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Journal für die reine und angewandte Mathematik

Managing Editor: Weissauer, Rainer

Ed. by Colding, Tobias / Cuntz, Joachim / Huybrechts, Daniel / Hwang, Jun-Muk

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Filtered spectra arising from permutative categories

Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal). Volume 2007, Issue 604, Pages 73–136, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102, DOI: 10.1515/CRELLE.2007.020, June 2007

Publication History

Received:
2005-04-18
Revised:
2006-01-03
Published Online:
2007-06-01

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 Spm(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 Weiss's 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.

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