×

A generalization of Snaith-type filtration. (English) Zbl 0945.55011

In [J. Lond. Math. Soc., II. Ser. 7, 577-583 (1974; Zbl 0275.55019)], V. P. Snaith showed that, for a connected CW complex \(X\), the space \(\Omega^mS^mX\) splits stably as an infinite wedge of certain other spaces. In the description of this decomposition, configuration spaces play an important role. This result has been generalized by numerous authors: For a compact parallelizable \(m\)-manifold \(M\) there is a stable splitting of \(\text{Map} (M,S^mX)\).
Consider the functor \(X\mapsto \Omega^\infty S^\infty \text{Map}(M,S^mX)\). It is analytic in the sense of Goodwillie’s “Calculus” for homotopy functors. The present paper starts from the observation that the above splitting of \(\Omega^\infty S^\infty \text{Map} (M,S^mX)\) is the “Taylor series” of this analytic functor. This opens up the road to a further generalization: Replace \(M\) by a finite CW complex \(K\) and describe the Taylor series of the functor \(X\mapsto\Omega^\infty S^\infty\text{Map}(K,X)\) on sufficiently highly connected spaces. This is what the author achieves.

MSC:

55P99 Homotopy theory
55P65 Homotopy functors in algebraic topology
55P42 Stable homotopy theory, spectra

Citations:

Zbl 0275.55019
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] C.-F. Bödigheimer, Stable splittings of mapping spaces, Algebraic topology (Seattle, Wash., 1985) Lecture Notes in Math., vol. 1286, Springer, Berlin, 1987, pp. 174 – 187.
[2] Gunnar Carlsson, A survey of equivariant stable homotopy theory, Topology 31 (1992), no. 1, 1 – 27. · Zbl 0759.55001
[3] W. G. Dwyer and D. M. Kan, Function complexes for diagrams of simplicial sets, Nederl. Akad. Wetensch. Indag. Math. 45 (1983), no. 2, 139 – 147. · Zbl 0524.55021
[4] Thomas G. Goodwillie, Calculus. I. The first derivative of pseudoisotopy theory, \?-Theory 4 (1990), no. 1, 1 – 27. · Zbl 0741.57021
[5] Thomas G. Goodwillie, Calculus. II. Analytic functors, \?-Theory 5 (1991/92), no. 4, 295 – 332. · Zbl 0776.55008
[6] T.G. Goodwillie, Calculus III: the Taylor series of a homotopy functor, in preparation. · Zbl 1067.55006
[7] J. M. Boardman and R. M. Vogt, Homotopy invariant algebraic structures on topological spaces, Lecture Notes in Mathematics, Vol. 347, Springer-Verlag, Berlin-New York, 1973. J. P. May, The geometry of iterated loop spaces, Springer-Verlag, Berlin-New York, 1972. Lectures Notes in Mathematics, Vol. 271. · Zbl 0285.55012
[8] J. P. May, Weak equivalences and quasifibrations, Groups of self-equivalences and related topics (Montreal, PQ, 1988) Lecture Notes in Math., vol. 1425, Springer, Berlin, 1990, pp. 91 – 101.
[9] Dusa McDuff, Configuration spaces of positive and negative particles, Topology 14 (1975), 91 – 107. · Zbl 0296.57001
[10] Graeme Segal, Categories and cohomology theories, Topology 13 (1974), 293 – 312. · Zbl 0284.55016
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.