Document Zbl 0902.55001 - zbMATH Open

The authors investigate the Morava \(K\)-homology of certain \(H\)-spaces with only finitely many non-trivial homotopy groups. Given a connected \(p\)-local space \(X\) with \(\pi_{k}(X)\) finitely generated over \({\mathbb Z}_{(p)}\), \(k>1\), and non-zero for only finitely many \(k\), they show that for \(n>0\) the Hopf algebra \(K(n)_{*}(\Omega X)\) has a natural increasing filtration by normal sub-Hopf algebras \(K(n)_{*} \cong F_{n+2} \subset \dots \subset F_{1} \subset F_{0} \cong K(n)_{*}( \Omega X)\) with subquotients \(F_{q}/ /F_{q+1} \cong K(n)_{*}(K(\pi_{q}(\Omega X),q))\). Furthermore, if \(\Omega X\) is homotopy commutative, then \(K(n)_{*}(\Omega X) \cong \bigotimes_{q=0}^{n+1} K(n)_{*}(K(\pi_{q}(\Omega X),q))\) as Hopf algebras over \(K(n)_{*}\), and the splitting is natural if either all \(\pi_{k}(X)\), \(k>1\), are finite or if they are all free. The main theorem in the paper actually is a little bit more general. The authors draw several consequences from this. For example, they show how their result can be used to compute the \(K(n)\)-homology of the spaces in the \(\Omega\)-spectra \(P(m)\) and \(k(m)\) for \(m>n\). To obtain the splitting result above, the authors need some structure theory on bicommutative Hopf algebras over \(K(n)_{*}\). This is developed in the last section of the paper, and it is of interest in its own right.
For the entire collection see [Zbl 0890.00047].

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