Kathryn Lesh

Filtered spectra arising from permutative categories

Abstract Given a special Γ-category 𝒞 satisfying some mild hypotheses, we construct a sequence of spectra interpolating between the spectrum associated to 𝒞 and the Eilenberg-Mac Lane spectrum Hℤ. Examples of categories to which our construction applies are: the category of finite sets, the category of finite-dimensional vector spaces, and the category of finitely-generated free modules over a reasonable ring. In the case of finite sets, our construction recovers the filtration of Hℤ by symmetric powers of the sphere spectrum. In the case of finite-dimensional complex vector spaces, we obtain an apparently new sequence of spectra, {Am }, that interpolate between bu and . We think of Am as a “bu-analogue” of Sp m (S) and describe far-reaching formal similarities between the two sequences of spectra. For instance, in both cases the mth subquotient is contractible unless m is a power of a prime, and in vk -periodic homotopy the filtration has only k + 2 non-trivial terms. There is an intriguing relationship between the bu-analogues of symmetric powers and Weisss orthogonal calculus, parallel to the not yet completely understood relationship between the symmetric powers of spheres and the Goodwillie calculus of homotopy functors. We conjecture that the sequence {Am }, when rewritten in a suitable chain complex form, gives rise to a minimal projective resolution of the connected cover of bu. This conjecture is the bu-analogue of a theorem of Kuhn and Priddy about the symmetric power filtration. The calculus of functors provides substantial supporting evidence for the conjecture.

The unstable Adams spectral sequence for two-stage towers

Abstract Let KP denote a generalized mod 2 Eilenberg–MacLane space and let Y be the fiber of a map X→KP to which the Massey–Peterson theorem applies. We study the relationship of the mod 2 unstable Adams spectral sequence (UASS) for X and for Y. Given conditions on X, we split the E2-term for Y, and we use a primary level calculation to compute d2 for Y up to an error term. If the UASS for X collapses at E2 (for example, if X is an Eilenberg–MacLane space), the UASS for Y collapses at E3, and we have the entire UASS for Y. We also give examples and address a conjecture of Bousfield on the UASS for the Lie group SO.

Hybrid spaces with interesting cohomology

Let p be an odd prime, and let R be a polynomial algebra over the Steenrod algebra with generators in dimensions prime to p. To such an algebra is associated a p-adic pseudoreflection group W , and we assume that W is of order prime to p and irreducible. Adjoin to R a one-dimensional element z, and give R[z] an action of the Steenrod algebra by βz = 0 and βx = (|x|/2)zx for an even dimensional element x. We show that the subalgebra of elements of R[z] consisting of elements of degree greater than one is realized uniquely, up to homotopy, as the cohomology of a p-complete space. This space can be thought of as a cross between spaces studied by Aguade, Broto, and Notbohm, and the Clark-Ewing examples, further studied by Dwyer, Miller, and Wilkerson.