#### Algebraic and Geometric Topology 2 (2002),
paper no. 28, pages 591-647.

## Product and other fine structure in polynomial resolutions of mapping spaces

### Stephen T. Ahearn, Nicholas J. Kuhn

**Abstract**.
Let Map_T(K,X) denote the mapping space of continuous based functions
between two based spaces K and X. If K is a fixed finite complex, Greg
Arone has recently given an explicit model for the Goodwillie tower of
the functor sending a space X to the suspension spectrum \Sigma
^\infty Map_T(K,X).

Applying a generalized homology theory h_* to
this tower yields a spectral sequence, and this will converge strongly
to h_*(Map_T(K,X)) under suitable conditions, e.g. if h_* is
connective and X is at least dim K connected. Even when the
convergence is more problematic, it appears the spectral sequence can
still shed considerable light on h_*(Map_T(K,X)). Similar comments
hold when a cohomology theory is applied.

In this paper we study
how various important natural constructions on mapping spaces induce
extra structure on the towers. This leads to useful interesting
additional structure in the associated spectral sequences. For
example, the diagonal on Map_T(K,X) induces a `diagonal' on the
associated tower. After applying any cohomology theory with products
h^*, the resulting spectral sequence is then a spectral sequence of
differential graded algebras. The product on the E_\infty -term
corresponds to the cup product in h^*(Map_T(K,X)) in the usual way,
and the product on the E_1-term is described in terms of group
theoretic transfers.

We use explicit equivariant S-duality maps
to show that, when K is the sphere S^n, our constructions at the fiber
level have descriptions in terms of the Boardman-Vogt little n-cubes
spaces. We are then able to identify, in a computationally useful way,
the Goodwillie tower of the functor from spectra to spectra sending a
spectrum X to \Sigma ^\infty \Omega ^\infty X.
**Keywords**.
Goodwillie towers, function spaces, spectral sequences

**AMS subject classification**.
Primary: 55P35.
Secondary: 55P42.

**DOI:** 10.2140/agt.2002.2.591

**E-print:** `arXiv:math.AT/0109041`

Submitted: 29 January 2002.
Accepted: 25 June 2002.
Published: 25 July 2002.

Notes on file formats
Stephen T. Ahearn, Nicholas J. Kuhn

Department of Mathematics, Macalester College

St.Paul, MN 55105, USA

and

Department of Mathematics, University of Virginia

Charlottesville, VA 22903, USA

Email: ahearn@macalester.edu, njk4x@virginia.edu

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