 
Summary: CIRCLEEQUIVARIANT CLASSIFYING SPACES AND THE RATIONAL
EQUIVARIANT SIGMA GENUS
MATTHEW ANDO AND J.P.C.GREENLEES
Abstract. We analyze the circleequivariant spectrum MStringC which is the equivariant ana
logue of the cobordism spectrum MU 6 of stably almost complex manifolds with c1 = c2 = 0.
In [Gre05], the second author showed how to construct the ring Tspectrum EC representing the
Tequivariant elliptic cohomology associated to a rational elliptic curve C. In the case that C is a
complex elliptic curve, we construct a map of ring Tspectra
MStringC EC
which is the rational equivariant analogue of the sigma orientation of [AHS01]. Our method gives
a proof of a conjecture of the first author in [And03b].
Contents
1. Introduction. 1
Part 1. Equivariant classifying spaces and characteristic classes. 4
2. Classifying spaces for equivariant vector bundles. 4
3. Characteristic classes of equivariant bundles. 10
4. Characteristic classes for Tvector bundles. 14
5. The cohomology of covers of BU × Z. 20
Part 2. Elliptic cohomology and the sigma orientation. 25
6. Properties of equivariant elliptic cohomology. 25
