SIGMA 21 (2025), 093, 19 pages      arXiv:2412.07955      https://doi.org/10.3842/SIGMA.2025.093

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

Massimiliano Puglisi ^a, Thomas Schick ^b and Vito Felice Zenobi ^c
a) Dipartimento di Matematica, Sapienza Università di Roma, Italy
^b) Mathematisches Institut, Universität Göttingen, Germany
^c) Istituto Nazionale di Alta Matematica, Piazzale Aldo Moro 5, 00185 Roma, Italy

Received February 11, 2025, in final form October 19, 2025; Published online October 30, 2025

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.

Key words: positive scalar curvature; universal space for proper actions; spin bordism; fundamental groupoid.

pdf (595 kb)   tex (188 kb)  

References

1. Davis J.F., Lück W., Spaces over a category and assembly maps in isomorphism conjectures in $K$- and $L$-theory, $K$-Theory 15 (1998), 201-252.
2. Hanke B., Positive scalar curvature with symmetry, J. Reine Angew. Math. 614 (2008), 73-115, arXiv:math.GT/0512284.
3. Mayer K.H., $G$-invariante Morse-Funktionen, Manuscripta Math. 63 (1989), 99-114.
4. Milnor J., Lectures on the $h$-cobordism theorem, Princeton University Press, Princeton, NJ, 1965.
5. Piazza P., Schick T., Rho-classes, index theory and Stolz' positive scalar curvature sequence, J. Topol. 7 (2014), 965-1004, arXiv:1210.6892.
6. Piazza P., Schick T., Zenobi V.F., On positive scalar curvature bordism, Comm. Anal. Geom. 30 (2022), 2049-2058, arXiv:1912.09168.
7. Piazza P., Schick T., Zenobi V.F., Mapping analytic surgery to homology, higher rho numbers and metrics of positive scalar curvature, Mem. Amer. Math. Soc. 309 (2025), v+145 pages, arXiv:1905.11861.
8. Schick T., The topology of positive scalar curvature, in Proceedings of the International Congress of Mathematicians - Seoul 2014. Vol. II, Kyung Moon Sa, Seoul, 2014, 1285-1307, arXiv:1405.4220.
9. Schick T., Zenobi V.F., Positive scalar curvature due to the cokernel of the classifying map, SIGMA 16 (2020), 129, 12 pages, arXiv:2006.15965.
10. Stolz S., Concordance classes of positive scalar curvature metrics, Preprint, University of Notre Dame, 1999, https://web.archive.org/web/20240430033820/https://www3.nd.edu/ stolz/concordance.ps.
11. tom Dieck T., Orbittypen und äquivariante Homologie. I, Arch. Math. (Basel) 23 (1972), 307-317.
12. tom Dieck T., Transformation groups, De Gruyter Stud. Math., Vol. 8, Walter de Gruyter & Co., Berlin, 1987.
13. Wasserman A.G., Equivariant differential topology, Topology 8 (1969), 127-150.
14. Xie Z., Yu G., Positive scalar curvature, higher rho invariants and localization algebras, Adv. Math. 262 (2014), 823-866, arXiv:1302.4418.
15. Xie Z., Yu G., Zeidler R., On the range of the relative higher index and the higher rho-invariant for positive scalar curvature, Adv. Math. 390 (2021), 107897, 24 pages, arXiv:1712.03722.
16. Zeidler R., Positive scalar curvature and product formulas for secondary index invariants, J. Topol. 9 (2016), 687-724, arXiv:1412.0685.