The Sullivan Conjecture on Maps from Classifying Spaces

Miller, Haynes

doi:10.2307/2007071

Cited by 259 publications

(180 citation statements)

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“…Our first result is a description of H * (T (n, j); Z/2) as a module over the mod 2 Steenrod algebra A. Following the lead of others in the n = 1 case [Ca,Mi2,LZ1], we describe the bigraded object H * (T (n, * ); Z/2), with the extra structure afforded by Ψ * and Φ * . We need first to define variants on the category U of unstable A modules, and the category K of unstable A algebras.…”

Section: Evaluation On Loopspaces Induces Mapsmentioning

confidence: 99%

“…These finite complexes were explored in the 1970's and 1980's in work by M.Mahowald, E.Brown, S.Gitler, F.Peterson, R. Cohen, G.Carlsson, H.Miller, J.Lannes, and P.Goerss, among others. (Entries into the extensive literature include [Mah,BC,Ca,Mi2,L2,GLM,HK].) They played an essential role in a number of the major achievements in homotopy theory during this time: Mahowald's construction [Mah] of an infinite family of 2-primary elements in π S * (S 0 ) having Adams filtration 2; Goerss, Lannes, and F.Morel's work [GLM] on representing mod 2 homology by maps from (desuspensions of) the T (j)'s; and Miller's proof of the Sullivan conjecture [Mi2].…”

Section: Introductionmentioning

confidence: 99%

“…[Mah] is based on two facts: that as modules over the Steenrod algebra, the cohomology of the T (j) are dual Brown-Gitler modules, and that one can construct maps T (j) − → T (2j) realizing on cohomology certain canonical maps between these. [Ca,Mi2] are then based on the connection, just on the level of cohomology, between the classifying spaces BV of elementary abelian 2 groups V , and the homotopy colimits of the sequences…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Evaluation On Loopspaces Induces Mapsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Kuhn

2001

Cohomological Methods in Homotopy Theory

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“…When p = 2, this is [M,Thm.2.6]; when p is odd, this is the content of the 10 page second chapter of [C]. Our indexing has been chosen to be consistant with the modern literature (for example [Mi,§7]), so that H * (T (n); Z/p) is the injective envelope of H * (S n ; Z/p) in the category of unstable A-modules 1 . In particular, H * (T (2p j ); Z/p) has bottom classes in dimensions 1 and 2 linked by the Bockstein, and top class in dimension 2p j .…”

Section: Three Key Propositionsmentioning

confidence: 99%

Mahowaldean families of elements in stable homotopy groups revisited

Hunter

Kuhn

1999

Math. Proc. Camb. Phil. Soc.

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In the mid 1970s Mark Mahowald constructed a new infinite family of elements in the 2-component of the stable homotopy groups of spheres, ηj∈πSj2 (S0)(2) [M]. Using standard Adams spectral sequence terminology (which will be recalled in Section 3 below), ηj is detected by h1hj∈Ext2,*[Ascr ] (Z/2, Z/2). Thus he had found an infinite family of elements all having the same Adams filtration (in this case, 2), thus dooming the so-called Doomsday Conjecture. His constructions were ingenious: his elements were constructed as composites of pairs of maps, with the intermediate spaces having, on one hand, a geometric origin coming from double loopspace theory and, on the other hand, mod2 cohomology making them amenable to Adams Spectral Sequence analysis and suggesting that they were related to the new discovered Brown–Gitler spectra [BG].In the years that followed, various other related 2-primary infinite families were constructed, perhaps most notably (and correctly) Bruner's family detected by h2h2j∈ Ext3,*[Ascr ](Z/2, Z/2) [B]. An odd prime version was studied by Cohen [C], leading to a family in πS∗(S0)(p) detected by h0bj∈ Ext3,*[Ascr ] (Z/p, Z/p) and a filtration 2 family in the stable homotopy groups of the odd prime Moore space. Cohen also initiated the development of odd primary Brown–Gitler spectra, completed in the mid 1980s, using a different approach, by Goerss [G], and given the ultimate ‘modern’ treatment by Goerss, Lannes and Morel in the 1993 paper [GLM]. Various papers in the late 1970s and early 1980s, e.g. [BP, C, BC], related some of these to loopspace constructions.Our project originated with two goals. One was to see if any of the later work on Brown–Gitler spectra led to clarification of the original constructions. The other was to see if taking advantage of post Segal Conjecture knowledge of the stable cohomotopy of the classifying space BZ/p would help in constructing new families at odd primes, in particular a conjectural family detected by h0hj∈ Ext2,*[Ascr ] (Z/p, Z/p). (This followed a paper [K1] by one of us on 2 primary families from this point of view.)

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“…He showed that H * RP ∞ splits off of the direct limit of the cohomologies of the Brown-Gitler spectra. The first author formulated Carlsson's work as saying that H * RP ∞ is an injective unstable A-module (actually the work was done in homology), and used it in his proof of the Sullivan conjecture [71].…”

Section: The Memoir the Metastable Homotopy Of S Nmentioning

confidence: 99%