1. Algebraic Topology Simplicial Homology Wayne Lawton
Department of Mathematics
National University of Singapore
S , 1

2. Homology Groups A differential group is an abelian group and an
endomorphism
that satisfies
is the differential or boundary operator
subgroup of cycles
subgroup of boundaries
Remark Clearly
Definition
is the homology
group and its elements are homology classes. 2

3. Oriented Simplices Definition An orientation of a d-dimensional
simplex
is one of two equivalence
classes of orderings (permutations) of its vertices
where two permutations are equivalent if they
differ by an even permutations.
denotes the equivalence class for permutation
An oriented simplex is a simplex with an orientation.
Examples
Result Two permutations are equivalent iff one can be obtained from the other using an even number of transpositions. 3

4. Group of q-Chains Definition Let be a simplicial complex and let
denote the abelian group
generated by the oriented
simplices in
whenever
together with the relations
and
are oriented simplices having the same
simplices but inequivalent orientations.
Example
is a free abelian group whose rank equals
the number of
simplices in
and is
called the group of
chains in 4

5. Boundary Operator Definition The group of chains of a simplicial
complex is
Lemma The homomorphisms
given by
where we
define
are well
defined, satisfy
and define a boundary
operator by
Proof 1st claim: a direct tedious computation gives
whence the
claim follows since every permutation is a product
of transpositions (permutation switching 2 things). 5

6. Boundary Operator
Lets do this tedious work since it is necessary ! 6

7. Boundary Operator Proof 2nd claim: it suffices to show that We compute
The result now follows since 7

8. Boundary Operator Proof 3rd claim: follows since Remark
(the trivial group) and
is a graded group and
is a graded homomorphism of degree 8

9. Computation of Homology Groups
Definition
is the
th homology group of
Result
Example 1.
Example 2. 9

10. Computation of Homology Groups
Example 3.
is represented by the following 6 x 4 matrix
Linear Algebra 
where
and hence
The sphere has one 3-dim hole 10

11. Computation of Homology Groups
Example 4.
represented 6 x 4 matrix M on preceding page
and hence
The ball does NOT have a 3-dim hole 11

12. Smith Normal Form http://en.wikipedia.org/wiki/Smith_normal_form
Definition A commutative ring R is principle if
every ideal in R has the form Rg for some g in R
Examples Z = ring of (rational) integers,
F[z] = ring of polynomials over a field F
Theorem [1] Every m x n matrix A over a PID R equals
where
Proof and Algorithm: see the very informative article
which also discusses applications to homology and the
structure theorem for finitely generated modules over pid
1] Smith, H. M. S., On systems of indeterminate equations and congruences, Philos. Trans., 151(1861), 12

13. MATLAB Code MATLAB Code available at
>> help smith
Smith normal form of an integer matrix.
[U,S,V] = smith(A) returns integer matrices U, S, and V such that
A = U*S*V',
S is diagonal and nonnegative, S(i,i) divides S(i+1,i+1) for all i,
det U =+-1, and det V =+-1.
s = smith(A) just returns diag(S).
Uses function ehermite.
[U,S,V] = smith(A);
This function is in some ways analogous to SVD.
Remark The matrix V is the transpose of the matrix
appearing in the previous page! 13

14. Example >> A = [2 4 4;-6 6 12;10 -4 -16] A = 2 4 4 -6 6 12
>> U
U =
>> V'
ans =
>> U*S*V'
>> A = [2 4 4; ; ]
A =
>> [U,S,V] = smith(A);
>> S
S = 14

15. Finitely Generated Abelian Groups
Theorem Every finitely generated abelian group
where
Proof If
is generated by elements
then
there exists a unique homomorphism
such that
Since
is surjective
the 1st
for groups gives
where
Clearly
is any matrix whose columns generate
Let
be the Smith form of
where
The result follows by 15

16. Smith Form in Homology Consider and choose a fixed bases for then and
is represented by
and
is represented
with Smith form
where
so we let
be its
be the lower submatrix of
and
Smith form with
Then 16

