Goodwillie towers and chromatic homotopy: an overview

This paper is based on talks I gave in Nagoya and Kinosaki in August of 2003. I survey, from my own perspective, Goodwillie's work on towers associated to continuous functors between topological model categories, and then include a discussion of applications to periodic homotopy as in my work and the work of Arone-Mahowald.Comment: This is the version published by Geometry & Topology Monographs on 29 January 200

The Krull filtration of the category of unstable modules over the Steenrod algebra

In the early 1990's, Lionel Schwartz gave a lovely characterization of the Krull filtration of U, the category of unstable modules over the mod p Steenrod algebra. Soon after, this filtration was used by the author as an organizational tool in posing and studying some topological nonrealization conjectures. In recent years the Krull filtration of U has been similarly used by Castellana, Crespo, and Scherer in their study of H--spaces with finiteness conditions, and Gaudens and Schwartz have given a proof of some of my conjectures. In light of these topological applications, it seems timely to better expose the algebraic properties of the Krull filtration.Comment: 21 page

Product and other fine structure in polynomial resolutions of mapping spaces

Let Map_T(K,X) denote the mapping space of continuous based functions between two based spaces K and X. If K is a fixed finite complex, Greg Arone has recently given an explicit model for the Goodwillie tower of the functor sending a space X to the suspension spectrum \Sigma^\infty Map_T(K,X). Applying a generalized homology theory h_* to this tower yields a spectral sequence, and this will converge strongly to h_*(Map_T(K,X)) under suitable conditions, e.g. if h_* is connective and X is at least dim K connected. Even when the convergence is more problematic, it appears the spectral sequence can still shed considerable light on h_*(Map_T(K,X)). Similar comments hold when a cohomology theory is applied. In this paper we study how various important natural constructions on mapping spaces induce extra structure on the towers. This leads to useful interesting additional structure in the associated spectral sequences. For example, the diagonal on Map_T(K,X) induces a `diagonal' on the associated tower. After applying any cohomology theory with products h^*, the resulting spectral sequence is then a spectral sequence of differential graded algebras. The product on the E_\infty -term corresponds to the cup product in h^*(Map_T(K,X)) in the usual way, and the product on the E_1-term is described in terms of group theoretic transfers. We use explicit equivariant S-duality maps to show that, when K is the sphere S^n, our constructions at the fiber level have descriptions in terms of the Boardman-Vogt little n-cubes spaces. We are then able to identify, in a computationally useful way, the Goodwillie tower of the functor from spectra to spectra sending a spectrum X to \Sigma ^\infty \Omega ^\infty X.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol2/agt-2-28.abs.htm

Primitives and central detection numbers in group cohomology

Henn, Lannes, and Schwartz have introduced two invariants, d_0(G) and d_1(G), of the mod p cohomology of a finite group G such that H^*(G) is detected and determined by H^d(C_G(V)) for d no bigger than d_0(G) and d_1(G), with V < G p-elementary abelian. We study how to calculate these invariants. We define a number e(G) that measures the image of the restriction of H^*(G) to its maximal central p-elementary abelian subgroup. Using Benson--Carlson duality, we show that when $G$ has a p-central Sylow subgroup P, d_0(G) = d_0(P) = e(P), and a similar exact formula holds for d_1(G). In general, we show that d_0(G) is bounded above by the maximum of the e(C_G(V))'s, if Benson's Regularity Conjecture holds. In particular, the inequality holds for all groups such that the p--rank of G minus the depth of H^*(G) is at most 2. When we look at examples with p=2, we learn that d_0(G) is at most 7 for all groups with 2--Sylow subgroup of order up to 64, unless the Sylow subgroup is isomorphic to that of either Sz(8) (and d_0(G) = 9) or SU(3,4) (and d_0(G)=14). Enroute we recover and strengthen theorems of Adem and Karagueuzian on essential cohomology, and Green on depth essential cohomology, and prove theorems about the structure of cohomology primitives associated to central extensions.Comment: 51 pages Prop. 8.1 now given a correct proo

The Whitehead Conjecture, the Tower of S^1 Conjecture, and Hecke algebras of type A

In the early 1980's the author proved G.W. Whitehead's conjecture about stable homotopy groups and symmetric products. In the mid 1990's, Arone and Mahowald showed that the Goodwillie tower of the identity had remarkably good properties when specialized to odd dimensional spheres. In this paper we prove that these results are linked, as has been long suspected. We give a state-of-the-art proof of the Whitehead conjecture valid for all primes, and simultaneously show that the identity tower specialized to the circle collapses in the expected sense. Key to our work is that Steenrod algebra module maps between the primitives in the mod p homology of certain infinite loopspaces are determined by elements in the mod p Hecke algebras of type A. Certain maps between spaces are shown to be chain homotopy contractions by using identities in these Hecke algebras.Comment: 27 pages. As accepted for publication by the Journal of Topology. New: section 2 has been expanded, section 8 has been improved, and a dedication has been adde