THE VANISHING PROBLEM OF THE STRING CLASS WITH DEGREE-3

Let xi be an SO(n)-bundle over a simply connected manifold M with a sp in structure Q --> M. The string class is an obstruction to lift the s tructure group LSpin(n) of the loop group bundle LQ --> LM to the univ ersal central extension of LSpin(n) by the circle. We prove that tine 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 produ ct space of those manifolds. This result is deduced by using the Eilen berg-Moore spectral sequence converging to the mod p cohomology of LM whose E-2-term is the Hochschild homology of the mod p cohomology alge bra of M. The key to the consideration is existence of a morphism of a lgebras, which is injective below degree 3, from an important graded c ommutative algebra into the Hochschild homology of a certain graded co mmutative algebra.