Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 0903.3762

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

Nigel Higson·John Roe·Thomas Schick
View original source ↗Read source document ↗Export metadata ↓

Abstract

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.

Explore related subjects

math.KTmath.ATmath.OAmath.SP

Keep this discovery

BibTeXRIS

Nigel Higson, John Roe, Thomas Schick. 2009-03-22. Spaces with vanishing $l\sp 2$-homology and their fundamental groups (after Farber and Weinberger). https://doi.org/10.1023/a%3A1012018013481

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

The Oka principle for \'etale Chow groups

The celebrated theorems of Shilov, Arens--Royden, and Forster give direct descriptions of the first three integral cohomology groups of the Gelfand spectrum of a commutative complex Banach algebra. In his 1974 ICM address, Taylor asked whether the higher cohomology groups admit descriptions in terms of the underlying ring. We give a solution to this question in even degrees: The \'etale (aka Lichtenbaum) Chow group in every codimension is canonically isomorphic to the corresponding even integral cohomology group of the Gelfand spectrum.

math.KT↗
arXiv2026

General linear and Steinberg groups over the Leavitt algebra $L_{\mathbb F_2}(1,2)$

Let $R=L_{\F_2}(1,2)$. We prove that $\GL_r(R)$ is integrally acyclic for every $r\geq1$ and that the canonical map $\St_r(R)\to\GL_r(R)$ is an isomorphism for every $r\geq3$. The homology calculation combines simultaneous extensions of ordered frames with scalar actions of the multiplicative groups of finite fields on their stabilizers. The presentation associated with the same frame complex defines a surjective section of the Steinberg map. An explicit finite presentation of $R^\times$ then follows from the theorem of Krsti\'c and McCool. We formulate separate criteria for acyclicity and for the Steinberg comparison over other rings.

math.KT↗
arXiv2026

The K-theory of uniform Roe algebras for coarse structures generated by finite-rank free abelian subgroups

For a uniformly locally finite coarse space $X$, the uniform Roe algebra $C_u^*(X)$ is the operator norm closure of the controlled operators on $\ell^2(X)$. The $K$-theory of uniform Roe algebras is known in asymptotic dimension zero, but it is not fully understood in higher dimensions. We compute $K_0(C_u^*(G,\mathcal E))$ and $K_1(C_u^*(G,\mathcal E))$ for every countable discrete abelian group $G$ and every finite-rank free abelian subgroup $H\leq G$, where $\mathcal E$ is the coarse structure generated by $H$. We use the Proietti--Yamashita spectral sequence to express the $K$-theory in terms of $H_*(H;\ell^\infty(G,\mathbb Z))$, which we then compute.

math.KT↗