Algebraic & Geometric Topology 1 (2001) 519–548

DOI: 10.2140/agt.2001.1.519

arXiv: math.AT/0110230

Abstract

This paper proves a particular case of a conjecture of N Kuhn. Consider the Gabriel–Krull filtration of the category U of unstable modules. Let U_n, n ≥ 0, denote the n^th term of this filtration. The category U is the smallest thick subcategory that contains all subcategories U_n and is stable under colimit [L Schwartz, Unstable modules over the Steenrod algebra and Sullivan's fixed point set conjecture, Chicago Lectures in Mathematics (1994)]. The category U₀ is the category of locally finite modules, that is, the modules that direct limits of finite modules.

The conjecture is as follows: Let X be a space, then either H^*X is locally finite, or H^*X not in U_n, for all n. For example, the cohomology of a finite-dimensional space, or that of the loop space of a finite space are always in U₀. The cohomology of the classifying space of a finite group of even order does not belong to any subcategory U_n.

We prove this conjecture, modulo the additional hypothesis that all quotients of the nilpotent fibration are finitely generated. This condition implies in particular that the cohomology is finite-dimensional in each degree. This is necessary to apply Lannes' theorem on the cohomology of mapping spaces, which is required for N Kuhn's reduction of the problem [N Kuhn, On topologically realizing modules over the Steenrod algebra, Ann. of Math. 141 (1995) 321–347].

For simplicity, this article does not consider the case p=2.

Keywords

Steenrod operations, nilpotent modules, Eilenberg–Moore spectral sequence

Mathematical Subject Classification

Primary: 55S10

Secondary: 57S35

References

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Publication

Received: 9 October 2000
Revised: 4 July 2001
Accepted: 30 September 2001
Published: 5 October 2001

Authors