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

Higson, Nigel; Roe, John; Schick, Thomas

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

Authors

Higson, Nigel

Roe, John

Schick, Thomas

Description

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.
11 pages, 2001 paper with updated references