Geometry & Topology, Vol. 9 (2005) Paper no. 11, pages 341--373.

The index of projective families of elliptic operators

Varghese Mathai, Richard B Melrose, Isadore M Singer

Abstract. An index theory for projective families of elliptic pseudodifferential operators is developed. The topological and the analytic index of such a family both take values in twisted K-theory of the parametrizing space, X. The main result is the equality of these two notions of index when the twisting class is in the torsion subgroup of H^3(X;Z). The Chern character of the index class is then computed.

Keywords. Projective vector bundles, twisted K-theory, projective families of elliptic operators, Index theorem, determinant lines, twisted Chern character

AMS subject classification. Primary: 19K56. Secondary: 58J20.

DOI: 10.2140/gt.2005.9.341

E-print: arXiv:math.DG/0206002

Submitted to GT on 7 December 2004. Paper accepted 28 February 2005. Paper published 1 March 2005.

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Varghese Mathai, Richard B Melrose, Isadore M Singer
Department of Pure Mathematics, University of Adelaide
Adelaide 5005, Australia
Department of Mathematics, Massachusetts Institute of Technology
Cambridge, Mass 02139, USA
Department of Mathematics, Massachusetts Institute of Technology
C ambridge, Mass 02139, USA

Email: vmathai@maths.adelaide.edu.au, rbm@math.mit.edu, ims@math.mit.edu

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