MAT 531 (Spring 1996) Topology/Geometry II

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 tex2html_wrap_inline25 and tex2html_wrap_inline27 are smoothly homotopic, then tex2html_wrap_inline29 .

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 tex2html_wrap_inline37 which fit, together with the maps induced by the inclusions tex2html_wrap_inline39 , tex2html_wrap_inline41 , tex2html_wrap_inline43 , tex2html_wrap_inline45 , into an exact sequence (the Mayer-Vietoris sequence)


Example. Calculation of tex2html_wrap_inline51 . We will need the following easy consequence of the Homotopy Theorem: if tex2html_wrap_inline53 as smooth deformation retract, then tex2html_wrap_inline55 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 tex2html_wrap_inline65 is an open annulus. The discs have points as deformation retracts, and the annulus has a circle, so using the homotopy theorem we know tex2html_wrap_inline67 , tex2html_wrap_inline69 and tex2html_wrap_inline71 in the Mayer-Vietoris sequence. We need the additional elementary piece of information: tex2html_wrap_inline73 . Then the beginning of the sequence is




Exactness implies that the map tex2html_wrap_inline79 is onto and therefore that the map tex2html_wrap_inline81 is the zero map. The next map in the sequence




must therefore be injective, so tex2html_wrap_inline87 . In the next part of the sequence




exactness implies tex2html_wrap_inline93 .

Class exercise: Use the same method to calculate tex2html_wrap_inline134 .

The homotopy theorem.

Proposition A: Let tex2html_wrap_inline136 be the inclusions at levels 0 and 1. So tex2html_wrap_inline138 , tex2html_wrap_inline140 . Then tex2html_wrap_inline142

Note: this proposition implies the homotopy theorem, if we take as definition of smooth homotopy between tex2html_wrap_inline25 and tex2html_wrap_inline146 the existence of tex2html_wrap_inline148 with tex2html_wrap_inline150 and tex2html_wrap_inline152 . Because then tex2html_wrap_inline154 and tex2html_wrap_inline156 , so that tex2html_wrap_inline158 .

Proof of Proposition A. A local coordinate system tex2html_wrap_inline160 on M gives a local coordinate system tex2html_wrap_inline164 on tex2html_wrap_inline166 . In terms of these coordinates, any p-form on tex2html_wrap_inline166 may be written as tex2html_wrap_inline172 , where the multi-index I ranges over all p-tuples tex2html_wrap_inline178 , and the multi-index J ranges over all (p-1)-tuples tex2html_wrap_inline184 . For for such an I, tex2html_wrap_inline188 means tex2html_wrap_inline190 , and similarly for J.
Let the linear map tex2html_wrap_inline194 be defined by tex2html_wrap_inline196 , with tex2html_wrap_inline198 as above. First we check that this definition is independent of the choice of coordinate system tex2html_wrap_inline160 . In another system tex2html_wrap_inline202 suppose tex2html_wrap_inline204 . 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 tex2html_wrap_inline214 where the tex2html_wrap_inline216 are appropriate tex2html_wrap_inline218 minors of the matrix tex2html_wrap_inline220 ; and similarly for tex2html_wrap_inline222 ; so calculating tex2html_wrap_inline224 in the v-cordinates gives




this last step because the tex2html_wrap_inline216 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 tex2html_wrap_inline260 and integration with respect to t makes this term equal and opposite to the second term in tex2html_wrap_inline264 . So


Now the inclusion tex2html_wrap_inline268 clearly satisfies tex2html_wrap_inline270 and tex2html_wrap_inline272 ; consequently tex2html_wrap_inline274 ; similarly tex2html_wrap_inline276 . This proves the formula.Finally suppose tex2html_wrap_inline198 is a closed form representing a class in tex2html_wrap_inline280 ; since tex2html_wrap_inline282 , the formula gives tex2html_wrap_inline284 ; the two pulled-back forms differ by a coboundary: they are in the same cohomology class.

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Tony Phillips
Wed Mar 20 16:45:52 EST 1996