The McCord model for the tensor product of a space and a commutative ring spectrum

Nicholas J. Kuhn

Published 2002-02-05Version 1

This paper begins by noting that, in a 1969 paper in the Transactions, M.C.McCord introduced a construction that can be interpreted as a model for the categorical tensor product of a based space and a topological abelian group. This can be adapted to Segal's very special Gamma--spaces, and then to a more modern situation: (K tensor R) where K is a based space and R is a unital, augmented, commutative, associative S--algebra. The model comes with an easy-to-describe filtration. If one lets K = S^n, and then stabilize with respect to n, one gets a filtered model for the Topological Andre--Quillen Homology of R. When R = Omega^{infty} Sigma^{infty} X, one arrives at a filtered model for the connective cover of a spectrum X, constructed from its 0th space. Another example comes by letting K be a finite complex, and R the S--dual of a finite complex Z. Dualizing again, one arrives at G.Arone's model for the Goodwillie tower of the functor sending Z to the suspension spectrum of Map(K,Z). Applying cohomology with field coefficients, one gets various spectral sequences for deloopings with known E_1--terms. A few nontrivial examples are given. In an appendix, we describe the construction for unital, commutative, associative S--algebras not necessarily augmented.