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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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

Author: Mark A. Mostow
Journal: Trans. Amer. Math. Soc. 240 (1978), 163-182
MSC: Primary 57F30; Secondary 57D30
MathSciNet review: 0474357
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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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Keywords: Bott-Haefliger continuous cohomology, space with two topologies, smooth cohomology, foliation, Haefliger classifying space, differentiable space, de Rham theorem, integration of forms, variation of simplices, category of morphisms, geometric realization, simplicial space, Milnor classifying space, variation of characteristic classes, characteristic classes of foliations
Article copyright: © Copyright 1978 American Mathematical Society