Circle actions and scalar curvature

Wiemeler, Michael

doi:10.1090/tran/6666

Circle actions and scalar curvature
HTML articles powered by AMS MathViewer

by Michael Wiemeler PDF

Trans. Amer. Math. Soc. 368 (2016), 2939-2966 Request permission

Abstract:

We construct metrics of positive scalar curvature on manifolds with circle actions. One of our main results is that there exist $S^1$-invariant metrics of positive scalar curvature on every $S^1$-manifold which has a fixed point component of codimension $2$. As a consequence we can prove that there are non-invariant metrics of positive scalar curvature on many manifolds with circle actions. Results from equivariant bordism allow us to show that there is an invariant metric of positive scalar curvature on the connected sum of two copies of a simply connected semi-free $S^1$-manifold $M$ of dimension at least six provided that $M$ is not $\text {spin}$ or that $M$ is $\text {spin}$ and the $S^1$-action is of odd type. If $M$ is spin and the $S^1$-action of even type, then there is a $k>0$ such that the equivariant connected sum of $2^k$ copies of $M$ admits an invariant metric of positive scalar curvature if and only if a generalized $\hat {A}$-genus of $M/S^1$ vanishes.

References

J. F. Adams, On the groups $J(X)$. IV, Topology 5 (1966), 21–71. MR 198470, DOI 10.1016/0040-9383(66)90004-8
D. W. Anderson, E. H. Brown Jr., and F. P. Peterson, The structure of the Spin cobordism ring, Ann. of Math. (2) 86 (1967), 271–298. MR 219077, DOI 10.2307/1970690
L. Bérard Bergery, Scalar curvature and isometry group, Spectra of Riemannian Manifolds, Kaigai Publications, Tokyo, 1983, pp. 9–28.
Lucília Daruiz Borsari, Bordism of semifree circle actions on Spin manifolds, Trans. Amer. Math. Soc. 301 (1987), no. 2, 479–487. MR 882700, DOI 10.1090/S0002-9947-1987-0882700-0
Glen E. Bredon, A $\Pi _\ast$-module structure for $\Theta _\ast$ and applications to transformation groups, Ann. of Math. (2) 86 (1967), 434–448. MR 221518, DOI 10.2307/1970609
Glen E. Bredon, Introduction to compact transformation groups, Pure and Applied Mathematics, Vol. 46, Academic Press, New York-London, 1972. MR 0413144
V. M. Bukhshtaber and N. Raĭ, Toric manifolds and complex cobordisms, Uspekhi Mat. Nauk 53 (1998), no. 2(320), 139–140 (Russian); English transl., Russian Math. Surveys 53 (1998), no. 2, 371–373. MR 1639388, DOI 10.1070/rm1998v053n02ABEH000011
Victor M. Buchstaber and Nigel Ray, Tangential structures on toric manifolds, and connected sums of polytopes, Internat. Math. Res. Notices 4 (2001), 193–219. MR 1813798, DOI 10.1155/S1073792801000125
P. E. Conner and E. E. Floyd, Differentiable periodic maps, Ergebnisse der Mathematik und ihrer Grenzgebiete, (N.F.), Band 33, Academic Press, Inc., Publishers, New York; Springer-Verlag, Berlin-Göttingen-Heidelberg, 1964. MR 0176478
Harold Donnelly and Reinhard Schultz, Compact group actions and maps into aspherical manifolds, Topology 21 (1982), no. 4, 443–455. MR 670746, DOI 10.1016/0040-9383(82)90022-2
PawełGajer, Riemannian metrics of positive scalar curvature on compact manifolds with boundary, Ann. Global Anal. Geom. 5 (1987), no. 3, 179–191. MR 962295, DOI 10.1007/BF00128019
Mikhael Gromov and H. Blaine Lawson Jr., The classification of simply connected manifolds of positive scalar curvature, Ann. of Math. (2) 111 (1980), no. 3, 423–434. MR 577131, DOI 10.2307/1971103
Bernhard Hanke, Positive scalar curvature with symmetry, J. Reine Angew. Math. 614 (2008), 73–115. MR 2376283, DOI 10.1515/CRELLE.2008.003
Akio Hattori and Tomoyoshi Yoshida, Lifting compact group actions in fiber bundles, Japan. J. Math. (N.S.) 2 (1976), no. 1, 13–25. MR 461538, DOI 10.4099/math1924.2.13
Michael Joachim and Thomas Schick, Positive and negative results concerning the Gromov-Lawson-Rosenberg conjecture, Geometry and topology: Aarhus (1998), Contemp. Math., vol. 258, Amer. Math. Soc., Providence, RI, 2000, pp. 213–226. MR 1778107, DOI 10.1090/conm/258/04066
Vappala J. Joseph, Smooth actions of the circle group on exotic spheres, Pacific J. Math. 95 (1981), no. 2, 323–336. MR 632190
Katsuo Kawakubo, The theory of transformation groups, Translated from the 1987 Japanese edition, The Clarendon Press, Oxford University Press, New York, 1991. MR 1150492
H. Blaine Lawson Jr. and Shing Tung Yau, Scalar curvature, non-abelian group actions, and the degree of symmetry of exotic spheres, Comment. Math. Helv. 49 (1974), 232–244. MR 358841, DOI 10.1007/BF02566731
John Lott, Signatures and higher signatures of $S^1$-quotients, Math. Ann. 316 (2000), no. 4, 617–657. MR 1758446, DOI 10.1007/s002080050347
J. P. May, Equivariant homotopy and cohomology theory,\nopunct: With contributions by M. Cole, G. Comezaña, S. Costenoble, A. D. Elmendorf, J. P. C. Greenlees, L. G. Lewis, Jr., R. J. Piacenza, G. Triantafillou, and S. Waner,\nopunct, CBMS Regional Conference Series in Mathematics, vol. 91, published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1996. MR1413302 (97k:55016)
Karl Heinz Mayer, $G$-invariante Morse-Funktionen, Manuscripta Math. 63 (1989), no. 1, 99–114 (German, with English summary). MR 975472, DOI 10.1007/BF01173705
John W. Milnor, Remarks concerning spin manifolds, Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), Princeton Univ. Press, Princeton, N.J., 1965, pp. 55–62. MR 0180978
Kaoru Ono, $\alpha$-invariant and $S^1$ actions, Proc. Amer. Math. Soc. 112 (1991), no. 2, 597–600. MR 1043417, DOI 10.1090/S0002-9939-1991-1043417-1
Thomas Schick, A counterexample to the (unstable) Gromov-Lawson-Rosenberg conjecture, Topology 37 (1998), no. 6, 1165–1168. MR 1632971, DOI 10.1016/S0040-9383(97)00082-7
R. Schoen and S. T. Yau, On the structure of manifolds with positive scalar curvature, Manuscripta Math. 28 (1979), no. 1-3, 159–183. MR 535700, DOI 10.1007/BF01647970
Reinhard Schultz, Circle actions on homotopy spheres not bounding spin manifolds, Trans. Amer. Math. Soc. 213 (1975), 89–98. MR 380853, DOI 10.1090/S0002-9947-1975-0380853-6
Stephan Stolz, Concordance classes of positive scalar curvature, preprint.
Fuichi Uchida, Cobordism groups of semi-free $S^{1}$- and $S^{3}$-actions, Osaka Math. J. 7 (1970), 345–351. MR 278338
Jaak Vilms, Totally geodesic maps, J. Differential Geometry 4 (1970), 73–79. MR 262984