MAT 531 (Spring 1996) Topology/Geometry II

Week 13
Singular homology theory, continued.

1. Proof that D_p o D_{p+1} = 0. The singular chain complex Delta_*(X) of a topological space X. Cycles, boundaries, the p-th singular homology group H_p(X).

Informal comparison of homology and cohomology.

Calculation of singular homology groups. Exactly as with the de Rham cohomology, a homotopy theorem and a Mayer-Vietoris theorem will allow the computation of these groups for many different spaces.
Suppose X = U union V is an open covering. Let i_1, i_2 be the inclusions of U intersect V into U and V, respectively, and let j_1, j_2 represent the inclusions of U and V, respectively, into X. Let Delta^{U,V}_*(X) represent the free abelian group generated by the simplexes which lie in either U or V. Then the Mayer-Vietoris theorem for singular homology will follow from the (obvious) exactness of the sequence

0 --> Delta_*(U intersect V) --> Delta_*(U) + Delta_*(V) --> Delta^{U,V}_*(X) --> 0

(the interior arrows represent the chain maps (i_1*, i_2*) and j_1* - j_2*), the diagram-chasing argument that leads from a short exact sequence of chain complexes to a long exact sequence of homology groups and the following proposition.

Proposition. The inclusion

Delta^{U,V}_*(X) --> Delta_*(X)

is a chain map which induces isomorphisms in homology.

2. For today, a ``small'' chain will be one made up of simplexes which lie in U or V. The proposition can be restated as saying that the homology groups of X can be calculated using small chains.

The proof consists in the construction of two linear operators on Delta_*(X). The first, Y (Upsilon in the text) is the ``subdivision'' operator. It is a chain map, and satisfies
a. supp(Y(s)) = supp (s) for any singular simplex s, where the support supp(s) of a singular simplex s : Delta_p --> X is the image s(Delta_p), and the support of a chain is the union of the supports of its simplexes.
b. For any singular simplex s : Delta_p --> X there is an integer k such that the chain Y^k(s) lies in Delta^{U,V}_*(X).

The second, T, is a chain homotopy between Y and the identity, i.e.
c. DT(s) + TDs = Y(s) - s for any singular simplex s. Here as usual D is the boundary map (usually, "partial").
It also satisfies
d. supp(T(s)) = supp (s) for any singular simplex s.

The existence of Y and T proves the proposition. In fact, an inclusion map of this type is automatically a chain map, and so induces a homomorphism of homology groups.
i. That homomorphism is one-one. Let z be a ``small'' cycle which bounds c in Delta_p(X). Suppose for the moment that Y(c) is small. then c. gives(*) DT(c) + TDc = Y(c) - c. Applying D gives DDT(c) + DTDc = DY(c) -Dc, or
DTz - DY(c) = z. Since z is ``small'' so is Tz, by d. This equation then exhibits z as the boundary of a ``small'' chain.
Suppose now that k = 2 for the chain c, so Y(c) may not be small, but Y^2(c) is. Then c. gives DT(Y(c)) + TD(Y(c)) = Y^2(c) - Y(c).
Adding this to the previous chain-homotopy relation(*) and using the facts that DY =YD (i.e. Y is a chain map), and that Dc = z gives
DT(Y(c)) + TY(D(c)) +DT(c) + TDc = Y^2(c) - Y(c) +Y(c) - c = Y^2(c) - c.
Applying D as before yields
D TY(z)+DT(z) = DY^2(c) - z,
which exhibits z as the boundary of a small chain. This argument easily extends to higher values of k.
ii. That homomorphism is onto. (Class exercise).

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