The Vanishing Problem of the String Class with Degree 3

Let 3⁄4 be an SO.n/-bundle over a simply connected manifold M with a spin structure Q ! M . The string class is an obstruction to lift the structure group LSpin.n/ of the loop group bundle L Q ! L M 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 3⁄4 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 L M 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. 1991 Mathematics subject classification (Amer. Math. Soc.): primary 57R20; secondary 55P35, 57T35.