Volume 13, issue 5 (2013)

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Homology equivalences of manifolds and zero-in-the-spectrum examples

Shengkui Ye

Algebraic & Geometric Topology 13 (2013) 2947–2965
Abstract

Working with group homomorphisms, a construction of manifolds is introduced, which preserves homology groups. The construction gives as special cases Quillen’s plus construction with handles obtained by Hausmann, the existence of the one-sided $h$–cobordism of Guilbault and Tinsley, and the existence of homology spheres and higher-dimensional knots proved by Kervaire. We also use it to recover counter-examples to the zero-in-the-spectrum conjecture found by Farber and Weinberger, and by Higson, Roe and Schick.

Keywords
Quillen's plus construction, homology spheres, homology equivalences, $G$–dense rings, zero-in-the spectrum conjecture
Mathematical Subject Classification 2010
Primary: 57N15
Secondary: 19D06, 14F35, 58J50