7.3 Singular Homology Groups

Transcription

1 184 CHAPTER 7. HOMOLOGY THEORY 7.3 Singular Homology Groups Cycles, Boundaries and Homology Groups We can define the singular p-chains with coefficients in a field K. Furthermore, we can define the multiplication of p-chains by elements of the field K, called the scalars by, for any a, b i K, r a b i σ i p = i=1 r ab i σ i p. The chain groups C p (M, K) with coefficients in a field K become infinitedimensional vector spaces. In this case the boundary homomorphism becomes a linear transformation (operator) in a vector space. Let M be a manifold and G an Abelian group. A singular p-chain z p in M whose boundary is 0 is called a singular p-cycle. The set of all p-cycles in M i=1 Z p (M; G) = {z p C p (M; G) z p = 0} is a subgroup of the chain group C p (M; G) called the p-cycle group. Obviously the p-cycle group is the kernel of the boundary homomorphism Z p (M; G) = Ker p, where p : C p (M; G) C p 1 (M; G). In the case, when G = K is a field, then Z p (M; K) is a vector subspace of the vector space C p (M; K). A singular p-chain b p in M that is the boundary of a singular (p + 1)-chain is called a p-boundary. diffgeom.tex; August 26, 2014; 11:32; p. 182

2 7.3. SINGULAR HOMOLOGY GROUPS 185 The set of all p-boundaries in M B p (M; G) = {b p C p (M; G) b p = c p+1 for some c p+1 C p+1 (M; G)} is a subgroup of the chain group C p (M; G) called the p-boundary group. Obviously the p-boundary group is the image of the boundary homomorphism B p (M; G) = Im p+1. Since every p-boundary is a p-cycle (because of 2 = 0) the group B p (M; G) is a subgroup of Z p (M; G). In the case, when G = K is a field, then B p (M; K) is a vector subspace of the vector space Z p (M; K). Let M be a manifold and G be an Abelain group. We say that two p-cycles are homologous if they differ by a boundary. The set of equivalence classes of p-cycles homologous to each other, that is, the quotient group is called the p-th homology group. H p (M; G) = Z p (M; G)/B p (M; G), In the case when the coefficient group G is a field G = K, all the groups, Z p, B p and H p are vector spaces Simplicial Homology The important fact is that if M is a compact manifold, then the vector space H p (M; K) is finite dimensional. (This can be proved but we will not do that). Let M be a compact n-dimensional manifold. Then there is a triangulation of M by finitely many n-simplexes diffeomorphic to the standard n-simplex Δ n. Thus, M is a union of finitely many n-simplexes, which are either disjoint of intersect along common r-simplexes with r = 0, 1,..., n 1. diffgeom.tex; August 26, 2014; 11:32; p. 183

3 186 CHAPTER 7. HOMOLOGY THEORY The set of all these simplexes form a finite simplicial complex with some coefficient group G. All the simplicial chain groups C p (M; G), Z p (M; G) and B p (M; G) are finitely generated Abelian groups. Therefore, the homology group H p (M; G) is a finitely generated group. Since any simplicial cycle can be described as a singular cycle, there are homomorphisms Z p Z p, B p B p and the induced homomorphism H p H p. Theorem Let M be a compact manifold and G be an Abelian group. Then the singular homology groups are isomorphic to the simplicial homology groups H p (M; G) = H p (M; G) and are finitely generated Abelian groups. Proof: Nontrivial. Corollary Let M be a compact manifold and K be field. Then the homology groups H p (M; K) are finite-dimensional vector spaces. Proof: Follows from above theorem Betti Numbers and Topological Invariants Let M be a compact manifold. The dimensions of the real homology groups H p (M; R) are called the Betti numbers, that is, B p (M) = dim H p (M; R). The Betti number B p is the maximal number of linearly independent p- cycles modulo a boundary. diffgeom.tex; August 26, 2014; 11:32; p. 184

