Positive scalar curvature and low-degree group homology

Abstract

Let $\mathrm{\Gamma }$ 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 $B\mathrm{\Gamma }$. The lower bounds are formulated in terms of the part of degree up to $2$ in the group homology of $\mathrm{\Gamma }$ with coefficients in the $ℂ\mathrm{\Gamma }$-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.