homology group of K with coefﬁcients in G.

Simplicial maps f : K Æ L induce homomorphisms

and one can again show that the homology groups H

(K;G) are topological invariants

and so we can associate a unique (up to isomorphism) group H

(X;G) to every poly-

hedron X. They are important new invariants associated to a space even though it

turns out that each is completely determined by H

(X;Z) and H

q-1

(X;Z) by the so-

called “universal coefﬁcient theorem.” A very important special case is the case where

G = Z

and it is worthwhile to look at that case more carefully.

Let K be simplicial complex. Since +1 is the same as -1 in Z

, there is no need to

orient the simplices of K to deﬁne C

(K;Z

). It follows that if T

is the set of all q-sim-

plices in K, then C

(K;Z

) can be identiﬁed with the set of all maps

that is, an element of C

(K;Z

) (usually called a mod 2 q-chain) can be thought of as

consisting of a linear sum of q-simplices of K. To add mod 2 q-chains we simply collect

the coefﬁcients of like simplices but must remember that s + s = 0 for every s Œ T

because 2 = 0 in Z

. For example, if K is the simplicial complex determined by the

simplex v

, then

:,Æ Z

fHKGHLG

qq q*

()

:; ;,

HKG

ZKG

BKG

;

()

∂

qq q

CKG C KG:; ;

()

-1

C K G f S G if f

;: .

()

=Æ ≥ -

()

{}

q 1, then f for all in S

aa a

406 7 Algebraic Topology

and

(7.5)

Recall our intuitive discussion of “holes” in Section 7.2.1. It is what we are doing now

that makes precise the ﬁrst approach to homology groups in that section.

The boundary map

satisﬁes

that is, ∂

sends the boundary of a q-simplex to the sum of all of its (q - 1)-dimen-

sional faces. Proving ∂

q-1

∂

= 0 is easy in this case because each (q - 2)-dimensional

face a q-simplex belongs to precisely two (q - 1)-dimensional faces of the simplex and

2 = 0 in Z

. The mod 2 homology groups of K, H

(K;Z

), are now deﬁned to be the

usual quotient group of the kernel of ∂

(called the mod 2 q-cycles) by the image of

∂

q+1

(called the mod 2 q-boundaries).

We can relate the mod 2 q-chains of K to subsets of ΩKΩ.

Deﬁnition. Let c Œ C

(K;G). The support of c, denoted by ΩcΩ, is the union of all the

q-simplices appearing in c with a nonzero coefﬁcient.

Observe that if we represent q-chains c in C

(K;Z

) by the subset ΩcΩ of ΩKΩ, which

is their support, then

(1) The subset Ωc + dΩ is the closure of the symmetric difference ΩcΩDΩdΩ of the

subsets ΩcΩ and ΩdΩ.

(2) The subset Ω∂

(c)Ω is the union of all the (q - 1)-simplices that are the face of

an odd number of q-simplices appearing in c.

Now, since Z

is a ﬁeld, the groups C

(K;Z

), Z

(K;Z

), B

(K;Z

), and H

(K;Z

) are

actually vector spaces over Z

Deﬁnition. The qth connectivity number of K, k

(K), is deﬁned to be the dimension

of the vector space H

(K;Z

) over Z

. If X is a polyhedron, then the qth connectivity

number of X, k

(X), is deﬁned to be the qth connectivity number of any simplicial

complex K that triangulates X.

Connectivity numbers are the mod 2 analogs of Betti numbers. They are well

deﬁned in the case of a polyhedron because the groups H

(X;Z

) are well deﬁned up

to isomorphism.

∂

vv v v v v

01 0

◊◊◊

()

= ◊◊◊ ◊◊◊

∂

qq q

CK C K:; ;ZZ

212

()

vv vv vv vv vv vv vv vv vv vv

01 12 20 12 23 31 01 31 23 20

()

+++

()

=+++.

vv vv vv vv vv vv Z

01 12 20 12 23 31 1 2

++ ++Œ

()

,;CK

7.2 Homology Theory 407

408 7 Algebraic Topology

7.2.6.1. Theorem. Let K be a simplicial complex. Then

Proof. The proof of this theorem is the same as that of Theorem 7.2.3.10. The only

difference is that one uses the mod 2 groups now and works with the dimensions of

these vector spaces rather than the ranks of the corresponding groups in the usual

homology theory with integer coefﬁcients. One also needs to use the fact that

7.2.6.2. Corollary. If X is a polyhedron, then

Connectivity numbers have a simple geometric interpretation in the special case

of surfaces.

7.2.6.3. Lemma. Let S be a closed and compact combinatorial surface. Then

(1) k

(S) = 1.

(2) k

(S) = 1.

(3) k

(S) = 2 -c(S).

Proof. Exercise 7.2.6.1(b) proves (1). Next, it is easy to see that H

(S;Z

) ª Z

because

the sum of all the 2-simplices of S is a mod 2 2-cycle that generates H

(S;Z

). (We

should point out that this fact is actually a special case of Theorem 7.5.1(1) in Section

7.5.) This proves (2). Parts (1), (2), and Corollary 7.2.6.2 now imply (3).

Lemma 7.2.6.3 shows that the ﬁrst connectivity number k

(S) (which is the only

one that is interesting for surfaces) does not depend on the orientability of the surface

S. This was certainly not the case with the ﬁrst Betti number b

(S) and more evidence

that mod 2 homology theory does not detect orientability properties of spaces. We

shall also see this later with respect to the top dimensional homology groups of

pseudomanifolds. The next theorem is the main result that we are after right now.

7.2.6.4. Theorem. The ﬁrst connectivity number k

(S) of a combinatorial surface S

equals the maximum number of distinct, but not necessarily disjoint, simple closed

curves in S along which one can cut and still have a connected set left over.

Outline of Proof. First of all, we need to explain the expression “simple closed

curves” in our current context. A collection of subsets X

of S will be called a collec-

tion of simple closed curves in S if each X

is homeomorphic to the circle S

and there

ckXX

()

() ()

q=0

Xdim

dim ; . CK nK

()

ckKK

()

() ()

q=0

Kdim