Spaces with vanishing $l\sp 2$-homology and their fundamental groups (after Farber and Weinberger)

Details Spaces with vanishing $l\sp 2$-homology and their fundamental groups (after Farber and Weinberger) Spaces with vanishing $l\sp 2$-homology and their fundamental groups (after Farber and Weinberger)

2009-03-22

arxiv.org/abs/0903.3762v1

Geom. Dedicata 87 (2001), no. 1-3, 335--343

Mathematics

K-Theory and Homology

11 pages, 2001 paper with updated references

Scientific paper

10.1023/A:1012018013481

The "zero in the spectrum conjecture" asserted (in its strongest form) that for any manifold M zero should be in the l2-spectrum of the Laplacian (on forms) of the universal covering of M, i.e. that at least one (unreduced) L2-cohomology group of (the universal covering of) M is non-zero. Farber and Weinberger gave the first counterexamples to this conjecture. In this paper, using their fundamental idea to show the following stronger version of this result: Let G be a finitely presented group and suppose that the homology groups H_k(G,\ell^2(G)) are zero for k=0,1,2. For every dimension n\ge 6 there is a closed manifold M of dimension n and with fundamental group G such that the L2-cohomology of (the universal covering of) M vanishes in all degrees.

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