Homological Obstructions to String Orientations

We observe that the Poincaré duality isomorphism for a string manifold is an isomorphism of modules over the subalgebra of the modulo 2 Steenrod algebra. In particular, the pattern of the operations Sq 1, Sq 2, and Sq 4 on the cohomology of a string manifold has a symmetry around the middle dimension. We characterize this kind of cohomology operation duality in terms of the annihilator of the Thom class of the negative tangent bundle, and in terms of the vanishing of top-degree cohomology operations. We also indicate how the existence of such an operation-preserving duality implies the integrality of certain polynomials in the Pontryagin classes of the manifold.