Twisted L^2 invariants of non-simply connected manifolds

Details Twisted L^2 invariants of non-simply connected manifolds Twisted L^2 invariants of non-simply connected manifolds

1996-10-29

arxiv.org/abs/dg-ga/9610018v1

Russian Journal of Mathematical Physics, 4 (1996) 499-527

Mathematics

Differential Geometry

AMS-LaTeX v1.2, 28 pages, to appear in special issue of Russ.Jour.of Math.Physics

Scientific paper

We develop the theory of twisted L^2-cohomology and twisted spectral invariants for flat Hilbertian bundles over compact manifolds. They can be viewed as functions on the first de Rham cohomology of M and they generalize the standard notions. A new feature of the twisted L^2-cohomology theory is that in addition to satisfying the standard L^2 Morse inequalities, they also satisfy certain asymptotic L^2 Morse inequalities. These reduce to the standard Morse inequalities in the finite dimensional case, and when the Morse 1-form is exact. We define the extended twisted L^2 de Rham cohomology and prove the asymptotic L^2 Morse-Farber inequalities, which give quantitative lower bounds for the Morse numbers of a Morse 1-form on M.

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