AI 中文总结
本文计算了$B_{C_2}O(1)$的参数化$\boldsymbol{\text{underline}\boldsymbol{\text{F}}_2}$上同调，细化了$RO(G)$分次Bredon上同调，还推导了可用于一般基上参数化上同调计算的示性类、定向定义等一般性结果。
AI 中文摘要
我们以常Mackey函子$\boldsymbol{\text{underline}\boldsymbol{\text{F}}_2}$为系数，计算实$C_2$-线丛分类空间$B_{C_2}O(1)$的参数化（或称扭曲）普通上同调。参数化上同调通过将所有局部系数的等变上同调组装成一个分次环，细化了$RO(G)$分次Bredon上同调。因此，我们的工作还编码了对$B_{C_2}O(1)$上实$C_2$-向量丛的所有Thom空间的$RO(C_2)$分次上同调的计算。在此过程中，我们证明了许多可应用于一般基$B$上参数化上同调计算的一般性结果，特别是引入了一系列示性类，给出了非齐次丛的定向定义，将等变Steenrod运算引入该语境，并给出了与基变换相关的一般性结果。
英文摘要
We compute the parametrized (or twisted) ordinary cohomology of the classifying space for real $C_2$-line bundles, $B_{C_2}O(1)$ with coefficients in the constant Mackey functor $\underline{\mathbb{F}}_2$. Parametrized cohomology refines $RO(G)$-graded Bredon cohomology by assembling equivariant cohomology for all local coefficients into a single graded ring. For this reason, our work also encodes a computation of the $RO(C_2)$-graded cohomology of all Thom spaces of real $C_2$-vector bundles over $B_{C_2}O(1)$. Along the way, we prove many general results that can be applied to computations of parametrized cohomology over general bases $B$. In particular, we introduce a collection of characteristic classes, give a definition of orientation for non-homogeneous bundles, import equivariant Steenrod operations to this context, and give general results relating to base change.
