2000]Primary 55P43; Secondary 18G55 THE MCCORD MODEL FOR THE TENSOR PRODUCT OF A SPACE AND A COMMUTATIVE RING SPECTRUM

We begin this paper 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 Γ–spaces indeed this is roughly what Segal did and then to a more modern situation: K ⊗ 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, and then stabilize with respect to n, one gets a filtered model for the Topological André–Quillen Homology of R. When R = ΩΣX, one arrives at a filtered model for the connective cover of a spectrum X, constructed from its 0 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 Σ MapT (K, Z). Applying cohomology with field coefficients, one gets various spectral sequences for deloopings with known E1–terms. A few nontrivial examples are given. In an appendix, we describe the construction for unital, commutative, associative S–algebras not necessarily augmented.