17. Homology of the Torus all 27 edges oriented all 18 faces oriented
counterclockwise
Question Compute the matrices for
and use
them with MATLAB to compute the homology groups of
Suggestion : I attempted to compute the matrix for
on the next page but did not check it for errors, the
matrix should have rank = 18 - dim ker = 17 17

18. Homology of the Torus 18

19. Computation of Theorem Proof are in the same connected
component of the polyhedron
iff there exist a path
such that
The
simplicial approximation theorem implies that this occurs
iff there exist
such that
Then the chain
satisfies
and hence
We leave it to the reader to prove the converse. 19

20. Why Elements in are Called Cycles We consider the case
Definition A loop in K is a chain
where
such that
Clearly
and hence
so every loop is a cycle.
Theorem Every element in
is a lin. comb. of loops.
Proof Express
with
then construct a loop
and
such that
has less coefficients, then use induction.
We suggest that the reader work out a detailed proof. 20

21. Normal Subgroups Definition Let be a group. A subgroup is
normal if for every
we have
Lemma If
is a normal subgroup of
then the set
of left cosets of
is a group (called the quotient
) under the binary operation defined by of by
Moreover, if
are normal subgroups then every homomorphism
that satisfies
induces a
homomorphism
defined by
Proof Left to the reader. 21

22. Commutator Subgroups Definition Let
be a group. The commutator subgroup
(also called the first derived subgroup and
is the subgroup generated
denoted by or ) of
by the set
Lemma For every homomorphism
Proof Left to the reader.
Corollary
is a normal subgroup of
Proof For each
construct by
Clearly
is a homomorphism
and hence the lemma implies that for every so
is normal. 22

23. The Abelianization Functor
is abelian.
Lemma If
is a group then
Proof It suffices to show that
such that
Given
choose
and
Then compute
Question For groups
and
and a homomorphism
define by
Show that
is a functor from the category of groups
to the category of abelian groups.
Question Prove is the left adjoint of the inclusion functor. See 23

24. The Abelianization Functor
Theorem If
is a closed surface with genus
then
when
is orientable and
when
is nonorientable.
Proof
Theorem
Proof To be given later.
Corollary
when
is orientable and
when
is nonorientable.
Proof The first assertion is left to the reader. Clearly if
is nonorientable then
where
whence the 2nd assertion. 24

25. Homology from Homotopy
Theorem If
is a simplicial complex whose polyhdron
is connected and
then
Proof Let
be the edge group defined on page 132
in Armstrong’s Basic Topology (BT). Theorem 6.1 in BT
proves (using the simplicial approximation theorem) that
and hence it suffices to prove that
Define by
Clearly equivalent edge loops e give homologous z(e), so
is well defined.
is surjective since every cycle is
a linear combination of ‘loops’ of the form z(g).
Moreover 25

26. Homology from Homotopy
For each
choose an edge path
joining to
and construct the edge loop
L is a
max tree
Since
and hence
Moreover
implies that
Therefore for every oriented edge (a,b)
that occurs k-times in
the oriented edge (b,a)
also occurs k-times. Since by Theorem 6.12 in BT
we have 26

27. Chain Complexes and Chain Maps
Chain Complex: differential graded abelian group
Chain Maps: deg 0 homomorphism
that gives a commutative diagram:
Theorem Chain complexes & chain maps are a category. Homology is a functor from it to the cat. of graded abelian groups&deg-0 homomorphisms
Proof Left to the reader. 27

28. Example: Simplicial Homology
Theorem If
are simplicial complexes and
is a simplicial map then the maps
defined by:
( = 0 if
are not distinct)
give a chain map
Proof Left to the reader.
Corollary
is a functor from the category of
simplicial complexes & simplicial maps to the
category of chain complexes and chain maps.
Corollary
is a functor from SIMP to GAB. 28

29. Example: Singular Homology
let
Definition For
be a standard ordered n-simplex and define
simplicial maps by
(this maps
onto )
the
face of
For any topological space
let
be the
free abelian group generated by maps
let the boundary operator be
Theorem
is a functor from category TOP to
category CHAIN and
= sing. hom. functor. 29

