Hostname: page-component-7bb8b95d7b-wpx69 Total loading time: 0 Render date: 2024-09-21T18:39:35.751Z Has data issue: false hasContentIssue false
  • English
  • Français

The Vanishing problem of the string class with degree 3

Part of: Homotopy theory Differential topology Homology and homotopy of topological groups and related structures

Published online by Cambridge University Press:  09 April 2009

Katsuhiko Kuribayashi  and
Toshihiro Yamaguchi
Katsuhiko Kuribayashi
Affiliation:
Department of Applied Mathematics, Okayama University of Science, 1-1 Ridai-cho, Okayama 700, Japan e-mail: kuri@geom.xmath.ous.ac.jp
Toshihiro Yamaguchi
Affiliation:
Graduate School of Science and Technology, Okayama University, Okayama 700, Japan e-mail: t_ yama@math.okayama-u.ac.jp
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let ξbe an SO(n)-bundle over a simple connected manifold M with a spin structure Q → M. The string class is an obstruction to h1 the structure group LSpin(n) of the loop group bundle LQ → LM to the universal central extension of LSpain(n) by the circle. We prove that the string class vanishes if and only if 1/2 the first Pontrjagin clsss of values when M is a compact simply connected homogeneous space of rank one, a simpiy connected 4 dimensional manifold or a finite product space of those manifolds. This result is deduced by using the Eclesberg spectral sequence converging to the mod p cohomology of LM whose E2-term to the Hochschild homology of the mod p cohomology algebra of M. The key to the consideration is existence of a morphism of algebras, which is injective below degree 3, from an important graded commutator algebra into the Hochschild homology of a certain graded commutative algebra.

MSC classification

Secondary: 57R20: Characteristic classes and numbers 55P35: Loop spaces 57T35: Applications of Eilenberg-Moore spectral sequences
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 65 , Issue 1 , August 1998 , pp. 129 - 142
Copyright
Copyright © Australian Mathematical Society 1998

References

[1]Burghelea, D. and Vigué-Poirrier, M., Cyclic homology of commutative algebras I, Lecture Notes Math. 1318 (Springer, New York, 1990).Google Scholar
[2]Dynkin, E. B., ‘Semisimple subalgebras of semisimple Lie algebras’, Mat. Sb. 30 (1952), 349–462 [Russian],Google Scholar
Amer Math. Soc. Transl. 6 (1957), 111–244.Google Scholar
[3]Kuribayashi, K., ‘On the mod p cohomology of spaces of free loops on the Grassmann and Stiefel manifolds’, J. Math. Soc. Japan 43 (1991), 331–346.CrossRefGoogle Scholar
[4]Kuribayashi, K., ‘On the vanishing problem of string classes’, J. Austral Math. Soc. (Series A) 61 (1996), 258–266.CrossRefGoogle Scholar
[5]Kuribayashi, K., ‘Module derivations and the adjoint action of finite loop space’, prepnnt Okayama University of Science, 1998.CrossRefGoogle Scholar
[6]Kuribayashi, K. and Yamaguchi, T., ‘The cohomology algebra of certain free loop spaces’, Fund. Math. 154(1997), 57–73.CrossRefGoogle Scholar
[7]McCleary, J. and Ziller, W., ‘On the free loop space of homogeneous spaces’, Amer J. Math. 109 (1987), 765–782.CrossRefGoogle Scholar
Correction: Amer. J. Math 113 (1991), 375–377.Google Scholar
[8]McLaughlin, D. A., ‘Orientation and string structures on loop space, Pacific J. Math. 155 (1992), 143–156.CrossRefGoogle Scholar
[9]Oniscik, A. L., ‘Transitive compact transformation groups’, Mat. Sb. 60(1663), 447–485 [Russian],Google Scholar
Amer Math. Soc. Transl. 55 (1966), 153–194.Google Scholar
[10]Smith, L., ‘Homological algebra and the Eilenberg-Moore spectral sequence’, Trans. Amer. Math. Soc. 129 (1967), 58–93CrossRefGoogle Scholar
[11]Smith, L., ‘On the characteristic zero cohomology of the free loop space’, Amer J. Math 103 (1981), 887–910.CrossRefGoogle Scholar