4 7.3. SINGULAR HOMOLOGY GROUPS 187 Let M and V be manifolds, G be a n Abelain group, F : M V be a map and F : C p (M; G) C p (V; G) be the induced homomorphism of chain groups. Since the induced homomorphism F commutes with the boundary homomorphism, the groups Z p (M; G) and B p (M; G) are closed under F. Therefore, the homomorphism F naturally acts on the homology groups F : H p (M; G) H p (V; G). If F : M V is a homeomorphism, then there is the inverse homeomorphism F 1 : V M and the inverse induced homomorphism F 1 : H p (V; G) H p (M; G). In this case, the induced homomorphism F is an isomorphism. Theorem Let M and V be compact homeomorphic manifolds and G be an Abelian group. Then their homology groups are isomorphic, that is, for any p H p (M; K) H p (V; G). Proof: Follows from above. Thus, homology groups are topological invariants. Corollary Let M and V be compact manifolds. If there is an Abelian group G and an integer p such that their homology groups are not isomorphic, that is, H p (M; K) H p (V; G), then the manifolds M and V are not homeomorphic to each other. Remark. The converse is not true. diffgeom.tex; August 26, 2014; 11:32; p. 185

5 188 CHAPTER 7. HOMOLOGY THEORY Some Theorems from Algebraic Topology A manifold M is path-connected (or just connected) if every two points in M can be connected by a piecewise-smooth curve. Let M be a manifold and G be an Abelian group. A point p in a manifold M is a 0-chain. Since p = 0, then each point is a 0-cycle (by definition). A smooth map C : [0, 1] M defines a singular 1-simplex and is a 0-chain. C = C(1) C(0) More generally, for any g G, then gc is a singular 1-simplex and (gc) = gc(1) gc(0). A piecewise-smooth map C : [0, 1] M defines a 1-chain (as a formal sum c 1 = r gc k k=1 of smooth pieces with the same coefficient) and is a 0-chain. c 1 = gc(1) gc(0) Theorem Let M be a compact connected manifold and G be an Abelian group. Then Proof: H 0 (M; G) = Gp = {gp g G, p M}. 1. In a connected manifold any two 0-simplexes with the same coefficient are homologous. 2. Moreover, for any point p M, g 0, and any element g G, there is no 1-chain c 1 such that c 1 = gp. diffgeom.tex; August 26, 2014; 11:32; p. 186

6 7.3. SINGULAR HOMOLOGY GROUPS Therefore, a multiple gp of a single point is not a boundary for any g Thus, any point p M is a 0-cycle that is not a boundary. 5. Moreover, for any g G and any p M the 0-chain gp is a 0-cycle that is not a boundary. In particular, and is a one-dimensional vector space. H 0 (M; Z) = {0, ±p, ±2p,... } H 0 (M; R) = R Corollary Let M be a compact connected manifold. Then the zero Betti number is equal to B 0 (M) = 1. Proof: Follows from above. Theorem Let M be a compact manifold consisting of k connected pieces M 1,... M k. Then H 0 (M; R) = Rp Rp k, where p i M i, i = 1,..., k, meaning k Rp Rp k = a i p i ai R, p i M i, i = 1,..., k. Proof: i=1 In this case is a k-dimensional vector space. H 0 (M; R) = R k diffgeom.tex; August 26, 2014; 11:32; p. 187

7 190 CHAPTER 7. HOMOLOGY THEORY Corollary Let M be a compact manifold consisting of k connected pieces. Then the zero Betti number is equal to B 0 (M) = k. Proof: Follows from above. Let V be a p-dimensional oriented closed (compact without boundary) manifold. Then a triangulation of V defines an integer p-cycle, denoted by [V] so that [V] = 0. This does not work for non-orientable closed manifolds. A triangulation of a non-orientable closed manifold V gives an integer p-chain, which is not a p-cycle since [V] 0. Example. Klein bottle. By changing the coefficient group G sometimes one can get a p-cycle even for non-orientable manifolds. Theorem Let M be an n-dimensional compact manifold and V be a closed oriented p-dimensional submanifold of M. Let G be an Abelian group and g be an element of G. Consider a triangulation of V in p-triangles and assign to each p-triangle in this triangulation of V the same coefficient g G. Then g[v] defines a p-cycle z p in H p (M; G). Proof: Without proof. Remark. A p-cycle is a generalization of the concept of a closed oriented submanifold. In the case of real homology groups, when G = R, the following theorem is true. diffgeom.tex; August 26, 2014; 11:32; p. 188

