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Simplicial Objects and Singular Homology

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Simplicial Objects and Singular Homology Simplicial Objects and Singular Homology Helen Gilmour 24th October 2002 The Simplicial Category and Simplicial Objects The simplicial category ∆ is the category with objects, all finite ordinal numbers n = 0, 1, . , n and morphisms f : m n all non-decreasing functions (i.e. all functions f{ for which} f(i) 6 f(j) for i 6 j). −→ To the simplicial category we can associate the opposite category ∆op. The objects of ∆op are the objects of ∆, the morphisms of ∆op are the morphisms f op, which are in one-to one correspondence f f op with the morphisms f of ∆. For each morphism f : m n of ∆, the domain and codomain7−→ of the corresponding f op are as in f op : n m, so the−→ direction is reversed. −→ If we take a category A, then a simplicial object S in A is defined to be the functor S : ∆op A i.e. a contavariant functor ∆ A. If A is the category of sets, then the functor S is known−→ as a simplicial set. Similarly we−→ have simplicial groups and simplicial rings etc. We have a presentation for the simplicial category; there are morphisms δn : n n + 1 δn 0, . , n = 0,...,ˆi, . , n + 1 i −→ i { } { } σn : n + 1 n σn(i) = σn(i + 1) i −→ i i n n where δi is an injection and σi a surjection. These morphisms generate the simplicial category ∆ and along with a list of relations, give a presentation of ∆. The simplicial category ∆, has a direct geometric interpretation by affine simplices which give a functor : ∆ Top 4 −→ Singular Homology Groups of a Topological Space The singular homology of a topological space is a classical example of using simplicial objects. In order to define the singular homology groups of a topological space X, it is necessary to introduce some notation and definitions. We take the n-dimensional Euclidean space Rn (this is the space of all n-tuples of real numbers (a1, a2, . , an)). n The line segment joining two points u, v R is the set of all points x0u + x1v where x0 , ∈ x1 are real numbers with x0 + x1 = 1, x0 > 0, x1 > 0. We can say that a subset C of Rn is convex if it contains the line segment joining any two n of its points. If u0, . , um are m + 1 points of R , the set of all points u = x0u0 + ... + xmum, xo + x1 + ... + xm = 1 xi > 0 is the smallest convex set containing the points u0, . , um and is called the convex hull of u0, . , um. Here the real numbers xi are the barycentric coordinates of u relative to u0, . , um. An m-simplex is defined to be the convex hull of m + 1 points, these points are the vertices of the simplex. 1 For each dimension n we can take a standard n-simplex n in the space Rn, labelling the vertices (0, 1, ..., n). This standard n-simplex is the convex4 hull of the standard basis of Rn along with the origin (labelled 0). The standard n-simplex can be expressed by barycentric coordinates relative to the origin and the standard basis of Rn. For any topological space X, a singular n-simplex T in X is just a continuous map T : n X. The word “singular” is used to emphasise that T need not be an embedding. 4 −→ The free abelian group generated by the singular n-simplices of X is denoted Sn(X) and is called the singular chain group of X in dimension n. We can think of the “boundary” of the standard n-simplex n as consisting of certain (n 1)-dimensional singular simplices which are the “faces” of n4. −In general n has n + 1 faces, with it’s ith face being4 the linear singular simplex i n 1 4n i n 1 ε : − , i = 0, . , n, where ε is a map that carries − by a linear homeomorphism onto4 the face−→ (0 4, 1,...,ˆi, . , n) of n, where the vertex i is to4 be omitted. 4 n So if we have any singular n-simplex T : X, it has n + 1 faces diT defined by the composite 4 −→ i n 1 diT = T ε : − X 4 −→ so, the ith face omits the ith vertex. The process of forming iterated faces satisfies the identity didjT = dj 1diT i j − ≤ We can now define the boundary homomorphism ∂ : Sn(X) Sn 1(X) −→ − as the sum of the face homomorphisms diT with alternating signs; that is n n i ∂T = d0T d1T + ... + ( 1) dnT = ( 1) diT n > 0 − − X − i=0 So, ∂T is a formal sum of singular simplices of dimension n 1 which are the “faces” of T . 