**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*) + *TD**s* = *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*) + *TD**c* = *Y*(*c*) -
*c*. Applying *D* gives
*D**DT*(*c*) + *D**TD**c* =
*D**Y*(*c*) -*D**c*, or

*D**T**z* - *D**Y*(*c*) = *z*.
Since *z* is ``small'' so is *T**z*, 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 *D**Y* =*Y**D*
(i.e. *Y* is a chain map), and that *Dc* = *z*
gives

*DT*(*Y*(*c*)) + *TY*(*D*(*c*))
+*DT*(*c*) + *TD**c* =
*Y^2*(*c*) -
*Y*(*c*) +*Y*(*c*) -
*c* = *Y^2*(*c*) - *c*.

Applying *D* as before yields

*D* *TY*(*z*)+*D**T*(*z*) =
*D**Y^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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