30. Chain Homotopy between two chain maps is a degree 1 homomorphism
such that
Lemma
Theorem
Proof 30

31. Chain Homotopy of chain Lemma Composites homotopic maps
are chain homotopic. This means that
a chain homotopy
such that
Proof Let
Then
is a degree 1 homomorphism and 31

32. Chain Homotopy Definition A chain equivalence between two chain
complexes is a chain map
such
that there exists a chain map
with
chain homotopic to
respectively.
Lemma Chain equivalence
same homology.
Definition A chain contraction of a chain complex
is a chain homotopy between
and
Lemma A contractible chain complex is acyclic
(all its homology groups equal 0).
Theorem A free acyclic chain comp.is contractible.
Proof
free
satisfies 32

33. Chain Homotopy Definition An augmentation of a chain complex with
is a chain complex
such that
Lemma If
is a simplicial complex then
has
a unique augmentation given by
Theorem If
is a cone (Lecture 2, slide 8)
then
is acyclic.
Proof Define a hom. by
and
Homework due 20 Oct: compute 33

34. Stellar Subdivision Lemma Barycentric subdivision can be obtained
by a sequence of stellar subdivisions of the form
where
is the barycenter of
Step 1
Lemma Stellar subdivision
defines a chain map
(the subdivision operator) by
Proof (Step 1) 34

35. Stellar Subdivision Theorem If is a stellar subdivision of
is a simplicial approximation of
that induces the chain map
defined by
then
Corollary
Proof (Step 1)
Theorem
Proof Subcomplex
of simplices containing
is a cone and
Finish it! 35

36. Contiguity Classes are Definition Simplicial maps contiguous if
(delete repeated vertices) and in the same
contiguity class if there exist simplicial maps
with
and
contiguous.
Lemma If
are simplicial approximations
to a map
then
and
are contiguous.
(review simplicial approx. on slide 17, lecture 3)
Proof 36

37. Contiguity Classes that are Lemma Simplicial maps
in the same contiguity class are homotopic.
Proof Without loss of generality we may assume
that
and
are contiguous.
Proof Define a homotopy by
Remark Let
be a simplicial complex and realize
its polyhedron
as a subset of Euclidean space
with norm
Let
be the Lebesque number
for the open cover of of
Hence
and 37

38. Contiguity Classes Lemma Let be maps such that where is the Lebesgue
number for the open cover of
then
is an open
cover of
and there exists
and a simplicial
approximation
to both
Proof The first assertion is obvious. Choose
such that
hence
Then construct
so that 38

39. Contiguity Classes Corollary If are homotopic then there exist
and simplicial approximations to to
and
such that
and
are in the same contiguity class.
Proof Homework Due Tuesday 27 October. 39

40. Contiguity Classes are simplicial maps Theorem If
that are in the same contiguity class then their
induced chain maps
are chain homotopic.
Proof We may assume that
and
are contiguous.
For
define
and
observe that
is a subset of a simplex (called
the carrier of
) in
and that
Assume we have homomorphisms
for
so that
and
is a chain in the the carrier of
For an
oriented n-simplex
let
Then
so let
Corollary 40

41. Invariance Lemma If and are simplicial complexes,
is a map with simplicial approx.
with
and
are subdivision
chain maps (by composing stellar subdiv. maps),
is a simplicial approx. of
and
chain maps induced by the simp. maps,
then
and
Proof. Since both
they are
are simplicial approx. to
contiguous so
Also so 41

42. Invariance Theorem 1. Every map induces a homomorphism where
by the rule:
is a simplicial approxroximation to
and
is the subdivision chain map.
Proof Preceding lemma  the rule is well defined. 42

43. Invariance satisfies Theorem 2. The identity and maps satisfy
be a simplicial approx.
Proof Let
then let be to
a simplicial approx. to be
Let
be subdivision chain maps and be
a simplicial approx. to
(a standard simp. map)
and
its induced chain map.
Since
are simp. approx. to
resp. 43

44. Invariance Theorem 3. If are homotopic then
Proof Homework due Friday 30 October 44

45. Applications Theorem Proof Corollary Proof
Theorem (Brouwer-Fixed Point) Every map
has a fixed point.
Proof Otherwise there exists a retraction
so that
Since
is contractible
However
and this ccontradiction concludes the proof. 45