TY - JOUR
T1 - Circle-equivariant classifying spaces and the rational equivariant sigma genus
AU - Ando, Matthew
AU - Greenlees, J. P.C.
N1 - 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 \u201CTopological Structures in Physics\u201D 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 -