Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Mobile Device Pairing
Green Open Access
Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Cobordism invariance of the homotopy type of the space of positive scalar curvature metrics


Author: Mark Walsh
Journal: Proc. Amer. Math. Soc. 141 (2013), 2475-2484
MSC (2010): Primary 53C21, 55P10
Published electronically: February 21, 2013
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ X$ and $ Y$ be a pair of smooth manifolds, each obtainable from the other by surgery in codimension at least three. We show that the corresponding spaces $ {\mathcal R}{\mathrm i}{\mathrm e}{\mathrm m}^{+}(X)$ and $ {\mathcal R}{\mathrm i}{\mathrm e}{\mathrm m}^{+}(Y)$, respectively consisting of Riemannian metrics of positive scalar curvature on $ X$ and $ Y$, are homotopy equivalent. This result is originally due to V. Chernysh but remains unpublished.


References [Enhancements On Off] (What's this?)

  • 1. A. L. Besse, Einstein Manifolds, Springer-Verlag, Berlin, 1987. MR 867684 (88f:53087)
  • 2. V. Chernysh, On the homotopy type of the space $ \mathcal {R}^{+}(M)$, Preprint, arXiv:math.GT/0405235
  • 3. M. Gromov and H. B. Lawson, Jr., The classification of simply connected manifolds of positive scalar curvature, Ann. of Math. (2) 111 (1980), 423-434. MR 577131 (81h:53036)
  • 4. R. S. Palais, Homotopy theory of infinite dimensional manifolds, Topology 5 (1966), 1-16. MR 0189028 (32:6455)
  • 5. P. Peterson, Riemannian Geometry, 2nd Edition, Springer, 2006. MR 2243772 (2007a:53001)
  • 6. J. Rosenberg and S. Stolz, Metrics of positive scalar curvature and connections with surgery, Surveys on Surgery Theory, Vol. 2, Ann. of Math. Studies 149, Princeton Univ. Press, 2001. MR 1818778 (2002f:53054)
  • 7. R. Schoen and S.-T. Yau, On the structure of manifolds with positive scalar curvature, Manuscripta Math. 28 (1979), 159-183. MR 535700 (80k:53064)
  • 8. M. Walsh, Metrics of positive scalar curvature and generalised Morse functions, Part I, Memoirs of the American Mathematical Society. Volume 209, No. 983, January 2011. MR 2789750 (2012c:53049)
  • 9. M. Walsh, Metrics of positive scalar curvature and generalised Morse functions, Part II. arXiv.org/0910.2114. To appear in Trans. Amer. Math. Soc.
  • 10. H. Whitney, Differentiable Manifolds, Ann. of Math. (2) 37 (1936), 645-680. MR 1503303

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 53C21, 55P10

Retrieve articles in all journals with MSC (2010): 53C21, 55P10


Additional Information

Mark Walsh
Affiliation: Department of Mathematics, Oregon State University, Corvallis, Oregon 97331
Address at time of publication: Department of Mathematics, Statistics and Physics, Wichita State University, Wichita, Kansas 67260
Email: walsh@math.wichita.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-2013-11647-3
PII: S 0002-9939(2013)11647-3
Received by editor(s): October 10, 2011
Published electronically: February 21, 2013
Communicated by: Daniel Ruberman
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.