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. PDF file size: 106K © Copyright 1998, Australian Mathematical Society