(CO)HOMOLOGY AND CRITICAL POINT THEORY - Math - IPM

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374 CHAPTER 3. (CO)HOMOLOGY AND CRITICAL POINT THEORY Exercise 3.1.3 Show that the homology of two circles joined at a point (i.e. figure ∞) is Z⊕Z in dimension 1. More generally, show that if X ∨ Y is two simplicial complexes X and Y joined at one vertex, then for j > 0 Hj(X ∨ Y ; R) = Hj(X; R) ⊕ Hj(Y ; R). For reasons that will become clear later it is essential to extend the notion of homology to the relative case, i.e., define Hn(X, A; R) whee X is a simplicial complex and A ⊂ X is a subcomplex. Then by construction Cn(A; R) ⊆ Cn(X; R), and we define Cn(X, A; R) = Cn(X; R) Cn(A; R) . Since ∂n+1(Cn+1(A; R)) ⊆ Cn(A; R), we have the induced boundary map ∂n+1 : Cn+1(X, A; R) → Cn(X, A; R) which we are denoting by the same letter ∂n+1 It clearly satisfies (3.1.1). We refer to Cn(X, A; R) as the R-module of relative chains. Similarly the R-modules relative cycles and boundaries are defined as We define Hn(X, A; R) as (3.1.7) as before. Zn(X, A; R) = ker∂n, Bn(X, A; R) = Im∂n+1. Hn(X, A; R) = Zn(X, A; R) Bn(X, A; R) . Exercise 3.1.4 Let X be the disc B 2 ⊂ R 2 and A = ∂B 2 be its boundary. By realizing B 2 and a simplicial complex and A as a subcomplex show that H1(X, A; R) R, Hj(X, A; R) = 0 for j = 1. Let X and Y be simplicial complexes and f : X → Y be a simplicial map. Therefore it gives R-module homomorphisms fn : Cn(X; R) −→ Cn(Y ; R). Then fn’s commutes with the boundary operators ∂n, i.e., ∂ Y n fn = fn−1∂ X n where the superscripts X or Y refer to the simplicial complex relative to which the boundary operator is defined. It follows that f induces a homomorphism f⋆ = fn⋆ : Hn(X; R) −→ Hn(Y ; R). Similar considerations apply to simplicial maps of pairs of simplicial complexes f : (X, A) → (Y, B). This means that A ⊂ X and B ⊂ Y are subcomplexes, f is a simplicial map of X to Y mapping A to B, and we have induced maps f⋆ = fn⋆ : Hn(X, A; R) −→ Hn(Y, B; R).
3.1. BASIC NOTIONS OF HOMOLOGY 375 In reality continuous maps between manifolds or topological spaces (admitting of triangulations) are not given as simplicial maps. A general devise for attaching an induced map on homology groups (or modules) is by approximating the given continuous with a simplicial map. The following theorem explains this approximation method: Theorem 3.1.1 (Simplicial Approximation Theorem) - Let f : |XV| → |YU| be a continuous map of topological spaces associated to simplicial complexes. Then after passing to a finer subdivisions of X and Y , we can approximate f by a simplicial map f ′ (see below for explanation). f and f ′ are homotopic. Let us explain the meaning of this theorem. We say a simplicial map f ′ approximates or is simplicial approximation to f if f(x) ∈ |s|, where s is a simplex of YU, implies |f ′ |(x) ∈ |s|. The proof of theorem 1.1 actually gives more than the statement above. In fact, it shows that there is a homotopy F : |X| × I → |Y | between f and the map |f ′ | such that if f(x) ∈ s then throughout the homotopy the image of x lies in s, i.e., F (x, t) ∈ s for all t ∈ I. The fact that f and |f ′ | are homotopic implies that at the level of homology the induced maps are identical. Therefore this theorem means that in dealing with homology we can treat continuous maps of topological spaces, with implicit structure of simplicial complexes, as if they were maps of simplicial complexes without further ado. We shall omit the proof of this basic theorem since its proof is not useful for our purposes. 3.1.2 The Axioms and Singular Theory In order to make more efficient use of homology theory it is useful to introduce it axiomatically. First we need some definitions. Let R be a commutative ring with identity. For our purposes, it often suffices to take R to be Q, R, C, Z, or Z/p. Let A1, A2, · · · be R-modules and gi : Ai → Ai−1 a homomorphism. We say the sequence · · · −→ Ai+1 gi+1 −→ Ai gi gi−1 −→ Ai−1 −→ · · · is exact if ker gi = Imgi+1 for all i. In particular, the sequence 0 → A g → B → 0 is exact if and only if g : A → B is an isomorphism. An exact sequence of the form 0 → A → B → C → 0 is called a short exact sequence. For such a sequence we may regard A as a submodule of B, and C is isomorphic to B/A. We let (X, A) denote a pair of Hausdorf topological spaces with A ⊆ X. By a homology theory we mean an assignment of abelian groups Hi(X, A) to every pair (X, A) and integer i ≥ 0, and homomorphism Hi(f) = f⋆ : Hi(X, A) → Hi(Y, B) (called induced homomorphism) to every continuous map f : (X, A) → (Y, B) such that the following conditions are satisfied: 1. Covariance - If f : (X, A) → (Y, B) and g : (Y, B) → (Z, C), then Hi(g · f) = Hi(g) · Hi(f); 2. Homotopy - If f, g : (X, A) → (Y, B) are homotopic, then the induced maps Hi(f) and Hi(g) are identical; 3. Long Exact Sequence - There is the exact sequence · · · → Hi(A) → Hi(X) → Hi(X, A) δi → Hi−1(A) → · · · → H◦(X) → H◦(X, A),
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3.4. APPLICATIONS 437 For M = M ′
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3.4. APPLICATIONS 439 Proposition 3
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3.4. APPLICATIONS 441 where ω is t
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3.4. APPLICATIONS 443 3.4.5 Zeta Fu
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3.4. APPLICATIONS 445 and consequen
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3.5. SOME ALGEBRAIC CONSIDERATIONS
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