Let

be an

*SO*(

*n*)-bundle over a simply connected manifold

*M* with a
spin structure

.
The string class is an obstruction to lift the
structure group

of the loop group bundle

to the
universal central extension of

by the circle. We prove that the
string class vanishes if and only if 1/2 the first Pontrjagin class of

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

*E*_{2}-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.