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
Exactness implies that the map is onto and therefore that the map is the zero map. The next map in the sequence
must therefore be injective, so . In the next part of the sequence
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,
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
the two pulled-back forms differ by a coboundary: they are
in the same cohomology class.
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