论文标题
简单连接空间的表示同源
Representation homology of simply connected spaces
论文作者
论文摘要
让$ g $是一个特征零的字段$ k $定义的仿射代数组。我们在$ x $的基数上研究了G-Local系统的衍生模量空间。此派生的模量空间由Aggine DG方案RLOC $ _G（X，*）$：我们称为RLOC $ _G（x，*）$ $ g $中的$ x $的代表同源性的结构捆（CO）同源性，并用hr $ _*（x，g）$表示。 HR $ _0（x，g）$是代表变量的坐标环$ _g [π_1（x）] $的坐标环，$ g $的$ x $ in $ g $的基本组是$ g $的$ x $ - 一个众所周知的代数数字不变$ x $，其中许多应用于拓扑。当X被简单地连接的情况似乎较少研究：在这种情况下，HR $ _0（x，g）$是微不足道的，但较高的表示同源性仍然是$ x $的有趣理性不变性，具体取决于代数组$ g $。在本文中，我们使用理性同义理论来计算任意连接的空间$ x $（有限理性类型）的HR $ _*（x，g）$（其Quillen和Sullivan代数模型）。当$ g $还原时，我们还计算了代表同源性的$ g $ invariant部分，hr $ _*（x，g）^g $，并研究hr $ _*（x，g）^g $何时没有本地有限的类型作为分级的交换性代数。事实证明，这个问题与所谓的强大麦克唐纳猜想密切相关，这是B. Feigin和P. Hanlon在1980年代提出的代表理论（作为一种猜想）的著名结果，并由S. Fishel，I。Grojnowski和I. Grojnowski和C. teleman证明了2008年的强大概述，$。 HR $ _*（x，g）^g $是任何复杂的还原组$ g $的分级对称代数。
Let $G$ be an affine algebraic group defined over field $k$ of characteristic zero. We study the derived moduli space of G-local systems on a pointed connected CW complex X trivialized at the basepoint of $X$. This derived moduli space is represented by an affine DG scheme RLoc$_G(X,*)$: we call the (co)homology of the structure sheaf of RLoc$_G(X,*)$ the representation homology of $X$ in $G$ and denote it by HR$_*(X,G)$. The HR$_0(X,G)$ is isomorphic to the coordinate ring of the representation variety Rep$_G[π_1(X)]$ of the fundamental group of $X$ in $G$ -- a well-known algebro-geometric invariant of $X$ with many applications in topology. The case when X is simply connected seems much less studied: in this case, the HR$_0(X,G)$ is trivial but the higher representation homology is still an interesting rational invariant of $X$ depending on the algebraic group $G$. In this paper, we use rational homotopy theory to compute the HR$_*(X,G)$ for an arbitrary simply connected space $X$ (of finite rational type) in terms of its Quillen and Sullivan algebraic models. When $G$ is reductive, we also compute the $G$-invariant part of representation homology, HR$_*(X,G)^G$, and study the question when HR$_*(X,G)^G$ is free of locally finite type as a graded commutative algebra. This question turns out to be closely related to the so-called Strong Macdonald Conjecture, a celebrated result in representation theory proposed (as a conjecture) by B. Feigin and P. Hanlon in the 1980s and proved by S. Fishel, I. Grojnowski and C. Teleman in 2008. Reformulating the Strong Macdonald Conjecture in topological terms, we give a simple characterization of spaces $X$ for which HR$_*(X,G)^G$ is a graded symmetric algebra for any complex reductive group $G$.
