Homological obstructions to string orientations
Abstract
We observe that the Poincare duality isomorphism for a string manifold is an isomorphism of modules over the subalgebra A(2) 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 term of the annihilator of the Thom class of the negative tangent bundle, and in terms of the vanishing of topdegree cohomology operations. We also indicate how the existence of such an operationpreserving duality implies the integrality of certain polynomials in the Pontryagin classes of the manifold.
 Publication:

arXiv eprints
 Pub Date:
 October 2008
 arXiv:
 arXiv:0810.2131
 Bibcode:
 2008arXiv0810.2131D
 Keywords:

 Mathematics  Algebraic Topology;
 Mathematics  Geometric Topology;
 57R15;
 55P25 (Primary) 55S05;
 57T15 (Secondary)
 EPrint:
 Int. Math. Res. Notices 18 (2011), 40744088