Differential characters and the Steenrod squares

TL;DR

This paper explores the algebraic structure of differential characters, showing how their squares relate to Steenrod squares, with implications for five-dimensional Chern-Simons theory and invariance properties on spin manifolds.

Contribution

It demonstrates that the squaring map on differential characters factors through Steenrod squares, linking differential topology with algebraic cohomology operations.

Findings

The squaring map reduces to a homomorphism of ordinary cohomology groups.

The homomorphism factors through Steenrod squares.

SL(2, Z)-invariance in five-dimensional Chern-Simons theory on spin manifolds.

Abstract

The groups of differential characters of Cheeger and Simons admit a natural multiplicative structure. The map given by the squares of degree 2k differential characters reduces to a homomorphism of ordinary cohomology groups. We prove that the homomorphism factors through the Steenrod squaring operation of degree 2k. A simple application shows that five-dimensional Chern-Simons theory for pairs of B-fields is SL(2, Z)-invariant on spin manifolds.