8 7.3. SINGULAR HOMOLOGY GROUPS 191 Theorem Let M be an n-dimensional compact manifold. Every real p-cycle z p in H p (M; R) is homologous to a finite formal sum z p r a k V k k=1 of closed oriented p-dimensional submanifolds V k of M with real coefficients a k. Proof: Nontrivial. Theorem Let M be an n-dimensional manifold and G be an Abelian group. Let z p and z p be two cycles in H p (M; G) that can be deformed into each other. Then they are homologous to each other Proof: z p z p. 1. Since the deformation defines a deformation chain c p+1 such that c p+1 = z p z p. Proposition Let M be an n-dimensional closed manifold and G be an Abelian group. Then for p > n the singular homology groups H p (M; G) are trivial H p (M; G) = 0. Proof: 1. Singular homology groups are isomorphic to the simplicial homology groups. 2. Since there are no simplicial complexes of dimension p > n then all simplicial homology groups are trivial for p > n. diffgeom.tex; August 26, 2014; 11:32; p. 189

9 192 CHAPTER 7. HOMOLOGY THEORY Examples. Sphere S n. From the facts that S n is connected, orientable and closed it follows that H 0 (S n ; G) = H n (S n ; G) = G, H p (S n ; G) = 0, for p 0, n, B 0 (S n ) = B n (S n ) = 1, B p (S n ) = 0, for p 0, n. Torus T 2. H 0 (T 2 ; G) = H 2 (T 2 ; G) = G, H 1 (T 2 ; G) = GA + GB, B 0 (T 2 ) = 1, B 1 (T 2 ) = 2, B 2 (T 2 ) = 1, where A and B are the basic 1-cycles. Klein Bottle K 2. Since K 2 is connected closed non-orientable it follows that H 0 (K 2 ; Z) = Z, H 2 (K 2, Z) = 0, H 1 (K 2 ; Z) = ZA + Z 2 B, where A and B are the basic 1-cycles, and H 0 (K 2 ; R) = R, H 2 (K 2, R) = 0, H 1 (K 2 ; R) = RA, B 0 (K 2 ) = 1, B 1 (K 2 ) = 1, B 2 (K 2 ) = 0. Real Projective Plane RP 2. RP 2 is connected closed non-orientable. H 0 (RP 2 ; Z) = Z, H 2 (RP 2, Z) = 0, H 1 (RP 2 ; Z) = Z 2 A, diffgeom.tex; August 26, 2014; 11:32; p. 190

10 7.3. SINGULAR HOMOLOGY GROUPS 193 where A is the basic 1-cycle, and H 0 (RP 2 ; R) = R, H 2 (RP 2, R) = 0, H 1 (RP 2 ; R) = 0, B 0 (RP 2 ) = 1, B 1 (RP 2 ) = 0, B 2 (RP 2 ) = 0. Torus T 3. T 3 is a connected closed orientable manifold. H 0 (T 3 ; Z) = H 3 (T 3, Z) = Z, H 1 (T 3 ; Z) = ZA + ZB + ZC, H 2 (T 3 ; Z) = ZD + ZE + ZF, where A, B and C are basic 1-cycles, and D, E and F are basic 2-cyles, B 0 (T 3 ) = 1, B 1 (T 3 ) = B 2 (T 3 ) = 3, B 2 (T 3 ) = 1. Real Projective Space RP 3. RP 3 is connected closed orientable. H 0 (RP 3 ; R) = H 3 (RP 3, R) = R, H 1 (RP 3 ; R) = H 2 (RP 3 ; R) = 0, B 0 (RP 3 ) = B 3 (RP 3 ) = 1, B 1 (RP 3 ) = B 2 (RP 3 ) = 0. diffgeom.tex; August 26, 2014; 11:32; p. 191