Iterates of the suspension map and Mitchell’s finite spectra with $A_k$-free cohomology

Abstract

Abstract. We study certain cross-effects of the unstable homotopy of spheres. These cross-effects were constructed by Weiss, for different purposes, in the context of "Orthogonal calculus". We show that Mithchell's finite spectra with A kfree cohomology (constructed in [Mt85]) arise naturally as stabilizations of Weiss' cross-effects. In particular, we find that after a suitable Bousfield localization, our cross-effects, which capture meaningful information about the unstable homotopy of spheres, are homotopy equivalent to the infinite loop spaces associated with Mitchell's spectra. This last result is a partial generalization of the main result of Mahowald and the author in [AM97]. IntroductionLet X be a topological space. We are interested in studying the difference between the homotopy type of X and that of S 2 X, the double suspension of X (for technical reasons the statements turn out to be a little cleaner if one works with double suspensions rather than single suspensions). One naive way to compare X and S 2 X would be by means of the map X → S 2 X given by taking the smash product of the identity map on X with the inclusion S 0 → S 2 . Of course, this map is null-homotopic and thus is unlikely to provide useful information about the difference between X and S 2 X. A much better idea is to consider the Freudental (double) suspension map w 1 : X → Ω 2 S 2 X. Let F 1 (X) be the homotopy fiber of this map. We regard F 1 (X) as measuring the difference between the homotopy type of X and that of S 2 X. We would like to iterate the process and find higher analogues of the suspension map and higher iterated differences of the suspension map. Thus at the second stage we would like to find a suitable way to compare F 1 (X) with F 1 (S 2 X). Obviously, there exists a natural map F 1 (X) → Ω 2 F 1 (S 2 X). However, this map turns out to be null-homotopic, just as the map X → S 2 X is null-homotopic. It follows from the 1991 Mathematics Subject Classification. 55P40, 55P42, 55P65.

Suspension spectra of matrix algebras, the rank filtration, and rational noncommutative CW-spectra

“…Except for the statement about the chromatic type, this is [AL1,proposition 9.6], which in turn relies on [Ar1]. The statement about the chromatic type is part of [Ar2,Theorem 2.2].…”

On the Balmer spectrum for compact Lie groups

“…Note that when is the trivial group, an equality in part (ii) would mean that the type of is always the same as the type of . Counterexamples to such an equality are given by the following complexes, which play a central role in [BHN + 19] and were studied extensively by Arone, Lesh, and Mahowald [Aro98, AM99, AL17]. We refer to [BHN + 19, § 4] for more details.…”

“…Since is a non-trivial wedge of spheres (see [OT92, 4.109]), the underlying non-equivariant spectrum of , i.e., , has type . On the other hand, by work of Arone–Mahowald [AM99] and Arone [Aro98], the underlying non-equivariant spectrum of has type , see also the proof of [BHN + 19, Theorem 2.2].…”

“…They further determined which inclusions occur for and deduced a complete description of the Balmer spectrum for all finite groups of square-free order. This classification was extended to all finite abelian groups in [BHN + 19], by computing the height shift of the generalized Tate construction for Lubin–Tate theory as well as making use of fixed-point properties of explicit finite equivariant complexes previously considered by Arone [Aro98] and Arone–Lesh [AL17].…”

The Balmer spectrum of the equivariant homotopy category of a finite abelian group

“…. This is easily recognized to be, up to a shift, one of the spectra figuring in [Aro98] (where Arone uses K n to denote the suspension of P ⋄ n ). In [Aro98, Thm.…”