Title:
Filtered spectra arising from permutative categories
Authors:
Gregory Arone
University of Virginia
zga2m@virginia.edu
Kathryn Lesh
Union College
leshk@union.edu
Abstract: Given a special Gamma-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, A_{m}, that interpolate between bu
and HZ. We think of A_{m} as a ``bu-analogue'' of the m'th symmetric
power of the sphere and describe far-reaching formal similarities
between the two sequences of spectra. For instance, in both cases the
m'th subquotient is contractible unless m is a power of a prime, and
in v_{k}-periodic homotopy the filtration has only k+2 nontrivial
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 {A_{m}}, 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.
This is a revision of a preprint previously submitted to Hopf.
The paper has been accepted for publication in
Journal für die reine und angewandte Mathematik (Crelle's Journal).