Vol. 3, No. 3, 2018

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ISSN: 2379-1691 (e-only)
ISSN: 2379-1683 (print)
Positive scalar curvature and low-degree group homology

Noé Bárcenas and Rudolf Zeidler

Vol. 3 (2018), No. 3, 565–579
Abstract

Let Γ be a discrete group. Assuming rational injectivity of the Baum–Connes assembly map, we provide new lower bounds on the rank of the positive scalar curvature bordism group and the relative group in Stolz’ positive scalar curvature sequence for BΓ. The lower bounds are formulated in terms of the part of degree up to 2 in the group homology of Γ with coefficients in the Γ-module generated by finite order elements. Our results use and extend work of Botvinnik and Gilkey which treated the case of finite groups. Further crucial ingredients are a real counterpart to the delocalized equivariant Chern character and Matthey’s work on explicitly inverting this Chern character in low homological degrees.

Keywords
positive scalar curvature, secondary index theory, $\rho$-invariant, equivariant Chern character, group homology
Mathematical Subject Classification 2010
Primary: 58D27, 58J22
Secondary: 19K33, 19L10, 19L47, 55N91
Milestones
Received: 3 October 2017
Revised: 21 December 2017
Accepted: 6 January 2018
Published: 16 July 2018
Authors
Noé Bárcenas
Centro de Ciencias Matemáticas
UNAM Campus Morelia
Morelia
Mexico
Rudolf Zeidler
Mathematisches Institut
Westfälische Wilhelms-Universität Münster
Münster
Germany