\(K(2)\)-homology of some infinite loop spaces
We compute the second Morava \(K\)-theory of \(\Omega^\infty \Sigma^\infty S^{2m}\). As an application, we show a decomposition theorem for \(K(2)\)-homology of some infinite loop spaces. Let a sequence of integers \(I= (I_1,\dots, I_n)\) with \(1\leq I_j\leq p+ 1\), for \(p\) an odd prime, be called allowable, whenever \(I_j= 1\), \(I_{j- 1}= 1\). Then we have Theorem: Let \(p> 2\). Then \(K(2)_*(CS^0)\cong K(2)_*\) \([Q_I \iota_0\): \(I\) is an allowable sequence], \(K(2)_*(QS^0)\cong K(2)_*\) \([Q_I \iota_0\): \(I\) is an allowable sequence]\(\otimes_{K(2)_*} K(2)_*[\iota^{- 1}_0]\), \(K(2)_*(QS^{2n})\cong K(2)_*\) \([Q_I \iota_{2n}\): \(I\) is an allowable sequence]. Corollary: Let \(X\) be a spectrum with stable cells in non-negative even degrees only. Then \(K(2)_*(\underline X_0)\cong \otimes K(2)_* (QS^{2n})\) corresponding to the stable cell decomposition of \(X\).
scientific article; zbMATH DE number 784085
scientific article