Journal of the Australian Mathematical Society

Research Article

A LOOP SPACE FORMULATION FOR GEOMETRIC LIFTING PROBLEMS

KONRAD WALDORFa1 p1

a1 Department of Mathematics, University of California, 970 Evans Hall #3840, Berkeley, CA 94720, USA

Abstract

We review and then combine two aspects of the theory of bundle gerbes. The first concerns lifting bundle gerbes and connections on those, developed by Murray and by Gomi. Lifting gerbes represent obstructions against extending the structure group of a principal bundle. The second is the transgression of gerbes to loop spaces, initiated by Brylinski and McLaughlin and with recent contributions of the author. Combining these two aspects, we obtain a new formulation of lifting problems in terms of geometry on the loop space. Most prominently, our formulation explains the relation between (complex) spin structures on a Riemannian manifold and orientations of its loop space.

(Received July 29 2010)

(Accepted December 16 2010)

(Online publication June 09 2011)

2010 Mathematics subject classification

  • primary 53C08; secondary 55P35;
  • 53C27

Keywords and phrases

  • lifting problem;
  • bundle gerbe;
  • transgression;
  • loop space

Correspondence:

p1 Current address: Fakultät für Mathematik, Universität Regensburg, Universitätsstraße 31, 93053 Regensburg, Germany (email: konrad.waldorf@mathematik.uni-regensburg.de)

Footnotes

Communicated by V. Mathai

Dedicated to Alan Carey, on the occasion of his 60th birthday