Positive scalar curvature and odd order abelian fundamental groups
HTML articles powered by AMS MathViewer
- by Reinhard Schultz PDF
- Proc. Amer. Math. Soc. 125 (1997), 907-915 Request permission
Abstract:
If a smooth manifold has a Riemannian metric with positive scalar curvature, it follows immediately that the universal covering also has such a metric. The paper establishes a converse if the manifold in question is closed of dimension at least 5 and the fundamental group is an elementary abelian $p$-group of rank 2, where $p$ is an odd prime.References
- M. F. Atiyah, $K$-theory, 2nd ed., Advanced Book Classics, Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA, 1989. Notes by D. W. Anderson. MR 1043170
- Nils Andreas Baas, Bordism theories with singularities, Proceedings of the Advanced Study Institute on Algebraic Topology (Aarhus Univ., Aarhus, 1970) Various Publ. Ser., No. 13, Mat. Inst., Aarhus Univ., Aarhus, 1970, pp. 1–16. MR 0346823
- J. C. Becker and R. E. Schultz, Equivariant function spaces and stable homotopy theory. I, Comment. Math. Helv. 49 (1974), 1–34. MR 339232, DOI 10.1007/BF02566716
- Richard Holzsager, Stable splitting of $K(G,\,1)$, Proc. Amer. Math. Soc. 31 (1972), 305–306. MR 287540, DOI 10.1090/S0002-9939-1972-0287540-6
- Thomas W. Hungerford, Multiple Künneth formulas for abelian groups, Trans. Amer. Math. Soc. 118 (1965), 257–275. MR 175953, DOI 10.1090/S0002-9947-1965-0175953-3
- R. Jung, Ph. D. Thesis, Univ. Mainz, in preparation.
- Sławomir Kwasik and Reinhard Schultz, Positive scalar curvature and periodic fundamental groups, Comment. Math. Helv. 65 (1990), no. 2, 271–286. MR 1057244, DOI 10.1007/BF02566607
- —, Fake spherical spaceforms with constant positive scalar curvature, Comment. Math. Helv. 71 (1996), 1–40.
- Peter S. Landweber, Künneth formulas for bordism theories, Trans. Amer. Math. Soc. 121 (1966), 242–256. MR 192503, DOI 10.1090/S0002-9947-1966-0192503-7
- Peter S. Landweber, Complex bordism of classifying spaces, Proc. Amer. Math. Soc. 27 (1971), 175–179. MR 268885, DOI 10.1090/S0002-9939-1971-0268885-1
- Stephen A. Mitchell and Stewart B. Priddy, Stable splittings derived from the Steinberg module, Topology 22 (1983), no. 3, 285–298. MR 710102, DOI 10.1016/0040-9383(83)90014-9
- Jonathan Rosenberg, $C^{\ast }$-algebras, positive scalar curvature, and the Novikov conjecture, Inst. Hautes Études Sci. Publ. Math. 58 (1983), 197–212 (1984). MR 720934, DOI 10.1007/BF02953775
- J. Rosenberg, $C^\ast$-algebras, positive scalar curvature and the Novikov conjecture. II, Geometric methods in operator algebras (Kyoto, 1983) Pitman Res. Notes Math. Ser., vol. 123, Longman Sci. Tech., Harlow, 1986, pp. 341–374. MR 866507
- Jonathan Rosenberg, $C^\ast$-algebras, positive scalar curvature, and the Novikov conjecture. III, Topology 25 (1986), no. 3, 319–336. MR 842428, DOI 10.1016/0040-9383(86)90047-9
- J. Rosenberg and S. Stolz, A “stable” version of the Gromov-Lawson conjecture, Contemp. Math., vol. 181, Amer. Math. Soc., Providence, RI, 1995, pp. 405–418.
- Russell J. Rowlett, Free actions of a $p$-group on two generators, Indiana Univ. Math. J. 26 (1977), no. 5, 885–889. MR 458453, DOI 10.1512/iumj.1977.26.26071
Additional Information
- Reinhard Schultz
- Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
- Address at time of publication: Department of Mathematics, University of California, Riverside, California 92521
- MR Author ID: 157165
- Email: schultz@math.ucr.edu
- Received by editor(s): February 13, 1995
- Received by editor(s) in revised form: September 13, 1995
- Communicated by: Thomas Goodwillie
- © Copyright 1997 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 125 (1997), 907-915
- MSC (1991): Primary 53C21, 55N15, 57R75; Secondary 53C20, 57R85
- DOI: https://doi.org/10.1090/S0002-9939-97-03683-6
- MathSciNet review: 1363184