On the zero-in-the-spectrum conjecture. (English) Zbl 0992.58012

Farber, Michael; Weinberger, Shmuel

Ann. Math. (2) 154, No. 1, 139-154 (2001).

The authors answer a question of J. Lott [Enseign. Math. (2) 42, 341-376 (1996; Zbl 0874.58086)] by showing:
Theorem 1: For any \(n\geq 6\), there exists a closed \(n\) dimensional smooth manifold \(M\) so that for any \(p=1,\dots ,n\), \(0\) does not belong to the spectrum of the Laplacian acting on the space of \(L^2\) forms on the universal covering of \(M\).
The authors prove Theorem 1 by restating it in an equivalent form using extended \(L^2\) homology:
Theorem 2: For any \(n\geq 6\), there exists a closed orientable smooth \(n\) dimensional manifold \(M\) so that the extended \(L^2\) homology \(H_p(M;\ell^2(\pi))\) vanishes for all \(p\) where \(\pi\) is the fundamental group of \(M\) and \(\ell^2(\pi)\) is the \(L^2\) completion of the group ring \(C[\pi]\).
This yields \(L^2\) invisible manifolds. Theorem 2 rests in turn on
Theorem 3: There exists a finite \(3\) dimensional polyhedron \(Y\) with fundamental group \(\pi=F\times F\times F\) where \(F\) is the free group with two generators such that the extended \(L^2\) homology \(H_p(Y;\ell^2(\pi))\) vanishes for all \(p=0,1,\dots \).
This yields as a corollary
Theorem 4: There exists an aspherical \(3\) dimensional finite polyhedron \(Z\) and a normal subgroup \(H\) in \(\pi=\pi_1(Z)\) such that the extended \(L^2\) homology \(H_p(Z;\ell^2(\pi/H))\) vanishes for all \(p=0,1,\dots\;\).

Reviewer: Peter B.Gilkey (Eugene)

MSC:

58J50

Spectral problems; spectral geometry; scattering theory on manifolds

Keywords:

extended \(L^2\) homology; spectrum of the Laplacian; \(L^2\) invisible manifolds; Laplace-Beltrami operator; \(L^2\) Hopf exact sequence

Citations:

Zbl 0874.58086