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

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.

💡 Research Summary

The paper addresses the “zero in the spectrum” conjecture, which in its strongest form predicts that for any closed manifold M the Laplacian on the universal cover (\widetilde M) has 0 in its ℓ²‑spectrum, i.e. at least one unreduced ℓ²‑cohomology group of (\widetilde M) is non‑zero. Farber and Weinberger produced the first counter‑examples: a 3‑dimensional finite CW‑complex and a 6‑dimensional smooth closed manifold whose universal covers have vanishing ℓ²‑homology in every degree.

The present note shows that the same construction works for any finitely presented group G satisfying a very natural ℓ²‑homological vanishing condition: \