论文标题
在$ \ mathrm {eo} $ - 向量捆绑包的可方向性
On the $\mathrm{EO}$-orientability of vector bundles
论文作者
论文摘要
我们研究了矢量捆绑包的可方向性相对于一个称为$ \ mathrm {eo} $理论的共同体理论家族。 $ \ mathrm {eo} $ - 理论是真正的$ \ mathrm {k} $的高度类似物 - 理论$ \ mathrm {ko} $。对于每个$ \ mathrm {eo} $ - 理论，我们证明，任何向量捆绑包的$ i $副本的直接总和均为$ \ mathrm {eo} $ - 对于某些特定的整数$ i $。使用拆分主体，我们将$ \ mathbb {cp}^{\ infty} $上的规范线束的情况减少。我们的方法涉及了解Morava稳定器组的顺序$ P $子组对Morava $ \ Mathrm {E} $ - $ \ Mathbb {cp}^{\ infty} $的理论。我们的计算还有另一个应用程序：我们确定$ \ mathrm {s}^{1} $ - tate Spectrum与所有$ \ Mathrm {s}^{1} $相关的$ \ mathrm {s}^{1} $ - tate Spectrum的同质类型。
We study the orientability of vector bundles with respect to a family of cohomology theories called $\mathrm{EO}$-theories. The $\mathrm{EO}$-theories are higher height analogues of real $\mathrm{K}$-theory $\mathrm{KO}$. For each $\mathrm{EO}$-theory, we prove that the direct sum of $i$ copies of any vector bundle is $\mathrm{EO}$-orientable for some specific integer $i$. Using a splitting principal, we reduce to the case of the canonical line bundle over $\mathbb{CP}^{\infty}$. Our method involves understanding the action of an order $p$ subgroup of the Morava stabilizer group on the Morava $\mathrm{E}$-theory of $\mathbb{CP}^{\infty}$. Our calculations have another application: We determine the homotopy type of the $\mathrm{S}^{1}$-Tate spectrum associated to the trivial action of $\mathrm{S}^{1}$ on all $\mathrm{EO}$-theories.
