TY - JOUR

T1 - Circle-equivariant classifying spaces and the rational equivariant sigma genus

AU - Ando, Matthew

AU - Greenlees, J. P.C.

N1 - Funding Information:
We began this project when we were participants in the program New contexts for stable homotopy at the Isaac Newton Institute in Autumn 2002. Some of the work was carried out while Ando was a participant in the program “Topological Structures in Physics” at MSRI. Ando was partially supported by NSF grants DMS-0306429 and DMS-0705233, and Greenlees by EPSRC grant EP/C52084X/1.

PY - 2011/12

Y1 - 2011/12

N2 - The circle-equivariant spectrum MStringℂ is the equivariant analogue of the cobordism spectrum MU 〈6〉 of stably almost complex manifolds with c1 = c2 = 0. Given a rational elliptic curve C, Greenlees (Topology 44:1213-1227, 2005) constructs a ring T-spectrum EC representing the associated T-equivariant elliptic cohomology. The core of the present paper is the construction, when C is a complex elliptic curve, of a map of ring T-spectra MStringℂ → EC which is the rational equivariant analogue of the sigma orientation of Ando et al. (Invent. Math. 146:595-687, 2001). We support this by a theory of characteristic classes for calculation, and a conceptual description in terms of algebraic geometry. In particular, we prove a conjecture the first author made in Ando (Geom. Topol. 7:91-153, 2003).

AB - The circle-equivariant spectrum MStringℂ is the equivariant analogue of the cobordism spectrum MU 〈6〉 of stably almost complex manifolds with c1 = c2 = 0. Given a rational elliptic curve C, Greenlees (Topology 44:1213-1227, 2005) constructs a ring T-spectrum EC representing the associated T-equivariant elliptic cohomology. The core of the present paper is the construction, when C is a complex elliptic curve, of a map of ring T-spectra MStringℂ → EC which is the rational equivariant analogue of the sigma orientation of Ando et al. (Invent. Math. 146:595-687, 2001). We support this by a theory of characteristic classes for calculation, and a conceptual description in terms of algebraic geometry. In particular, we prove a conjecture the first author made in Ando (Geom. Topol. 7:91-153, 2003).

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U2 - 10.1007/s00209-010-0773-7

DO - 10.1007/s00209-010-0773-7

M3 - Article

AN - SCOPUS:81555196048

SN - 0025-5874

VL - 269

SP - 1021

EP - 1104

JO - Mathematische Zeitschrift

JF - Mathematische Zeitschrift

IS - 3-4

ER -