MA5209 Algebraic Topology

1
MA5209 Algebraic Topology
Lecture 5. Simplicial Homology (18, 29 September,
2, 13, 16, 20, 23 October 2009)
Wayne Lawton Department of Mathematics National
University of Singapore S14-04-04,
matwml_at_nus.edu.sg http//math.nus.edu.sg/matwml
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.
3
Oriented Simplices
Definition An orientation of a d-dimensional
is one of two equivalence
simplex
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.
4
Group of q-Chains
Definition Let
be a simplicial complex and let
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
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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).
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Boundary Operator
Lets do this tedious work since it is necessary !
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Boundary Operator
Proof 2nd claim it suffices to show that
We compute
The result now follows since
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Boundary Operator
Proof 3rd claim follows since
Remark
(the trivial group) and
is a graded group and
is a graded homomorphism of degree
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Computation of Homology Groups
Definition
is the
th homology group of
Result
Example 1.
Example 2.
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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
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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
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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, Fz
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
http//en.wikipedia.org/wiki/Smith_normal_form
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), 293--326.
http//en.wikipedia.org/wiki/Henry_John_Stephen_Sm
ith
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MATLAB Code
MATLAB Code available at
http//mathforum.org/kb/thread.jspa?forumID80thr
eadID257763messageID835342835342
gtgt help smith Smith normal form of an integer
matrix. U,S,V smith(A) returns integer
matrices U, S, and V such that A USV', S
is diagonal and nonnegative, S(i,i) divides
S(i1,i1) 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!
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Example
gtgt U U 1 0 0 -12 -3 -2
17 4 3 gtgt V' ans 1 2
2 -1 -3 -2 0 0 -1 gtgt
USV' ans 2 4 4 -6 6
12 10 -4 -16
gtgt A 2 4 4-6 6 1210 -4 -16 A 2
4 4 -6 6 12 10 -4 -16 gtgt
U,S,V smith(A) gtgt S S 2 0
0 0 6 0 0 0 12
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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
http//en.wikipedia.org/wiki/Isomorphism_theorem
where
Clearly
is any matrix whose columns generate
Let
be the Smith form of
where
The result follows by
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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
and
be the lower submatrix of
Smith form with
Then
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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
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Homology of the Torus
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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.
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Why Elements in
are Called Cycles
We consider the case
Definition A loop in K is a chain
where
such that
Clearly
so every loop is a cycle.
and hence
Theorem Every element in
is a lin. comb. of loops.
Proof Express
with
then construct a loop
such that
and
has less coefficients, then use induction.
We suggest that the reader work out a detailed
proof.
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Normal Subgroups
Definition Let
is
be a group. A subgroup
normal if for every
we have
is a normal subgroup of
then the set
Lemma If
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.
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Commutator Subgroups
Definition Let
be a group. The commutator subgroup
(also called the first derived subgroup and
is the subgroup generated
or
denoted by
)
of
by the set
Lemma For every homomorphism
Proof Left to the reader.
is a normal subgroup of
Corollary
construct
Proof For each
by
Clearly
is a homomorphism
and hence the lemma implies that for every
so
is normal.
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The Abelianization Functor
is abelian.
Lemma If
is a group then
Proof It suffices to show that
such that
Given
choose
Then compute
and
Question For groups
and a homomorphism
and
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 http//en.wikipedia.org/wi
ki/Commutator_subgroup
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The Abelianization Functor
Theorem If
is a closed surface with genus
then
is orientable and
when
when
is nonorientable.
Proof http//wapedia.mobi/en/Fundamental_polygon
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.
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Homology from Homotopy
is a simplicial complex whose polyhdron
Theorem If
is connected and
then
Proof Let
be the edge group defined on page 132
in Armstrongs 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 surjective since every cycle is
is well defined.
a linear combination of loops of the form z(g).
Moreover
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Homology from Homotopy
choose an edge path
joining
For each
and construct the edge loop
to
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
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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 groupsdeg-0 homomorphisms
Proof Left to the reader.
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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.
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Example Singular Homology
let
Definition For
be a standard ordered n-simplex and define
simplicial maps
by
(this maps
onto
)
face of
the
For any topological space
be the
let
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.
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Chain Homotopy
between two chain maps
is a degree 1 homomorphism
such that
Lemma
Theorem
Proof
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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
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Chain Homotopy
Definition A chain equivalence between two chain
such
complexes is a chain map
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
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Chain Homotopy
Definition An augmentation of a chain complex
is a chain complex
with
such that
is a simplicial complex then
Lemma If
has
a unique augmentation given by
is a cone (Lecture 2, slide 8)
Theorem If
then
is acyclic.
Proof Define a hom.
by
and
Homework due 20 Oct compute

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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)

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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!

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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
then
to a map
and
are contiguous.
(review simplicial approx. on slide 17, lecture 3)
Proof

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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
as a subset of Euclidean space
its polyhedron
be the Lebesque number
with norm
Let
of
for the open cover of
Hence
and

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Contiguity Classes
Lemma Let
be maps such that
is the Lebesgue
where
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

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

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Contiguity Classes
are simplicial maps
Theorem If
that are in the same contiguity class then their
induced chain maps
are chain homotopic.
and
are contiguous.
Proof We may assume that
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

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Invariance
Lemma If
are simplicial complexes,
and
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

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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.

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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.

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

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