De Rham Cohomology From Infinitesimal Handles

Abstract: Compact smooth n-dimensional manifolds admit handle decompositions. Here a k-handleis a product of disks of dimension k and (n-k) and the decomposition says how the handles areglued together. If the manifold is connected it has a handle decomposition with only one0-handle. The dimension k de Rham cohomology of a manifold is its closed k-forms divided byits exact k-forms. We compute the de Rham cohomology from a handle decomposition, bytreating all handles as having infinitesimal size. This gives a description in terms of forms on the0-handle that satisfies certain conditions at the boundary. We work this out for spheres of anydimension and for orientable surfaces.