Stolz Positive Scalar Curvature Structure Groups, Proper Actions and Equivariant 2-Types

TL;DR

This paper extends Stolz's positive scalar curvature structure groups to equivariant settings involving proper group actions, introducing fundamental groupoid functors and classifying spaces to analyze their properties.

Contribution

It defines equivariant Stolz R-groups using fundamental groupoid functors and shows these groups depend only on the associated fundamental groupoid, unifying equivariant and classical cases.

Findings

Dependence of equivariant R-groups on fundamental groupoid functors

Construction of classifying spaces for fundamental groupoid functors

Unified approach covering classical non-equivariant case

Abstract

In this note, we study equivariant versions of Stolz' $R$-groups, the positive scalar curvature structure groups $R^{\rm spin}_n(X)^G$, for proper actions of discrete groups $G$. We define the concept of a fundamental groupoid functor for a $G$-space, encapsulating all the fundamental group information of all the fixed point sets and their relations. We construct classifying spaces for fundamental groupoid functors. As a geometric result, we show that Stolz' equivariant $R$-group $R^{\rm spin}_n(X)^G$ depends only on the fundamental groupoid functor of the reference space $X$. The proof covers at the same time in a concise and clear way the classical non-equivariant case.