Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 17 (2021), 034, 27 pages      arXiv:2005.03073      https://doi.org/10.3842/SIGMA.2021.034
Contribution to the Special Issue on Scalar and Ricci Curvature in honor of Misha Gromov on his 75th Birthday

Homotopy Invariance of the Space of Metrics with Positive Scalar Curvature on Manifolds with Singularities

Boris Botvinnik ^a and Mark G. Walsh ^b
a) Department of Mathematics, University of Oregon, Eugene, OR, 97405, USA
^b) Department of Mathematics and Statistics, Maynooth University, Maynooth, Ireland

Received June 16, 2020, in final form March 24, 2021; Published online April 02, 2021

Abstract
In this paper we study manifolds, $X_{\Sigma}$, with fibred singularities, more specifically, a relevant space ${\mathcal R}^{\rm psc}(X_{\Sigma})$ of Riemannian metrics with positive scalar curvature. Our main goal is to prove that the space ${\mathcal R}^{\rm psc}(X_{\Sigma})$ is homotopy invariant under certain surgeries on $X_{\Sigma}$.

Key words:positive scalar curvature metrics; manifolds with singularities; surgery.

pdf (625 kb)   tex (86 kb)  

References

1. Bérard-Bergery L., Scalar curvature and isometry group, in Spectra of Riemannian Manifolds, Kaigai Publications, Tokyo, 1983, 9-28.
2. Besse A.L., Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), Vol. 10, Springer-Verlag, Berlin, 1987.
3. Botvinnik B., Manifolds with singularities accepting a metric of positive scalar curvature, Geom. Topol. 5 (2001), 683-718, arXiv:math.DG/9910177.
4. Botvinnik B., Ebert J., Randal-Williams O., Infinite loop spaces and positive scalar curvature, Invent. Math. 209 (2017), 749-835, arXiv:1411.7408.
5. Botvinnik B., Piazza P., Rosenberg J., Positive scalar curvature on simply connected spin pseudomanifolds, arXiv:1908.04420.
6. Botvinnik B., Piazza P., Rosenberg J., Positive scalar curvature on spin pseudomanifolds: the fundamental group and secondary invariants, arXiv:2005.02744.
7. Botvinnik B., Rosenberg J., Positive scalar curvature on manifolds with fibered singularities, arXiv:1808.06007.
8. Chernysh V., On the homotopy type of the space $\mathcal{R}^+(M)$, arXiv:math.GT/0405235.
9. Ebert J., Frenck G., The Gromov-Lawson-Chernysh surgery theorem, arXiv:1807.06311.
10. Ebert J., Randal-Williams O., Infinite loop spaces and positive scalar curvature in the presence of a fundamental group, Geom. Topol. 23 (2019), 1549-1610, arXiv:1711.11363.
11. Ebert J., Randal-Williams O., The positive scalar curvature cobordism category, arXiv:1904.12951.
12. Gromov M., Lawson Jr. H.B., The classification of simply connected manifolds of positive scalar curvature, Ann. of Math. 111 (1980), 423-434.
13. Perlmutter N., Cobordism categories and parametrized Morse theory, arXiv:1703.01047.
14. Perlmutter N., Parametrized Morse theory and positive scalar curvature, arXiv:1705.02754.
15. Stolz S., Simply connected manifolds of positive scalar curvature, Ann. of Math. 136 (1992), 511-540.
16. Tuschmann W., Wraith D.J., Moduli spaces of Riemannian metrics, Oberwolfach Seminars, Vol. 46, Birkhäuser Verlag, Basel, 2015.
17. Walsh M., Metrics of positive scalar curvature and generalised Morse functions, Part I, Mem. Amer. Math. Soc. 209 (2011), xviii+80 pages, arXiv:0811.1245.
18. Walsh M., Cobordism invariance of the homotopy type of the space of positive scalar curvature metrics, Proc. Amer. Math. Soc. 141 (2013), 2475-2484, arXiv:1109.6878.
19. Walsh M., $H$-spaces, loop spaces and the space of positive scalar curvature metrics on the sphere, Geom. Topol. 18 (2014), 2189-2243, arXiv:1301.5670.
20. Walsh M., The space of positive scalar curvature metrics on a manifold with boundary, New York J. Math. 26 (2020), 853-930, arXiv:1411.2423.