Geometry & Topology, Vol. 7 (2003) Paper no. 3, pages 91--153.

The sigma orientation for analytic circle-equivariant elliptic cohomology

Matthew Ando

Abstract. We construct a canonical Thom isomorphism in Grojnowski's equivariant elliptic cohomology, for virtual T-oriented T-equivariant spin bundles with vanishing Borel-equivariant second Chern class, which is natural under pull-back of vector bundles and exponential under Whitney sum. It extends in the complex-analytic case the non-equivariant sigma orientation of Hopkins, Strickland, and the author. The construction relates the sigma orientation to the representation theory of loop groups and Looijenga's weighted projective space, and sheds light even on the non-equivariant case. Rigidity theorems of Witten-Bott-Taubes including generalizations by Kefeng Liu follow.

Keywords. Sigma orientation, equivariant elliptic cohomolgy, rigidity

AMS subject classification. Primary: 55N34. Secondary: 55N22, 57R91.

DOI: 10.2140/gt.2003.7.91

E-print: arXiv:math.AT/0201092

Submitted to GT on 1 February 2002. (Revised 18 October 2002.) Paper accepted 19 November 2002. Paper published 17 February 2003.

Notes on file formats

Matthew Ando
Department of Mathematics
University of Illinois at Urbana-Champaign
Urbana IL 61801, USA

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