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Summary: CIRCLE-EQUIVARIANT CLASSIFYING SPACES AND THE RATIONAL
EQUIVARIANT SIGMA GENUS
MATTHEW ANDO AND J.P.C.GREENLEES
Abstract. We analyze the circle-equivariant 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 T-spectrum EC representing the
T-equivariant 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 T-spectra
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 T-vector 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
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