4.2 Simplicial Homology Groups

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1 4.2. SIMPLICIAL HOMOLOGY GROUPS Simplicial Homology Groups Simplicial Complexes Let p 0, p 1,... p k be k + 1 points in R n, with k n. We identify points in R n 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 v i j = p j p i are linearly independent. The k-simplex σ k = p 0, p 1,..., p k is the compact subset of R n defined by k k σ k = x Rn x = c i p i, with c i 0, c i = 1 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 R n 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. i=0 topicsdiffgeom.tex; October 23, 2014; 10:32; p. 92

2 94 CHAPTER 4. HOMOLOGY THEORY An unoriented k-simplex p 0, p 1,..., p k can be oriented as follows. An oriented k-simplex (p 0, p 1,..., p k ) changes sign under a permutation of any two points. Let ϕ be a permutation of points {p 0, p 1,..., p k }. Then (p ϕ(0), p ϕ(1),..., p ϕ(k) ) = (sign ϕ)(p 0, p 1,..., p k ), where sign ϕ is the parity of the permutation ϕ 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 N p c = c i σ p,i where σ p,i are p-simplexes in K and c i Z. Remark. We can define chains over any Abelian group, for example, R or Z 2. 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, By definition C p (K) = 0 for p > n. N p C p (K) Z The boundary operator is a homomorphism defined as follows. p : C p (K) C p 1 (K) topicsdiffgeom.tex; October 23, 2014; 10:32; p. 93

3 4.2. SIMPLICIAL HOMOLOGY GROUPS 95 The boundary of an oriented p-simplex σ p = (p 0, p 1,..., p p ) is a (p 1)- chain defined by p σ p = p ( 1) i (p 0, p 1,..., ˆp i,..., p k ) i=0 where ˆp i is omitted. The boundary of a p-chain is defined by linearity. The chain complex is a sequence of free Abelian groups and homomorphisms 0 i C n (K) n C n 1 (K) n 1 1 C 0 (K) 0 0 where i : C n (K) is the inclusion map. A p-chain z such that is called a p-cycle. p z = 0 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 p- boundary 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) topicsdiffgeom.tex; October 23, 2014; 10:32; p. 94

4 96 CHAPTER 4. HOMOLOGY THEORY The p-homology group H p (K) is defined by It is not necessarily free Abelian. H p (K) = Z p (K)/B p (K) 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. H 0 (S 1 ) = H 1 (S 1 ) = Z. H 0 (S 2 ) = H 2 (S 2 ) = Z, H 1 (S 2 ) = 0. Theorem. For any connected simplicial complex K H 0 (K) = Z. Möbius Strip. H 0 (K) = Z, H 1 (K) = Z, H 2 (K) = 0. Real Projective Space RP 2. H 0 (RP 2 ) = Z, H 1 (RP 2 ) = Z 2, H 2 (RP 2 ) = 0. topicsdiffgeom.tex; October 23, 2014; 10:32; p. 95

5 4.2. SIMPLICIAL HOMOLOGY GROUPS 97 The homology group over Z is not necessarily free Abelian group but may include the torsion. Torus T 2. H 0 (T 2 ) = H 2 (T 2 ) = Z, H 1 (T 2 ) = Z Z. Surface Σ g of genus g. H 0 (Σ g ) = H 2 (Σ g ) = Z, H 1 (Σ g ) = 2g Z. Klein Bottle K 2. H 0 (K 2 ) = Z, H 2 (K 2 ) = 0, H 1 (K 2 ) = Z Z 2. 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 Corollary. For a complex K if and only if K is connected. H 0 (K) = m Z. H 0 (K) = Z A general homology group over Z has the form H p (K) = m Z r1 Z rk The number of generators of H p counts the number of (p + 1) dimensional holes in the polyhedron K. The torsion subgroup Z r1 Z rk measures the twisting in the polyhedron K. topicsdiffgeom.tex; October 23, 2014; 10:32; p. 96

6 98 CHAPTER 4. HOMOLOGY THEORY The homology groups over R or Z 2 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 χ(k) = ( 1) p dim C p (K, R) = ( 1) p N p. Theorem. The Euler characteristic of a simplicial complex K is equal to χ(k) = ( 1) p dim H p (K, R) = ( 1) p b p (K). 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. topicsdiffgeom.tex; October 23, 2014; 10:32; p. 97