Homology

1
Homology

• Selected Topics of Pattern Recognition, TU Wien
• Marco Ross

2
Chain Groups

• Chain is a formal sum of q-Simplices
• The formal sum of two chains a and b is defined
  as
• The q-Simplices and all possible linear
  combinations in a Simplicial Complex form the
  Chain Group Cq

3
Chain Groups example
e
1 Triangle and 2 Lines
t
b
c
d
a

• C1Ø,a,b,c,d,e,ab,ac,ad,ae,bc,bd,be,cd,ce,abc,abd
  ,abe,acd,ace,ade,bcd,bce,bde,cde,abcd,acde,abde,ab
  ce,bcde,abcde

4
Cycles

• The Group of q-Cycles is denoted by

e
Z1 Ø,abc,cde,abde
t
b
c
d
Kernel1 of the boundary operator ?1
a
1 (http//en.wikipedia.org/wiki/Simplicial_homol
ogy)
5
Boundaries

• The Group of q-Boundaries is denoted by

e
t
b
B1 Ø, cde
c
d
Image1 of the boundary operator ?2
a
1 (http//en.wikipedia.org/wiki/Simplicial_homol
ogy)
6
Homology Groups
The cosets define homology classes

• The q-th homology
• group of a Complex is
• defined as the quotient
• group

2 (Rocío González-Díaz Pedro Real On the
Cohomology of 3D Digital Images)
7
Quotient Group, Cosets
Z1 Ø,abc,cde,abde
B1 Ø, cde
Z1/B1 Ø,cde,abc,abde,Ø,cde,abde,abc
Ø
abc
cde
abde
2 Homology Classes Ø cde, abc abde
8
Homology Classes

• 2 Classes
• Ø cde Ø,cde
• abc abde abc,abde
• Two different kinds of cycles
• Chains of one class surroundthe hole
• the chains of the otherClass dont

e
t
b
c
d
a
9
Chain Complex

• A Chain complex is a sequence of abelian
  (commutative)
• groups Cq and homomorphisms ?q, such that for all
  q ?q1 ?q0

2 (Rocío González-Díaz Pedro Real On the
Cohomology of 3D Digital Images)
10
Chain Contraction

• A chain contraction from CCq, ?q to CCq,
  ?q is a set of three homomorphisms
• f C ? C and g C ? C are chain maps
• fg is the identity map of C
• fC ? C is a chain homotopy of the identity map
  idC of C to gf, that is f ? ? f idC gf
• Further Conditions
• C has fewer or the same number of generators as
  C
• C and C have isomorphic homology groups

2 (Rocío González-Díaz Pedro Real On the
Cohomology of 3D Digital Images)
11
Homotopy

• Formal
• Formally, a homotopy between two continuous
  functions f and g from a topological space X to a
  topological space Y is defined to be a continuous
  function H X 0,1 ? Y from the product of the
  space X with the unit interval 0,1 to Y such
  that, for all points x in X, H(x,0)f(x) and
  H(x,1)g(x).
• More intuitive
• If we think of the second parameter of H as
  "time", then H describes a "continuous
  deformation" of f into g at time 0 we have the
  function f, at time 1 we have the function g.

3 (http//en.wikipedia.org/wiki/Homotopy)
12
Homotopy
3 (http//en.wikipedia.org/wiki/Homotopy)
13
Group Generators

• a generating set of a group G is a subset S such
  that every element of G can be expressed as the
  product of finitely many elements of S and their
  inverses
• We denote the (sub)group generated by S as ltSgt
• If G ltSgt, we say S generates G and the elements
  in S are called generators

4 (http//en.wikipedia.org/wiki/Group_generator)
14
Group generators, example
e
t
Sa,b,c,d,e ltSgt C1
b
c
d
a

• C1Ø,a,b,c,d,e,ab,ac,ad,ae,bc,bd,be,cd,ce,abc,abd
  ,abe,acd,ace,ade,bcd,bce,bde,cde,abcd,acde,abde,ab
  ce,bcde,abcde

15
Chain Contraction

• A chain contraction from CCq, ?q to CCq,
  ?q is a set of three homomorphisms
• f C ? C and g C ? C are chain maps
• fg is the identity map of C
• fC ? C is a chain homotopy of the identity map
  idC of C to gf, that is f ? ? f idC gf
• Further Conditions
• C has fewer or the same number of generators as
  C
• C and C have isomorphic homology groups

2 (Rocío González-Díaz Pedro Real On the
Cohomology of 3D Digital Images)
16
Example Continued
r

• f t?Ø, a?a, b?b, c?c, d?c, e?c, p?p, q?q, r?r,
  s?r, f(ac1ßc2)? af(c1) ßf(c2) (a,ß?Z/Z2,
  dim(c1)dim (c2))
• g idC
• If ? 1 f(idC, ?) f
• else f(idC, ?) s? ?r(1- ?)s, sonst idC

e
s
??1
t
b
c
d
p
a
q
17
f ,g homomorphisms?
5
5 (http//en.wikipedia.org/wiki/Group_homomorphi
sm)
18
Homotopy, Generators?

• f is a homotopy? f(C,0) idC, f(C,1) fgf
• C contains fewer generators than C

19
Preserving Structure 1

• Homology Groups isomorphic?
• Z(C)0 ltp,q,r,sgt, B(C)0 ltpq,pr,qr,qs,rsgt
  
• H(C)0 p,q,r,s,qrs,prs,pqs,pqr,
  Ø,pq,pr,ps,qr,qs,rs,pqrs
• H(C)1 Ø,cde, abc,abde
• H(C)2
• Z(C)0 ltp,q,r,gt, B(C)0 ltpq,pr,qrgt
• H(C)0 p,q,r,pqr, Ø,pq,pr,qr
• H(C)1 Ø,c, abc,ab
• H(C)2

r
e
s
t
b
c
d
p
a
q
20
Preserving Structure 2

• f(c) f(c) for any cycle c in C

Z(C)1 Ø,abc,cde,abde
B(C)1 Ø, cde
Z(f(C))1 Ø,abc,c,ab
B(f(C))1 Ø, c
Z1/B1 H(C)1 Ø,cde,abc,abde,Ø,cde,abde,ab
c
Apply f to homolgy classes
Z1(f(C))/B1(f(C))H(f(C)) Ø,c,abc,ab,Ø,c,
abc,ab
21
Summary

• Same number of homology classes after chain
  contraction
• Mapped cycles determine mapped homolgy classes
• One-to-one match for homology classes
• Homolgy groups isomorphic 2

2 (Rocío González-Díaz Pedro Real On the
Cohomology of 3D Digital Images)
22
Conclusion

• Simplified representation of Complex
• Preserving structure (holes, connected
  components)
• Thank you for your attention
• Questions?