Product and other fine structure in polynomial resolutions of mapping spaces

Abstract

Let ${Map}_{\mathcal{T}}\left(K,X\right)$ 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 ${\Sigma }^{\infty }{Map}_{\mathcal{T}}\left(K,X\right)$.

Applying a generalized homology theory $h_{\ast }$ to this tower yields a spectral sequence, and this will converge strongly to $h_{\ast }\left({Map}_{\mathcal{T}}\left(K,X\right)\right)$ under suitable conditions, eg if $h_{\ast }$ is connective and $X$ is at least $\text{ dim }K$ connected. Even when the convergence is more problematic, it appears the spectral sequence can still shed considerable light on $h_{\ast }\left({Map}_{\mathcal{T}}\left(K,X\right)\right)$. 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 ${Map}_{\mathcal{T}}\left(K,X\right)$ induces a ‘diagonal’ on the associated tower. After applying any cohomology theory with products $h^{\ast }$, 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 $h^{\ast }\left({Map}_{\mathcal{T}}\left(K,X\right)\right)$ 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 ${\Sigma }^{\infty }{\Omega }^{\infty }X$.