论向量束的EO $\ mathm {EO}$ -可定向性

Pub Date : 2022-09-19 DOI:10.1112/topo.12265

P. Bhattacharya, H. Chatham

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引用次数: 1

摘要

我们研究了向量束在上同调理论(EO $\mathrm{EO}$ -理论)中的可定向性。EO $\mathrm{EO}$ -理论是真实K $\mathrm{K}$ -理论KO $\mathrm{KO}$的更高高度的类似物。对于每一个EO $\mathrm{EO}$ -理论，我们证明了对于特定整数i $i$，任意向量束的i $i$拷贝的直接和是EO $\mathrm{EO}$ -可定向的。利用分裂原理，我们简化到CP∞上正则线束$\mathbb {CP}^{\infty }$的情况。我们的方法包括理解Morava稳定群的p阶$p$子群对Morava E $\mathrm{E}$ - CP∞理论$\mathbb {CP}^{\infty }$的作用。我们的计算还有另一个应用:我们确定了s1 $\mathrm{S}^{1}$ -Tate谱在所有EO $\mathrm{EO}$ -理论上与s1的平凡作用$\mathrm{S}^{1}$相关的同伦类型。

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On the EO $\mathrm{EO}$ -orientability of vector bundles

We study the orientability of vector bundles with respect to a family of cohomology theories called $EO$ -theories. The $EO$ -theories are higher height analogues of real $K$ -theory $KO$ . For each $EO$ -theory, we prove that the direct sum of $i$ copies of any vector bundle is $EO$ -orientable for some specific integer $i$ . Using a splitting principal, we reduce to the case of the canonical line bundle over ${CP}^{\infty}$ . Our method involves understanding the action of an order $p$ subgroup of the Morava stabilizer group on the Morava $E$ -theory of ${CP}^{\infty}$ . Our calculations have another application: We determine the homotopy type of the $S^{1}$ -Tate spectrum associated to the trivial action of $S^{1}$ on all $EO$ -theories.

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