Vol. 249, No. 2, 2011

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String structures and canonical 3-forms

Corbett Redden

Vol. 249 (2011), No. 2, 447–484

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.

string structure, differential characters, positive Ricci curvature, elliptic cohomology
Mathematical Subject Classification 2000
Primary: 57R15, 58J28
Secondary: 58A14, 55N34, 53C05
Received: 11 December 2009
Accepted: 1 July 2010
Published: 1 February 2011
Corbett Redden
Department of Mathematics
Michigan State University
A-318 Wells Hall
East Lansing, MI 48824
United States