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

2013/01/31 by Shengkui Ye · 1 citation
Mathematics · #Algebraic structures and combinatorial models #Cobordism #Conjecture #Geometric and Algebraic Topology #Homology (biology) #Homomorphism #Homotopy and Cohomology in Algebraic Topology #Mathematics #Pure mathematics #Zero (linguistics) #math.AT #math.DG #math.GT #math.KT

paper · pdf · doi:10.2140/agt.2013.13.2947

published in Algebraic & Geometric Topology 13(5), 2947-2965 (Mathematical Sciences Publishers) · some typos are corrected. Final version, to appear in Algebraic & Geometric Topology

arxiv created 2013/05/04 · openalex publication_date 2013/07/30 · arxiv updated 2014/09/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

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 counterexamples to the zero-in-the-spectrum conjecture found by Farber and Weinberger, and by Higson, Roe and Schick.

Citations

• Surgery on Compact Manifolds
• Manifolds with a given homology and fundamental group
• Manifolds with non-stable fundamental groups at infinity, III