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Journal of the Australian Mathematical Society - Series A
Vol. 65 Part 1 (1998)

The vanishing problem of the string class
with degree 3

Katsuhiko Kuribayashi
Department of Applied Mathematics
Okayama University of Science
1-1 Ridai-cho, Okayama 700
Japan
e-mail: kuri@geom.xmath.ous.ac.jp
and
Toshihiro Yamaguchi
Graduate School of Science and Technology
Okayama University
Okayama 700
Japan
e-mail: t yama@math.okayama-u.ac.jp

Abstract:

Let $\xi$ be an SO(n)-bundle over a simply connected manifold M with a spin structure $Q \to M$. The string class is an obstruction to lift the structure group $\LSpin(n)$ of the loop group bundle $LQ \to LM$ to the universal central extension of $\LSpin(n)$ by the circle. We prove that the string class vanishes if and only if 1/2 the first Pontrjagin class of $\xi$ vanishes when M is a compact simply connected homogeneous space of rank one, a simply connected 4-dimensional manifold or a finite product space of those manifolds. This result is deduced by using the Eilenberg-Moore spectral sequence converging to the mod p cohomology of LM whose E2-term is 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 commutative algebra into the Hochschild homology of a certain graded commutative algebra.

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© Copyright 1998, Australian Mathematical Society
TeXAdel Scientific Publishing
1998-09-18