## 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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