**Week 9 - Part 1** (For a text-only page click here.)

1. *Elements of de Rham cohomology.*
There are two results which allow our elementary
calculations for the point and the circle to be extended
to a large collection of manifolds.

*Homotopy Theorem.* If and are
smoothly homotopic, then .

*Mayer-Vietoris Theorem.* Suppose a manifold *M*
can be written as a union of two open sets *U* and *V*. Then
there exist a set of linear
which fit, together with the maps induced by the inclusions
, ,
,
, into an exact sequence (the
*Mayer-Vietoris sequence*)

*Example.* Calculation of . We will need the following
easy consequence of the Homotopy Theorem: if as
smooth deformation retract, then
is an isomorphism.
Let *U* and *V* be the complements of the North and South poles
respectively; then *U* and *V* are open discs and is
an open annulus. The discs have points as deformation retracts,
and the annulus has a circle, so using the homotopy theorem we
know , and in the Mayer-Vietoris sequence.
We need the additional elementary piece of information: .
Then the beginning of the sequence is

i.e.

Exactness implies that the map is onto and therefore that the map is the zero map. The next map in the sequence

i.e.

must therefore be injective, so . In the next part of the sequence

i.e.

exactness implies .

*Class exercise:* Use the same method to calculate .

*The homotopy theorem. *

*Proposition A*: Let
be the inclusions at levels 0 and 1. So
, . Then

Note: this proposition implies the homotopy theorem, if we take as definition of smooth homotopy between and the existence of with and . Because then and , so that .

*Proof of Proposition A.* A local coordinate
system on *M* gives a local coordinate system
on . In terms of these coordinates,
any *p*-form on may be written as , where the multi-index *I* ranges over
all *p*-tuples , and the multi-index *J*
ranges over all *(p-1)*-tuples . For
for such an *I*, means , and
similarly for *J*.

Let the linear map be defined by
, with
as above. First we check that this definition is
independent of the choice of coordinate system . In another
system suppose . The change of coordinates from
the *(t,u)* system to the *(t,v)* system has the form

i.e. the *t*'s and the other coordinates transform independently.
Consequently where the are
appropriate minors of the matrix ; and similarly
for ; so calculating in the *v*-cordinates
gives

this last step because the are constant in *t*,

the same as the
calculation in the *u*-coordinates.
Next we verify the formula

. In fact,

so

On the other hand

Interchanging differentiation with respect to and
integration with respect to *t* makes this term equal and
opposite to the second term in . So

Now the inclusion clearly satisfies
and ; consequently ; similarly
. This proves the
formula.Finally suppose is a closed form representing
a class in ; since , the formula
gives ;
the two pulled-back forms differ by a coboundary: they are
in the same cohomology class.

Back to Main Toplogy-Geometry II Page.

Back to Tony's Home Page.

Wed Mar 20 16:45:52 EST 1996