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Let Map_T(K,X) denote the mapping space of continuous based functions betweentwo based spaces K and X. If K is a fixed finite complex, Greg Arone hasrecently given an explicit model for the Goodwillie tower of the functorsending a space X to the suspension spectrum \Sigma^\infty Map_T(K,X). Applyinga generalized homology theory h_* to this tower yields a spectral sequence, andthis 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 theconvergence is more problematic, it appears the spectral sequence can stillshed considerable light on h_*(Map_T(K,X)). Similar comments hold when acohomology theory is applied. In this paper we study how various importantnatural constructions on mapping spaces induce extra structure on the towers.This leads to useful interesting additional structure in the associatedspectral sequences. For example, the diagonal on Map_T(K,X) induces a`diagonal' on the associated tower. After applying any cohomology theory withproducts h^*, the resulting spectral sequence is then a spectral sequence ofdifferential graded algebras. The product on the E_\infty -term corresponds tothe cup product in h^*(Map_T(K,X)) in the usual way, and the product on theE_1-term is described in terms of group theoretic transfers. We use explicitequivariant S-duality maps to show that, when K is the sphere S^n, ourconstructions at the fiber level have descriptions in terms of theBoardman-Vogt little n-cubes spaces. We are then able to identify, in acomputationally useful way, the Goodwillie tower of the functor from spectra tospectra sending a spectrum X to \Sigma ^\infty \Omega ^\infty X.

Product and other fine structure in polynomial resolutions of mapping spaces