A dual Rothenberg-Steenrod spectral sequence
This paper constructs a strongly convergent cobar-type Eilenberg-Moore spectral sequence of the path-loop fibration \(\Omega^ n \Sigma^ n X\to P\Omega^{n-1} \Sigma^ n X\to \Omega^{n-1} \Sigma^ n X\) for any multiplicative homology theory \(h_ *\) and any pointed space \(X\) such that \((X, *)\) is a strong NDR pair. This is a spectral sequence of algebras over \(h_ * (*)\) which converges strongly to \(h_ * (\Omega^ n \Sigma^ n X)\). If \(h_ * (\Omega^{n-1} \Sigma^ n X)\) is \(h_ * (*)\)-flat, then there is an isomorphism of algebras over \(h_ * (*)\) \[ E^ 2 \approx \text{Cotor}^{h_ * (\Omega^{n -1} \Sigma^ n X)} (h_ * (*), h_ * (*)). \] If \(X\) is a co-H space, and if any \(h_ *(*)\)-module is free, then this isomorphism is a morphism of coalgebras. The construction of the spectral sequence is based on a new, geometrical, filtration of the little cube space \({\mathcal C}_ n (j)\). The author notes that some applications of this spectral sequence to the Morava \(K\)-theory of iterated loop spaces will appear elsewhere.
cobar-type Eilenberg-Moore spectral sequence
scientific article; zbMATH DE number 695900