论文标题
用于分类纤维空间的代数模型
Algebraic models for classifying spaces of fibrations
论文作者
论文摘要
我们证明了分类空间的合理同型类型的新结构结果$ b \ operatatorName {aut}（x）用光纤A简单连接的有限cw-complex $ x $。 我们首先研究$ b \ operatotorname {aut}（x）$的nilpotent封面，并表明他们的理性共同体学组是相关转换组的代数表示。对于普遍的覆盖，这会导致Sullivan-Wilkerson定理扩展到更高的同型和共同体学组。对于对应于同源性表示的内核的覆盖物，这证明了同型Torelli空间的同源物学的代数性。 为了使我们称之为正常的独立振动的封面，我们证明存在更强的结果，即在代数表示中存在nilpotent dg lie代数$ \ mathfrak g（x）$，该代数表示其eporiantiant有理同型类型。这导致了空间的代数模型$ b \ operatorname {aut}（x）$，并描述其理性同胞戒指是某种算术组$γ（x）$的共同体，并在chevalley-eilenberg coohomology of Chevalley-eilenberg Coohomology of Chevalley-eilenberg Coohomology of Chevalley-eilenberg coohomologient of $ \ Mathfrak G（x）$。这对共同学环具有强大的结构后果，在某些情况下，可以使用模块化形式的不变理论和计算完全确定它。我们用具体的例子说明了这些要点。 作为另一种应用，我们在伯格伦德（Berglund） - 马德森（Madsen）引起的高度连接的均匀歧管上的自我态度等效性的某些结果显着改善，我们证明在奇数维度中是平行的新结果。
We prove new structural results for the rational homotopy type of the classifying space $B\operatorname{aut}(X)$ of fibrations with fiber a simply connected finite CW-complex $X$. We first study nilpotent covers of $B\operatorname{aut}(X)$ and show that their rational cohomology groups are algebraic representations of the associated transformation groups. For the universal cover, this yields an extension of the Sullivan--Wilkerson theorem to higher homotopy and cohomology groups. For the cover corresponding to the kernel of the homology representation, this proves algebraicity of the cohomology of the homotopy Torelli space. For the cover that classifies what we call normal unipotent fibrations, we then prove the stronger result that there exists a nilpotent dg Lie algebra $\mathfrak g(X)$ in algebraic representations that models its equivariant rational homotopy type. This leads to an algebraic model for the space $B\operatorname{aut}(X)$ and to a description of its rational cohomology ring as the cohomology of a certain arithmetic group $Γ(X)$ with coefficients in the Chevalley-Eilenberg cohomology of $\mathfrak g(X)$. This has strong structural consequences for the cohomology ring and, in certain cases, allows it to be completely determined using invariant theory and calculations with modular forms. We illustrate these points with concrete examples. As another application, we significantly improve on certain results on self-homotopy equivalences of highly connected even-dimensional manifolds due to Berglund--Madsen, and we prove parallel new results in odd dimensions.
