Homological obstructions to string orientations

TL;DR

This paper explores the algebraic and topological structures underlying string manifolds, revealing symmetries in cohomology operations and their implications for Pontryagin classes and manifold orientations.

Contribution

It characterizes the cohomology operation duality in string manifolds using Steenrod algebra actions and links this to the integrality of polynomials in Pontryagin classes.

Findings

Poincare duality for string manifolds respects A(2) Steenrod algebra modules.

Symmetry in Sq^1, Sq^2, Sq^4 operations around the middle dimension.

Duality implies integrality conditions on Pontryagin class polynomials.

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 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.