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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