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Splitting Madsen-Tillmann spectra I. Twisted transfer maps. (English) Zbl 1431.55010

Let \(\pi \colon E \to B\) be a fibre bundle over a suitably nice space \(B\) with fibre a compact manifold \(F\). Given a vector bundle \(\zeta\) over \(B\), the authors first study the twisted Becker-Gottlieb transfer map \(t_{\pi}^{\zeta} \colon B^{\zeta} \to E^{\pi^*\zeta}\), showing that the composition \(B^{\zeta} \xrightarrow{t_{\pi}^{\zeta}} E^{\pi^* \zeta} \xrightarrow{\mathop{Th}^{\zeta}(\pi)} B^{\zeta}\) induces multiplication by \(\chi(F)\) in homology. Moreover, for a compact Lie group \(G\) and a closed subgroup \(K \subset G\), the twisted Becker-Gottlieb transfer factors through the Becker-Schultz-Mann-Miller-Miller transfer. Using this, the authors show that if \(\chi(F)\) is prime to \(p\), then \(p\)-localization \(B_{(p)}^{\zeta}\) splits off \(E_{(p)}^{\pi^*\zeta}\). In the compact Lie group case, if \(\eta\) is a vector bundle over \(BG\) and \(\chi(G/K)\) is prime to \(p\), they show that \(BG^{\eta}_{(p)}\) splits off \(BK_{(p)}^{\eta|_{K} \oplus \mathop{ad}_K-\mathop{ad}_{G \mid K}}\)
Let \(K(n)\) be a compact Lie group, then the Madsen-Tillmann spectrum \(MTK(n)\) is defined as \(BK(n)^{\gamma_n}\), the Thom spectrum of \(-\gamma_n\), where \(\gamma_n\) is the canonical bundle for the classifying space for \(n\)-dimensional \(K(n)\)-vector bundles. The results in the previous paragraph then imply numerous splitting results for \(MTK(n)\) and \(MTK(n)_{(p)}\). For example, if \((G(n),K(n))\) is one of the pairs \((\mathrm{O}(2n),\mathrm{SO}(2n+1)),(\mathrm{Pin}^+(4n),\mathrm{Spin}(4n+1)), (\mathrm{Pin}^{-}(4n+2),\mathrm{Spin}(4n+3))\), then \(BG_+\) stably splits off the Madsen-Tillmann spectrum \(MTK(n)\). In the particular case of \((O(2n),\mathrm{SO}(2n+1))\) the authors show that after localization at 2 this refines to a homotopy equivalence \(\mathrm{MTO}(2n) \simeq \mathrm{BO}(2n)_+\). This allows the identification of certain algebraically independent classes in the mod 2 cohomology of \(\Omega^{\infty}\mathrm{MTO}(2n)\).

MSC:

55P42 Stable homotopy theory, spectra
55P47 Infinite loop spaces
55R12 Transfer for fiber spaces and bundles in algebraic topology
57T25 Homology and cohomology of \(H\)-spaces
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