Stolz Positive Scalar Curvature Structure Groups, Proper Actions and Equivariant 2-Types
Massimiliano Puglisi, Thomas Schick, Vito Felice Zenobi
TL;DR
The paper develops an equivariant framework for Stolz's positive scalar curvature structure groups $R^{\rm spin}_n(X)^G$ under proper actions of a discrete group $G$. It introduces a fundamental groupoid functor $\Pi_1(X;G)$ and constructs classifying spaces that realize this functor, proving that $R^{\rm spin}_n(X)^G$ depends only on the equivalence class of $\Pi_1(X;G)$. An equivariant Stolz exact sequence is established, together with a 2-equivalence invariance result for $R^{\rm spin}_n(X)^G$ (for $n\ge6$), and a universal-space framework is developed to classify these invariants via $B\Pi$. The construction of a universal space $B\Pi$ and its associated realizations provides a principled way to compare equivariant and non-equivariant cases, with explicit examples and corollaries showing the dependence on the fundamental groupoid data. This work bridges equivariant topology, surgery theory, and index-theoretic methods to understand PSC metrics in settings with symmetry.
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.