2 − We can verify that ∂ = 0, that is that the family of groups Sn(X) and homomorphisms ∂ : Sn(X) Sn 1(X) for n 0 forms a chain complex, known as the singular chain complex of X and denoted−→ −S(X). ≥ So we want to prove that the composite ∂∂ : Sn Sn 2 is the zero homomorphism for n > 1. It is sufficient to prove ∂∂T = 0 for singular n-simplex−→ −T : n X. We have 4 −→ n ∂∂T = d0(d0T d1T + ... + ( 1) dnT ) − − n d1(d0T d1T + ... + ( 1) dnT ) − n − − n +( 1) dnT (d0T d1T + ... + ( 1) dnT ) − − − i+j = ( 1) didjT X − This can be split up as i+j i+j ∂∂T = ( 1) didjT + ( 1) didjT X − X − j 1<i i j − ≥ If we use the identity didjT = dj 1diT , i j and switch the labels i and j in the second sum, we get − ≤ i+j i+k ∂∂T = ( 1) dj 1diT + ( 1) dkdiT X − − X − j 1 i k i − ≥ ≥ The two sums are equal, except for sign and so cancel to give ∂∂ = 0 The homology groups of this chain complex are called the singular homology groups of X and are denoted Hn(X). 2 Computation of Singular Homology Groups There are several ways of defining homology groups. So far, we have considered singular groups, however, singular homology theory was introduced as a generalisation of simplicial homology theory. Simplicial homology groups are easy to understand and compute, however, they are defined only for particular triangulable spaces. Singular homology groups however are defined for arbitrary topological spaces and agree with the simplicial homology groups when they are both defined. It is difficult to compute the singular homology groups of even the simplest of spaces. The theory of CW-complexes allows singular homology to be computed fairly readily. As an example, we can compute the homology of a point; a very simple situation, where the answer is found via a straightforward calculation. n We want to calculate Hn( ). There is precisely one map . i.e. one singular n- ∗ 4 −→ ∗ simplex. Therefore, the singular chain group of , Sn( ) (the free abelian group generated by ∗ ∗ n-simplices of ) is Z. If we let αn be the standard generator of the group Sn, then the boundary homomorphism∗ ∂ : Sn( ) Sn 1( ) ∗ −→ − ∗ is given by n i αn 1 n even ∂αn = ( 1) αn 1 = − X − − 0 n odd i=0 Therefore ∂n = 0 when n is odd and ∂n is an isomorphism when n is even. For n = 0, we have Hn( ) = Z. Now assume that n > 0.∗ So the singular homology groups of , are the homology groups of the following singular chain complex; ∗ ∂n+1 ∂n ... Sn+1( ) Sn( ) Sn 1( ) ... −→ ∗ −→ ∗ −→ − ∗ −→ For n odd, then ∂n = 0 implies that Sn( ) = ker ∂n. Also ∂n+1 is an isomorphism (since n + 1 ∗ is even), hence surjective and so Sn( ) = im ∂n+1. Thus ∗ Hn( ) = ker ∂nim ∂n+1 = 0. ∗ For n > 0 and even, then ∂n is an isomorphism, hence injective and so, ker ∂n = 0. And Hn( ) =0 in this case also. ∗ So we have shown that if X is a one point space, then Hn(X) = 0 for all n > 0. This result is known as the Dimension Axiom for Singular Homology Theory. Homology groups of all contractible topological spaces coincide with those of a point. In general, if X is a space with n-components, then H0(X) is n copies of Z. Mayer-Vietoris Sequences If X is the union of two subspaces X1 and X2, we can under suitable hypotheses, produce an exact sequence relating the homology of X with that of X1 and X2. This exact sequence is called the Mayer-Vietoris sequence of the pair X1 and X2. Suppose we have X = X1 X2 and that X1,X2 is an excisive couple. Let A = X1 X2. Then there is an exact sequence∪ { } ∩ φ ψ ... Hp(A) ∗ Hp(X1) Hp(X2) ∗ Hp(X) Hp 1(A) ... −→ −→ ⊕ −→ −→ − −→ called the Mayer-Vietoris sequence of X1,X2 .The homomorphisms are defined by { } φ (a) = (i (a), j (a)) ∗ ∗ − ∗ ψ (x1, x2) = k (x1) + l (x2) ∗ ∗ ∗ 3 where the maps i A X1 −−−→ j k y l y X2 X −−−→ are inclusions. Homology of the Sphere If we let Sn Rn+1 be the n-sphere (of radius 1 and centre the origin), where n 0. Then ⊂ ≥ Z if p = n H (Sn) = p 0 if p = n e 6 n n Here Hp(S ) are called the reduced homology groups of S . These reduced homology groups are more readilye used and for i > 0 the groups Hi(X) and Hi(X) are isomorphic. n To prove the results above we induct onen 0 to show that Hp(S ) is as claimed for all ≥ p 0. e ≥For n = 0, we have the 0-sphere S0, we should note that it consists of two points 1, 1 and is therefore a discrete two-point space.
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