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An \(E_\infty\) splitting of spin bordism. (English) Zbl 1058.55001

Anderson, Brown and Peterson gave an additive splitting of the Spin bordism Thom spectrum \(MSpin\) in terms of connective real \(K\) theory, \(k_0\), and mod2 Eilenberg-MacLane spectra \(H\mathbb Z/2\). This additive presentation failed to capture the ring structure of \(MSpin\). The culprit is the \(H\mathbb Z/2\) part; however this does not survive localization with respect to mod2 \(K\)-theory (also known as \(K(1)\), the first Morava \(K\)-theory for the prime 2). This paper gives an \(E_\infty\) splitting of \(K(1)\)-localized \(M\)Spin. The generator \(\zeta\) of the generator of the -1st homotopy group of the \(K(1)\)-localized sphere induces a map from the \(E_\infty\)-cone of \(T_\zeta\) over \(\zeta\) to the \(K(1)\)-localized \(MSpin\), \[ \varphi:T_\zeta\to MSpin. \] For each odd positive integer \(k\), the author constructs spherical classes \(z_k\) in \(\pi_0 KO\wedge MSpin\). The author’s main theorem is that the \(E_\infty\) map \[ (\varphi, z_3, z_5,\dots): T_\zeta\wedge \bigwedge^\infty_{i=1} TS^0\to MSpin \] is a \(KO\) equivalence. \(TS^0\) is the free \(E_\infty\) spectrum generated by the sphere spectrum. Its \(K\)-homology is a free \(\theta\)-algebra on one generator. The same is true of the \(K\)-homology of \(T_\zeta\). A corollary to the main splitting theorem is that the homotopy of \(MSpin\) is a free \(\theta\)-algebra over \(\pi_*KO\) in infinitely many generators.

MSC:

55N22 Bordism and cobordism theories and formal group laws in algebraic topology
55P42 Stable homotopy theory, spectra
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