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Formal schemes and formal groups
 in honor of J.M. Boardman, volume 239 of Contemporary Mathematics
, 1999
"... 1.1. Notation and conventions 3 1.2. Even periodic ring spectra 3 2. Schemes 3 ..."
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Cited by 12 (6 self)
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1.1. Notation and conventions 3 1.2. Even periodic ring spectra 3 2. Schemes 3
Orbifold genera, product formulas and power operations
 Adv. Math
, 2006
"... Moore, Verlinde and Verlinde, expressing the orbifold elliptic genus of the symmetric powers of an almost complex manifold M in terms of the elliptic genus of M itself. We show that from the point of view of elliptic cohomology an analogous ptypical statement follows as an easy corollary from the f ..."
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Cited by 7 (4 self)
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Moore, Verlinde and Verlinde, expressing the orbifold elliptic genus of the symmetric powers of an almost complex manifold M in terms of the elliptic genus of M itself. We show that from the point of view of elliptic cohomology an analogous ptypical statement follows as an easy corollary from the fact that the map of spectra corresponding to the genus preserves power operations. We define higher chromatic versions of the notion of orbifold genus, involving htuples rather than pairs of commuting elements. Using homotopy theoretic methods we are able to prove an integrality result and show that our definition is independent of the representation of the orbifold. Our setup is so simple, that it allows us to prove DMVVtype product formulas for these higher chromatic orbifold genera in the same way that the product formula for the topological Todd genus is proved. More precisely, we show that any genus induced by an H∞map into one of the MoravaLubinTate cohomology theories Eh has such a product formula and that the formula depends only on h and not on the genus. Since the complex H∞genera into Eh have been classified in [And95], a large family of genera to which our results apply is completely understood. Loosely speaking, our result says that some Borcherds
Stable splittings and the diagonal
 in Homotopy Methods in Algebraic Topology ( Boulder, CO,1999), A.M.S. Cont.Math.Ser
"... Abstract. Many approximations to function spaces admit natural stable splittings, with a typical example being the stable splitting of a space CdX approximating ΩddX. With an eye towards understanding cup products in the cohomology of such function spaces, we describe how the diagonal interacts wi ..."
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Abstract. Many approximations to function spaces admit natural stable splittings, with a typical example being the stable splitting of a space CdX approximating ΩddX. With an eye towards understanding cup products in the cohomology of such function spaces, we describe how the diagonal interacts with the stable splitting. The description involves group theoretic transfers. In an appendix independent of the rest of the paper, we use ideas from Goodwillie calculus to show that such natural stable splittings are unique, and discuss three dierent constructions showing their existence. 1.
CHERN APPROXIMATIONS FOR GENERALISED GROUP COHOMOLOGY
"... Abstract. Let G be a finite group and E is a suitable generalised cohomology theory. We define and study a ring C(E,G) that is the best possible approximation to E 0 BG that can be built using knowledge of the complex representations of G. We give a description of Q⊗C(E,G) in terms of generalised ch ..."
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Abstract. Let G be a finite group and E is a suitable generalised cohomology theory. We define and study a ring C(E,G) that is the best possible approximation to E 0 BG that can be built using knowledge of the complex representations of G. We give a description of Q⊗C(E,G) in terms of generalised characters, and we study some cases in which the map C(E,G) − → E 0 BG is an isomorphism. Let G be a finite group, and let E ∗ be a generalised cohomology theory, subject to certain technical conditions (“admissibility ” in the sense of [5]) recalled in Section 1. Our aim in this paper is to define and study a certain ring C(E,G) that is in a precise sense the best possible approximation to E 0 BG that can be built using only knowledge of the complex representation theory of G. There is a natural map C(E,G) − → E 0 BG, whose image is the subring C(E,G) ≤ E 0 BG generated over E 0 by all Chern classes of all complex representations. There is ample precedent for considering this subring in the parallel case of ordinary cohomology; see for example [15, 16, 4]. However, although the generators of C(E,G) come from representation theory, the same cannot be said for the relations; one purpose of our construction is to remedy this. We also also develop a kind of generalised character theory which gives good information about Q ⊗ C(E,G).
Orbifold genera, product formulas and . . .
, 2005
"... We generalize the definition of orbifold elliptic genus, and introduce orbifold genera of chromatic level h, using htuples rather than pairs of commuting elements. We show that our genera are in fact orbifold invariants, and we prove integrality results for them. If the genus arises from an H∞map ..."
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We generalize the definition of orbifold elliptic genus, and introduce orbifold genera of chromatic level h, using htuples rather than pairs of commuting elements. We show that our genera are in fact orbifold invariants, and we prove integrality results for them. If the genus arises from an H∞map into the MoravaLubinTate theory Eh, then we give a formula expressing the orbifold genus of the symmetric powers of a stably almost complex manifold M in terms of the genus of M itself. Our formula is the ptypical analogue of the DijkgraafMooreVerlindeVerlinde formula for the orbifold elliptic genus [DMVV97]. It depends only on h and not on the genus.