Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Morava $ K$-theory and the free loop space


Authors: John McCleary and Dennis A. McLaughlin
Journal: Proc. Amer. Math. Soc. 114 (1992), 243-250
MSC: Primary 55P35; Secondary 19L99, 55N20
DOI: https://doi.org/10.1090/S0002-9939-1992-1079897-6
MathSciNet review: 1079897
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We generalize a result of Hopkins, Kuhn, and Ravenel relating the $ n$ th Morava $ K$-theory of the free loop space of a classifying space of a finite group to the $ (n + 1)$ st Morava $ K$-theory of the space. We show that the analogous result holds for any Eilenberg-Mac Lane space for a finite group. We also compute the Morava $ K$-theory of the free loop space of a suspension, and comment on the general problem.


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

  • [1] J.-L. Brylinski, Representations of loop groups, Dirac operators on loop space, and modular forms, Topology 29 (1990), 461-480. MR 1071369 (91j:58151)
  • [2] E. H. Brown and F. P. Peterson, A spectrum whose $ {\mathbb{Z}_p}$-cohomology is the algebra of reduced $ p$th powers, Topology 5 (1966), 149-154. MR 0192494 (33:719)
  • [3] G. E. Carlsson and R. L. Cohen, The cyclic groups and the free loop space, Comment. Math. Helv. 62 (1987), 423-449. MR 910170 (88j:55006)
  • [4] R. L. Cohen, A model for the free loop space of a suspension, Algebraic Topology (Proceedings, Seattle, 1985), Lecture Notes in Math., vol. 1286, Springer-Verlag, New York, 1987, pp. 193-207. MR 922928 (89d:55018)
  • [5] T. Ghazal, A new example in the $ K$-theory of loop spaces, Proc. Amer. Math. Soc. 107 (1989), 855-856. MR 984790 (90h:55006)
  • [6] T. Goodwillie, Cyclic homology, derivations, and the free loop space, Topology 24 (1985), 187-215. MR 793184 (87c:18009)
  • [7] M. J. Hopkins, N. J. Kuhn, and D. C. Ravenel, Generalized group characters and complex oriented cohomology theories (to appear). MR 1758754 (2001k:55015)
  • [8] D. C. Johnson and W. S. Wilson, BP operations and Morava's extraordinary $ K$-theories, Math. Z. 144 (1975), 55-75. MR 0377856 (51:14025)
  • [9] H. Miller, The elliptic character and the Witten genus, Algebraic Topology (Proc. of the Internat. Conf., Evanston, 1988), Contemp. Math. 96 (1989), 281-289. MR 1022688 (90i:55005)
  • [10] D. C. Ravenel and W. S. Wilson, The Morava $ K$-theory of Eilenberg-Mac Lane spaces and the Conner-Floyd conjecture, Amer. J. Math. 102 (1980), 691-748. MR 584466 (81i:55005)
  • [11] G. Segal, Elliptic cohomology, Astérique 161-162 (1988), 187-201. MR 992209 (91b:55005)
  • [12] N. Steenrod and D. B. A. Epstein, Cohomology operations, Princeton Univ. Press, Princeton, NJ, 1962. MR 0145525 (26:3056)
  • [13] E. Witten, Elliptic genera and quantum field theory, Comm. Math. Phys. 109 (1987), 525-536. MR 885560 (89i:57017)
  • [14] -, The index of the Dirac operator in loop space, Elliptic Curves and Modular Forms in Algebraic Topology (Proceedings, Princeton, 1986), Lecture Notes in Math., vol. 1326 Springer-Verlag, New York, 1988, pp. 161-181. MR 970288

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 55P35, 19L99, 55N20

Retrieve articles in all journals with MSC: 55P35, 19L99, 55N20


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1992-1079897-6
Keywords: Free loop space, Morava $ K$-theory
Article copyright: © Copyright 1992 American Mathematical Society

American Mathematical Society