Variations, characteristic classes, and the obstruction to mapping smooth to continuous cohomology

Mostow, Mark A.

doi:10.1090/S0002-9947-1978-0474357-2

Variations, characteristic classes, and the obstruction to mapping smooth to continuous cohomology
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Trans. Amer. Math. Soc. 240 (1978), 163-182 Request permission

Abstract:

In a recent paper, the author gave an example of a singular foliation on ${{\mathbf {R}}^2}$ for which it is impossible to map the de Rham cohomology ${T_{{\text {DR}}}}$ to the continuous singular cohomology ${T_{\text {c}}}$ (in the sense of Bott and Haefliger’s continuous cohomology of spaces with two topologies) compatibly with evaluation of cohomology classes on homology classes. In this paper the obstruction to mapping ${T_{{\text {DR}}}}$ to ${T_{\text {c}}}$ is pinpointed by defining a whole family of cohomology theories ${T_{k,m,n}}$, based on cochains which vary in a ${C^k}$ manner, which mediate between the two. It is shown that the obstruction vanishes on nonsingularly foliated manifolds. The cohomology theories are extended to Haefliger’s classifying space $(B{\Gamma _q} \to B{J_q})$, with its germ and jet topologies, by using a notion of differentiable space similar to those of J. W. Smith and K. T. Chen. The author proposes that certain of the ${T_{kmn}}$ be used instead of ${T_{\text {c}}}$ to study Bott and Haefliger’s conjecture that the continuous cohomology of $(B{\Gamma _q} \to B{J_q})$ equals the relative Gel’fand-Fuks cohomology ${H^\ast }({\mathfrak {a}_q},{O_q})$. It is shown that ${T_{kmn}}(B{\Gamma _q} \to B{J_q})$ may contain new characteristic classes for foliations which vary only in a ${C^k}$ manner when a foliation is varied smoothly.

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