Ganea分解的分类空间
Ganea decompositions of classifying spaces
AI总结：
本文研究了通过纤维-纤维束构造获得的紧连通李群分类空间BG的Ganea分解，确定了纤维束的上同调条件，证明了分解的sharp性，并构造了多个例子。
摘要：
我们研究通过(相对)纤维-纤维束构造获得的紧连通李群分类空间BG的同调分解。给定一对Borel纤维束F→E→BG和F'→E'→BG，这种构造会产生一个索引为Z_+的空问塔(望远镜)X_m(F,F') over BG，使得hocolim(X_m)在弱同伦意义下等价于BG。我们确定纤维束上的上同调条件，使得X_m(F,F')具有类似Weyl群准不变空间的性质。我们证明，在这些条件下，所得的BG分解是sharp的(在Q上)，X_m(F,F')在有理上是形式的和Cohen-Macaulay的，其上同调环是有限秩自由模H*(BG,Q)的。我们构造了许多例子，包括基本(最大 torus)纤维束G/T→BT→BG以及对于分类空间B_comG的通用元素纤维束E_comG_1→B_comG_1→BG，由Adem和Gómez引入。在大多数情况下，我们给出(等价)上同调环的显式表示，用特征类来计算(等价)K-理论。论文包含一个附录，其中我们重新审视拓扑纤维-纤维束构造，在抽象设置中证明了经典Ganea定理的∞-范畴扩展。
英文摘要：
We study homotopy decompositions of the classifying spaces $BG$ of compact connected Lie groups obtained by (relative) fiber-cofiber construction. Given a pair of Borel fibrations $ F \to E \to BG $ and $F' \to E' \to BG $, this construction yields a tower (telescope) of spaces $ X_{m}(F,F') $ over $BG$ indexed by $ \mathbb{Z}_+ $ that converges in the sense that $\text{hocolim} \,(X_{m})\,$ is weakly homotopy equivalent to $BG$. We determine cohomological conditions on the fibrations that produce the spaces $X_{m}(F,F')$ with properties similar to those of the spaces of quasi-invariants of Weyl groups constructed by the first and third authors. We prove that, under these conditions, the resulting homotopy decompositions of $BG$ are sharp (over $\mathbb{Q}$), the spaces $X_{m}(F,F')$ are rationally formal and Cohen-Macaulay, their cohomology rings being finite rank free modules over $H^*(BG, \mathbb{Q})$. We construct many examples which include the fundamental (maximal torus) fibration $ G/T \to BT \to BG $ as well as the universal fibration $\, E_{\rm com}G_{\bf 1} \to B_{\rm com}G_{\bf 1} \to BG \,$ for the classifying space $B_{\rm com}G$ of commuting elements in $G$ introduced by Adem and Gómez, as the first fibration in the pair. In most cases, we give an explicit presentation for the (equivariant) cohomology rings in terms of characteristic classes and compute the (equivariant) $K$-theory of the spaces involved. The paper contains an Appendix, where we re-examine the topological fiber-cofiber construction in an abstract setting, proving an $\infty$-categorical extension of the classical Ganea Theorem.
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