Adams filtration and generalized Hurewicz maps for infinite loopspaces (Q1989469)

The paper studies the Hurewicz map \[ h_*: \pi_*(X) \longrightarrow R_*(\Omega^\infty X), \] where \(R\) is a commutative \(S\)-algebra, and \(\Omega^\infty\) is the \(0^{th}\) space of a spectrum. The central result is that under the Hurewicz map, the \(R\)-based Adams filtration on \(\pi_*(X)\) is compatible with the augmentation ideal filtration of the augmented \(S\)-algebra \(\Sigma^\infty (\Omega^\infty X)_+.\) The proof uses an explicit construction of \(\Sigma^\infty(\Omega^\infty)_+\) via André-Quillen homology. This leads to the recovery of a classical result by Milnor for \(R=H\mathbb{Z}/2\) and \(X=BO\), namely that in this case the image of \(h_*\) is one-dimensional in degrees 1, 2, 4 and 8. The article extends this to \(X=tmf\) to obtain that \[ h_*: \pi_*(tmf) \longrightarrow H_*(\Omega^\infty tmf, \mathbb{Z}/2) \] has five-dimensional image generated by \(\eta, \eta^2, \nu, \nu^2\) and \(c_4\). For \(R=\mathbb{Z}/p\) and \(X=BP\langle {n}\rangle,\) the author furthermore uses the Hurewicz map to obtain a decomposition of \(BP\langle n \rangle\) into ``atomic'' parts.