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The sigma orientation for analytic circle-equivariant elliptic cohomology. (English) Zbl 1021.55004
The author studies in detail the circle-equivariant elliptic cohomology theory constructed by I. Grojnowski (unpublished). It seems that the main achievement of the author is a construction of a canonical Thom class for a certain class of $$S^1$$-equivariant vector bundles: this class generalizes the sigma-orientation constructed by M. Ando, M. J. Hopkins and N. P. Strickland [Invent. Math. 146, No. 3, 595–687 (2001; Zbl 1031.55005)].

##### MSC:
 55N34 Elliptic cohomology 55N22 Bordism and cobordism theories and formal group laws in algebraic topology 57R91 Equivariant algebraic topology of manifolds
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##### References:
 [1] M Ando, M Basterra, The Witten genus and equivariant elliptic cohomology, Math. Z. 240 (2002) 787 · Zbl 1027.55007 [2] M Ando, M J Hopkins, N P Strickland, Elliptic spectra, the Witten genus and the theorem of the cube, Invent. Math. 146 (2001) 595 · Zbl 1031.55005 [3] M Ando, Power operations in elliptic cohomology and representations of loop groups, Trans. Amer. Math. Soc. 352 (2000) 5619 · Zbl 0958.55016 [4] A Borel, Topology of Lie groups and characteristic classes, Bull. Amer. Math. Soc. 61 (1955) 397 · Zbl 0066.02002 [5] J L Brylinski, Representations of loop groups, Dirac operators on loop space, and modular forms, Topology 29 (1990) 461 · Zbl 0715.22023 [6] R Bott, H Samelson, Applications of the theory of Morse to symmetric spaces, Amer. J. Math. 80 (1958) 964 · Zbl 0101.39701 [7] R Bott, C Taubes, On the rigidity theorems of Witten, J. Amer. Math. Soc. 2 (1989) 137 · Zbl 0667.57009 [8] D S Freed, E Witten, Anomalies in string theory with D-branes, Asian J. Math. 3 (1999) 819 · Zbl 1028.81052 [9] V Ginzburg, M Kapranov, É Vasserot, Langlands reciprocity for algebraic surfaces, Math. Res. Lett. 2 (1995) 147 · Zbl 0914.11040 [10] J P C Greenlees, Rational $$\mathrm SO(3)$$-equivariant cohomology theories, Contemp. Math. 271, Amer. Math. Soc. (2001) 99 · Zbl 0995.55001 [11] I Grojnowski, Delocalized equivariant elliptic cohomology, unpublished manuscript (1994) · Zbl 1236.55008 [12] M J Hopkins, Characters and elliptic cohomology, London Math. Soc. Lecture Note Ser. 139, Cambridge Univ. Press (1989) 87 · Zbl 0729.55003 [13] M J Hopkins, Topological modular forms, the Witten genus, and the theorem of the cubeurich, 1994)”, Birkhäuser (1995) 554 · Zbl 0848.55002 [14] V G Kac, Infinite-dimensional Lie algebras, Cambridge University Press (1985) · Zbl 0574.17010 [15] K Liu, On modular invariance and rigidity theorems, J. Differential Geom. 41 (1995) 343 · Zbl 0836.57024 [16] E Looijenga, Root systems and elliptic curves, Invent. Math. 38 (1976/77) 17 · Zbl 0358.17016 [17] A Pressley, G Segal, Loop groups, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press (1986) · Zbl 0618.22011 [18] D Quillen, The spectrum of an equivariant cohomology ring I, II, Ann. of Math. $$(2)$$ 94 (1971) 549, 573 · Zbl 0247.57013 [19] I Rosu, Equivariant elliptic cohomology and rigidity, Amer. J. Math. 123 (2001) 647 · Zbl 0990.55002
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