String structures and canonical 3-forms

Corbett Redden

Vol. 249 (2011), No. 2, 447–484

DOI: 10.2140/pjm.2011.249.447

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.

Keywords

string structure, differential characters, positive Ricci curvature, elliptic cohomology

Mathematical Subject Classification 2000

Primary: 57R15, 58J28

Secondary: 58A14, 55N34, 53C05

Milestones

Received: 11 December 2009

Accepted: 1 July 2010

Published: 1 February 2011