De Rham theorem for extended L^2-cohomology

Details De Rham theorem for extended L^2-cohomology De Rham theorem for extended L^2-cohomology

1996-10-11

arxiv.org/abs/dg-ga/9610007v2

Mathematics

Differential Geometry

16 pages, author-supplied file available at ftp://ftp.math.neu.edu/Pub/faculty/Shubin_Mikhail/papers/DR19.tex This is a slig

Scientific paper

We prove an analogue of the de Rham theorem for the extended L^2-cohomology introduced by M. Farber. This is done by establishing that the de Rham complex over a compact closed manifold with coefficients in a flat Hilbert bundle E of A-modules over a finite von Neumann algebra A is chain-homotopy equivalent (with bounded morphisms and homotopy operators) to a combinatorial complex with the same coefficients. This is established by using the Witten deformation of the de Rham complex. We also prove that the de Rham complex is chain-homotopy equivalent to the spectrally truncated de Rham complex which is also finitely generated.

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