String structures and canonical 3-forms

TL;DR

This paper explores the relationship between string structures on spin bundles, cohomology classes, and canonical 3-forms, providing explicit calculations and discussing implications for geometric obstructions.

Contribution

It introduces a cohomological classification of string structures and links harmonic representatives to canonical 3-forms, with explicit examples and potential applications in geometry.

Findings

String classes correspond to degree 3 cohomology classes on bundle total spaces.

Harmonic representatives of string classes yield canonical 3-forms on the base.

Explicit calculation of 3-forms for homogeneous 3-spheres.

Abstract

Using basic homotopy constructions, we show that isomorphism classes of string structures on spin bundles are naturally given by certain degree 3 cohomology classes, which we call string classes, on the total space of the bundle. Using a Hodge isomorphism, we then show that the harmonic representative of a string class gives rise to a canonical 3-form on the base space, refining the associated differential character. We explicitly calculate this 3-form for homogeneous metrics on 3-spheres, and we discuss how the cohomology theory tmf could potentially encode obstructions to positive Ricci curvature metrics.