Stolz Positive Scalar Curvature Structure Groups, Proper Actions and Equivariant 2-Types
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866908621530464256 |
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| author | Puglisi, Massimiliano Schick, Thomas Zenobi, Vito Felice |
| author_facet | Puglisi, Massimiliano Schick, Thomas Zenobi, Vito Felice |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_07955 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Stolz Positive Scalar Curvature Structure Groups, Proper Actions and Equivariant 2-Types Puglisi, Massimiliano Schick, Thomas Zenobi, Vito Felice Geometric Topology Algebraic Topology Differential Geometry 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. |
| title | Stolz Positive Scalar Curvature Structure Groups, Proper Actions and Equivariant 2-Types |
| topic | Geometric Topology Algebraic Topology Differential Geometry |
| url | https://arxiv.org/abs/2412.07955 |