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The Vanishing problem of the string class with degree 3

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

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: [email protected]
Toshihiro Yamaguchi
Affiliation:
Graduate School of Science and Technology, Okayama University, Okayama 700, Japan e-mail: t_ [email protected]
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Abstract

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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

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