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In classical de Rham theory, integration of differential forms along simplices
provides an isomorphism of de Rham cohomology and singular cohomology of a smooth
manifold. For a Nash manifold, i.e. a manifold given locally as the common solutions
to a system of polynomial inequalities over R, one can consider variants of the de
Rham complex by considering differential forms which decay faster than any polynomial,
along with their derivative. The resulting complex is called the Schwartz-de Rham
complex. For a Nash manifold M, it was shown by Aizenbud and Gourevitch, and
independently by Luca Prelli, that the Schwartz-de Rham complex is quasi-isomorphic to
the complex of compactly supported differential forms on M, and hence its cohomologies
are identified with the cohomologies of M with compact support.
In my talk I will present a work in progress, joint with Avraham Aizenbud, to generalize this result to the relative situation, namely replacing the Nash manifold M with a Nash submersion f:M-->N. Unlike the absolute case, in general the compactly supported and Schwartz sections of the de Rham complex of f does not coincide, but rather form the compactly supported and Schwartz sections respectively of certain construcible sheaf constructed from f, parametrizing the compactly supported cohomology of the fibers of f. Our main tool in the formulation and proof of this fact is the notion of sections of a cosheaf valued in a constructible sheaf, along with its "derived" (or, more precisely, infinity categorical) version. I will explain this construction and how to obtain the relative de Rham theorem from its basic properties. |