Download 4.2 Simplicial Homology Groups

4.2. SIMPLICIAL HOMOLOGY GROUPS
4.2
4.2.1
93
Simplicial Homology Groups
Simplicial Complexes
• Let p0 , p1 , . . . pk be k + 1 points in Rn , with k ≤ n. We identify points in Rn
with the vectors that point to them.
• Assume that they are independent, that is, do not lie in a (k − 1)-dimensional
hyperplane, or that the vectors vi j = p j − pi are linearly independent.
• The k-simplex σk = �p0 , p1 , . . . , pk � is the compact subset of Rn defined by


k
k
��


�
�




n �
σk = 
c
p
,
with
c
≥
0,
c
=
1
x
=
x
∈
R

�
i
i
i
i




i=0
i=0
• For any j, 0 ≤ j ≤ k, a subset of j + 1 points defines a j-simplex called the
j-face.
• A 0-simplex is a point, called a vertex.
• A 1-simplex is a line segment, called an edge.
• A 2-simplex is the interior of a triangle.
• The 3-simplex is a tetrahedron.
• A simplicial complex is a set K of finitely many simplexes such that:
– every face of every simplex of K belongs to K,
– the intersection of any two simplexes in K is either empty or is a common face.
• A subset |K| of Rn which is the union of all simplexes in a complex K is
called a polyhedron.
• A simplicial complex K and a homeomorphism
F : |K| → X
to a topological space X is called a triangulation of X.
• A topological space X is called triangulable if there is a triangulation of X.
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CHAPTER 4. HOMOLOGY THEORY
• An unoriented k-simplex �p0 , p1 , . . . , pk � can be oriented as follows. An
oriented k-simplex (p0 , p1 , . . . , pk ) changes sign under a permutation of
any two points. Let ϕ be a permutation of points {p0 , p1 , . . . , pk }. Then
(pϕ(0) , pϕ(1) , . . . , pϕ(k) ) = (sign ϕ)(p0 , p1 , . . . , pk ),
where sign ϕ is the parity of the permutation ϕ.
4.2.2
Simplicial Homology Groups
• Let K be an n-dimensional simplicial complex.
• Let N p is the number of p-simplexes in K.
• A p-chains is a formal sum
c=
Np
�
ci σ p,i
i=1
where σ p,i are p-simplexes in K and ci ∈ Z.
• Remark. We can define chains over any Abelian group, for example, R or
Z2 .
• This allows to define the Abelian group structure: addition, zero, opposite.
• The p-chain group C p (K) of K is a free Abelian group generated by the
oriented k-simplexes of K,
C p (K) �
Np
�
Z
i=1
• By definition C p (K) = 0 for p > n.
• The boundary operator is a homomorphism
∂ p : C p (K) → C p−1 (K)
defined as follows.
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4.2. SIMPLICIAL HOMOLOGY GROUPS
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• The boundary of an oriented p-simplex σ p = (p0 , p1 , . . . , p p ) is a (p − 1)chain defined by
∂pσp =
p
�
(−1)i (p0 , p1 , . . . , p̂i , . . . , pk )
i=0
where p̂i is omitted.
• The boundary of a p-chain is defined by linearity.
• The chain complex is a sequence of free Abelian groups and homomorphisms
i
∂n
∂n−1
∂1
∂0
0 → Cn (K) → Cn−1 (K) → · · · → C0 (K) → 0
where i :�→ Cn (K) is the inclusion map.
• A p-chain z such that
∂pz = 0
is called a p-cycle.
• The p-cycles form a free Abelian subgroup of C p (K) called the p-cycle
group
Z p (K) = Ker ∂ p
• A p-chain b such that
b = ∂ p+1 c
for some (p + 1)-chain c, is called a p-boundary.
• The p-boundaries form a free Abelian subgroup of C p (K) called the pboundary group
B p (K) = Im ∂ p+1
• Proposition. The boundary of a boundary vanishes, that is,
∂ p ∂ p+1 = 0
• Corollary. Every boundary is a cycle, that is,
B p (K) ⊂ Z p (K)
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CHAPTER 4. HOMOLOGY THEORY
• The p-homology group H p (K) is defined by
H p (K) = Z p (K)/B p (K)
It is not necessarily free Abelian.
• We say that two p-cycles are homologous if they differ by a boundary.
• Homology is an equivalence relation.
• The equivalence classes of the homology are called homology classes.
• The homology groups are the sets of homology classes.
• Theorem. Homology groups are topological invariants. In particular,
– The homology groups of different triangulations of the same topological space are isomorphic.
– The homology groups of any triangulations of homeomorphic topological spaces are isomorphic.
• Therefore, the homology groups of a triangulable topological space (which
is not necessary a polyhedron) are defined to be the homology groups of
some triangulation.
• Spheres.
H0 (S 1 ) = H1 (S 1 ) = Z.
H0 (S 2 ) = H2 (S 2 ) = Z,
H1 (S 2 ) = 0.
• Theorem. For any connected simplicial complex K
H0 (K) = Z .
• Möbius Strip.
H0 (K) = Z,
H1 (K) = Z,
H2 (K) = 0 .
• Real Projective Space RP2 .
H0 (RP2 ) = Z,
H1 (RP2 ) = Z2 ,
H2 (RP2 ) = 0 .
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4.2. SIMPLICIAL HOMOLOGY GROUPS
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• The homology group over Z is not necessarily free Abelian group but may
include the torsion.
• Torus T 2 .
H0 (T 2 ) = H2 (T 2 ) = Z,
H1 (T 2 ) = Z ⊕ Z.
• Surface Σg of genus g.
H0 (Σg ) = H2 (Σg ) = Z,
H1 (Σg ) =
2g
�
Z.
i=1
• Klein Bottle K 2 .
H0 (K 2 ) = Z,
H2 (K 2 ) = 0,
H1 (K 2 ) = Z ⊕ Z2 .
• Theorem. The homology groups of a disconnected simplicial complex are
equal to the direct sum of the homology groups of its connected components.
• Corollary. If a complex K has m connected components, then
H0 (K) =
m
�
Z.
i=1
• Corollary. For a complex K
H0 (K) = Z
if and only if K is connected.
• A general homology group over Z has the form
H p (K) =
m
�
i=1
⊕Zr1 ⊕ · · · ⊕ Zrk
• The number of generators of H p counts the number of (p + 1) dimensional
holes in the polyhedron |K|.
• The torsion subgroup Zr1 ⊕· · ·⊕Zrk measures the twisting in the polyhedron
|K|.
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CHAPTER 4. HOMOLOGY THEORY
• The homology groups over R or Z2 do not have torsion.
• The homology groups H p (K, R) are finite-dimensional vector spaces.
• The dimension of the vector spaces H p (K, R) are called Betti numbers
b p (K) = dim H p (K, R)
• The Betti numbers are equal to the ranks of the free Abelian parts of the
homology groups over Z.
• The Euler characteristic of a simplicial complex K with N p p-simplexes
is an integer defined by
n
n
�
�
p
(−1) dim C p (K, R) =
(−1) p N p .
χ(K) =
p=0
p=0
• Theorem. The Euler characteristic of a simplicial complex K is equal to
χ(K) =
n
�
p=0
(−1) p dim H p (K, R) =
n
�
(−1) p b p (K).
p=0
• The Euler characteristic is a topological invariant.
• The Euler characteristic of a topological space does not depend on the triangulation, so, it can be defined for any triangulation.
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