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

In this note, we study equivariant versions of Stolz’ $R$-groups, the positive scalar curvature structure groups $R_n^{\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{n}}(X{)}^G$, 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 $R_n^{\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{n}}(X{)}^G$ 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.