Equivariant Stable Homotopy Theory

Lewis, L; May, J. P.; Steinberger, Mark

doi:10.1007/bfb0075778

Cited by 579 publications

(886 citation statements)

References 27 publications

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“…The free loopspace LX of a CW-space X is weakly T-homotopy equivalent to a T-CW-space, by a map which preserves the fixed-point structure [25] ( §1.1).…”

Section: An Isomorphism Of Pro-objects Moreover If the Fixedpoint Smentioning

confidence: 99%

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Thom prospectra for loopgroup representations

Kitchloo¹,

Morava²

2007

Elliptic Cohomology

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Abstract. This is very much an account of work in progress. We sketch the construction of an Atiyah dual (in the category of T-spaces) for the free loopspace of a manifold; the main technical tool is a kind of Tits building for loop groups, discussed in detail in an appendix. Together with a new localization theorem for T-equivariant K-theory, this yields a construction of the elliptic genus in the string topology framework of Chas-Sullivan, Cohen-Jones, Dwyer, Klein, and others. We also show how the Tits building can be used to construct the dualizing spectrum of the loop group. Using a tentative notion of equivariant K-theory for loop groups, we relate the equivariant K-theory of the dualizing spectrum to recent work of Freed, Hopkins and Teleman.

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“…The free loopspace LX of a CW-space X is weakly T-homotopy equivalent to a T-CW-space, by a map which preserves the fixed-point structure [25] ( §1.1).…”

Section: An Isomorphism Of Pro-objects Moreover If the Fixedpoint Smentioning

confidence: 99%

“…Let E denote the union of the one-point compactifications of all representations of T which do not contain the trivial representation (cf. [25]). Then the theory E T ∧ E satisfies a strong localization theorem.…”

Section: Theorem 54mentioning

confidence: 99%

Thom prospectra for loopgroup representations

Kitchloo¹,

Morava²

2007

Elliptic Cohomology

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“…Then C(n, j) is equivariantly homotopy equivalent to S nj − ∆(n, j) (the configuration space). Thus, by equivariant Alexander duality [LMMS,Theorem III.4.1], F (C(n, j) + , (Σ n X) [j] ) S nj /∆(n, j) ∧ X [j] as Σ j spectra. Now note that this latter spectrum is clearly Σ j -free, as S nj /∆(n, j) is, thus its fixed point spectrum is naturally equivalent to its orbit spectrum [LMMS,Theorem II.7.1].…”

Section: Towards the Conjecturesmentioning

confidence: 99%

“…Thus, by equivariant Alexander duality [LMMS,Theorem III.4.1], F (C(n, j) + , (Σ n X) [j] ) S nj /∆(n, j) ∧ X [j] as Σ j spectra. Now note that this latter spectrum is clearly Σ j -free, as S nj /∆(n, j) is, thus its fixed point spectrum is naturally equivalent to its orbit spectrum [LMMS,Theorem II.7.1]. We have proved Proposition A.1.D n,j (Σ n X) is naturally equivalent to ((S nj /∆(n, j)) ∧ X [j] ) Σj .…”

Section: Towards the Conjecturesmentioning

confidence: 99%

New relationships among loopspaces, symmetric products, and Eilenberg Maclane spaces

Kuhn

2001

Cohomological Methods in Homotopy Theory

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Abstract. Let T (j) be the dual of the j th stable summand of Ω 2 S 3 (at the prime 2) with top class in dimension j. Then it is known that T (j) is a retract of a suspension spectrum, and that the homotopy colimit of a certain sequence T (j) → T (2j) → . . . is an infinite wedge of stable summands of K(V, 1)'s, where V denotes an elementary abelian 2 group. In particular, when one starts with T (1), one gets K(Z/2, 1) = RP ∞ as one of the summands.I discuss a generalization of this picture using higher iterated loopspaces and Eilenberg MacLane spaces. I consider certain finite spectra T (n, j) for n, j ≥ 0 (with T (1, j) = T (j)), dual to summands of Ω n+1 S N , conjecture generalizations of the above, and prove that these conjectures are correct in cohomology. So, for example, T (n, j) has unstable cohomology, and the cohomology of the hocolimit of a certain sequence T (n, j) → T (n, 2j) → . . . agrees with the cohomology of the wedge of stable summands of K(V, n)'s corresponding to the wedge occurring in the n = 1 case above.One can also map the T (n, j) to each other as n varies, and here the cohomological calculations imply a homotopical conclusion: the hocolimits that are nonzero, T (∞, 2 k ), for k ≥ 0, map to each other, giving rise to a fitration of HZ/2 which is equivalent to the mod 2 symmetric powers of spheres filtration.Our homotopical constructions use Hopf invariant methods and loopspace technology. These are quite general and should be of independent interest.To study the action of the Steenrod operations on the cohomology of our spectra, we derive a Nishida formula for how χ(Sq i ) acts on Dyer-Lashof operations. This should be of use in other settings.In an appendix, we explain connections with recent work by Greg Arone and Mark Mahowald on the Goodwillie tower of the identity.

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“…In particular, (S G ) * is the equivariant stable homotopy groups of spheres and (S G ) 0 is isomorphic to the Burnside ring A(G). The ring A(G), and more so its localizations at subrings of the rationals, usually does have non-trivial idempotents [5,8].…”

mentioning

confidence: 99%

Idempotents and Landweber exactness in brave new algebra

May

2001

Homology, Homotopy and Applications

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We explain how idempotents in homotopy groups give rise to splittings of homotopy categories of modules over commutative S-algebras, and we observe that there are naturally occurring equivariant examples involving idempotents in Burnside rings. We then give a version of the Landweber exact functor theorem that applies to M U -modules.In 1997, not long after [6] was written, I gave an April Fool's talk on how to prove that BP is an E ∞ ring spectrum or equivalently, in the language of [6], a commutative S-algebra. Unfortunately, the problem of whether or not BP is an E ∞ ring spectrum remains open. However, two interesting remarks emerged and will be presented here. One concerns splittings along idempotents and the other concerns the Landweber exact functor theorem.One of the nicest things in [6] is its one line proof that KO and KU are commutative S-algebras. This is an application of the following theorem [6, VIII.2.2], or rather the special case that follows. Theorem 1. Let R be a cell commutative S-algebra, A be a cell commutative Ralgebra, and M be a cell R-module. Then the Bousfield localization λ : A −→ A M ofA at M can be constructed as the inclusion of a subcomplex in a cell commutative R-algebra. In particular, the commutative R-algebra A M is a commutative S-algebra by neglect of structure.The cell assumptions can always be arranged by use of the cofibrant replacement constructions in [6], so they result in no loss of generality. The theorem specializes as follows to algebraic localizations at elements of R * = π * (R) [6, VIII.4.2]. The connective real K-theory spectrum ko is a commutative S-algebra by multiplicative infinite loop space theory [11], and KO is the localization ko[β −1 ] obtained

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Equivariant Stable Homotopy Theory

Cited by 579 publications

References 27 publications

Thom prospectra for loopgroup representations

Thom prospectra for loopgroup representations

New relationships among loopspaces, symmetric products, and Eilenberg Maclane spaces

Idempotents and Landweber exactness in brave new algebra

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