Adams filtration and generalized Hurewicz maps for infinite loopspaces

Abstract

We study the Hurewicz map h from the homotopy groups of a spectrum X to the R-homology of its 0th space X(0), where R is a connective commutative S-algebra. We prove that the decreasing filtration of the domain of h associated to an R-based Adams resolution is compatible with the augmentation ideal filtration of the range associated to the suspension spectrum of X(0)_+, an augmented commutative S-algebra. The proof makes use of the interplay of this filtration with Topological Andre Quillen Homology. An application is a Connectivity Theorem: Localize at a prime p and suppose X is (c-1)-connected for some positive c. If f in pi_*(X) has Adams filtration s and |f| < cp^s, then f maps to zero in R_*(X(0)). An application of that is a Finiteness Theorem: If the mod p cohomology of X is finitely presented as a module over the Steenrod algebra, then the image of the mod p Hurewicz map for X(0) is finite. We illustrate these theorems with calculations of the mod 2 Hurewicz image of BO, its connected covers, and tmf(0), and the mod p Hurewicz image of all the spaces in the BP and BP spectra. Enroute, we get new proofs of theorems of Milnor and Wilson. In the special case when X is a suspension spectrum and R = HZ/2, we recover results announced by Lannes and Zarati in the 1980s (with a totally different proof), and generalizations to all primes p. We get a chromatic version of this for the Hurewicz map for Morava E theory and all X.