Classifying sufficiently connected PSC manifolds in 4 and 5 dimensions

Chodosh, Otis¹ ; Li, Chao² ; Liokumovich, Yevgeny³

¹ Department of Mathematics, Stanford University, Stanford, CA, United States
² Courant Institute of Mathematical Sciences, New York University, New York, NY, United States
³ Department of Mathematics, University of Toronto, Toronto ON, Canada

Geometry & topology, Tome 27 (2023) no. 4, pp. 1635-1655.

Résumé

We show that if $N$ is a closed manifold of dimension $n=4$ (resp. $n=5$) with ${\pi }_2(N)=0$ (resp. ${\pi }_2(N)={\pi }_3(N)=0$) that admits a metric of positive scalar curvature, then a finite cover $\widehat{N}$ of $N$ is homotopy equivalent to $S^n$ or connected sums of $S^{n-1}\times S^1$. Our approach combines recent advances in the study of positive scalar curvature with a novel argument of Alpert, Balitskiy and Guth.

Additionally, we prove a more general mapping version of this result. In particular, this implies that if $N$ is a closed manifold of dimensions $4$ or $5$, and $N$ admits a map of nonzero degree to a closed aspherical manifold, then $N$ does not admit any Riemannian metric with positive scalar curvature.

Détail

DOI : 10.2140/gt.2023.27.1635

Keywords: positive scalar curvature, Urysohn width, minimal surfaces, $\mu$–bubbles

Bibliographie

[1] H Alpert, A Balitskiy, L Guth, Macroscopic scalar curvature and codimension 2 width, preprint (2021)

[2] D Bolotov, About the macroscopic dimension of certain PSC-manifolds, Algebr. Geom. Topol. 9 (2009) 21 | DOI

[3] D Bolotov, A Dranishnikov, On Gromov’s scalar curvature conjecture, Proc. Amer. Math. Soc. 138 (2010) 1517 | DOI

[4] S Brendle, Ricci flow with surgery on manifolds with positive isotropic curvature, Ann. of Math. 190 (2019) 465 | DOI

[5] S Brendle, R Schoen, Manifolds with 1∕4–pinched curvature are space forms, J. Amer. Math. Soc. 22 (2009) 287 | DOI

[6] B L Chen, S H Tang, X P Zhu, Complete classification of compact four-manifolds with positive isotropic curvature, J. Differential Geom. 91 (2012) 41

[7] O Chodosh, C Li, Generalized soap bubbles and the topology of manifolds with positive scalar curvature, preprint (2020)

[8] M J Dunwoody, The accessibility of finitely presented groups, Invent. Math. 81 (1985) 449 | DOI

[9] D B A Epstein, Ends, from: "Topology of –manifolds and related topics" (editor M K Fort Jr.), Prentice-Hall (1962) 110

[10] A M Fraser, Fundamental groups of manifolds with positive isotropic curvature, Ann. of Math. 158 (2003) 345 | DOI

[11] M H Freedman, The topology of four-dimensional manifolds, J. Differential Geometry 17 (1982) 357

[12] S Gadgil, H Seshadri, On the topology of manifolds with positive isotropic curvature, Proc. Amer. Math. Soc. 137 (2009) 1807 | DOI

[13] M Gromov, Positive curvature, macroscopic dimension, spectral gaps and higher signatures, from: "Functional analysis on the eve of the 21st century, II" (editors S Gindikin, J Lepowsky, R L Wilson), Progr. Math. 132, Birkhäuser (1996) 1 | DOI

[14] M Gromov, No metrics with positive scalar curvatures on aspherical 5–manifolds, preprint (2020)

[15] M Gromov, Four lectures on scalar curvature, preprint (2021)

[16] M Gromov, H B Lawson Jr., Spin and scalar curvature in the presence of a fundamental group, I, Ann. of Math. 111 (1980) 209 | DOI

[17] M Gromov, H B Lawson Jr., Positive scalar curvature and the Dirac operator on complete Riemannian manifolds, Inst. Hautes Études Sci. Publ. Math. 58 (1983) 83 | DOI

[18] R S Hamilton, Four-manifolds with positive isotropic curvature, Comm. Anal. Geom. 5 (1997) 1 | DOI

[19] A Hatcher, Algebraic topology, Cambridge Univ. Press (2002)

[20] H Huang, Compact manifolds of dimension n ≥ 12 with positive isotropic curvature, preprint (2019)

[21] M Katz, The first diameter of 3–manifolds of positive scalar curvature, Proc. Amer. Math. Soc. 104 (1988) 591 | DOI

[22] M Kreck, W Lück, Topological rigidity for non-aspherical manifolds, Pure Appl. Math. Q. 5 (2009) 873 | DOI

[23] Y Liokumovich, D Maximo, Waist inequality for 3–manifolds with positive scalar curvature, preprint (2020)

[24] F C Marques, A Neves, Rigidity of min-max minimal spheres in three-manifolds, Duke Math. J. 161 (2012) 2725 | DOI

[25] M J Micallef, J D Moore, Minimal two-spheres and the topology of manifolds with positive curvature on totally isotropic two-planes, Ann. of Math. 127 (1988) 199 | DOI

[26] J Milnor, Lectures on the h–cobordism theorem, Princeton Univ. Press (1965)

[27] H T Nguyen, Isotropic curvature and the Ricci flow, Int. Math. Res. Not. 2010 (2010) 536 | DOI

[28] M Ramachandran, J Wolfson, Fill radius and the fundamental group, J. Topol. Anal. 2 (2010) 99 | DOI

[29] R Schoen, Manifolds with poinwise 1∕4-pinched curvature, preprint (2007)

[30] P Scott, T Wall, Topological methods in group theory, from: "Homological group theory" (editor C T C Wall), London Math. Soc. Lecture Note Ser. 36, Cambridge Univ. Press (1979) 137 | DOI

[31] J P Serre, Trees, Springer (2003) | DOI