arXiv 2007.108442020-07-19math.ATmath.KTmath.QAmath.RT

simply connected空间的表示同调

Representation homology of simply connected spaces

Yuri Berest, Ajay C. Ramadoss, Wai-Kit Yeung

备注：: 39 pages. arXiv admin note: substantial text overlap with arXiv:1703.03505

摘要：

设$G$是定义在特征为零的域$k$上的仿射代数群.研究了在$X$基点平凡化的点连通CW复形X上G-局部系统的导模空间。这个导出的模空间由仿射DG方案RLoc $_G(X,*)$表示：我们称RLoc $_G(X,*)$的结构层的（上）同调为$X$在$G$中的表示同调，并将其表示为HR $_*(X,G)$。HR $_0(X,G)$同构于$G$中的$X$基本群的表示簇Rep $_G[π_1(X)]$的坐标环--这是$X$的一个著名的代数几何不变量，在拓扑学中有许多应用。当X是单连通的情况下似乎研究得少得多：在这种情况下，HR $_0(X,G)$是平凡的，但更高的表示同调仍然是$X$的一个有趣的有理不变量，取决于代数群$G$。本文利用有理同伦理论计算了任意单连通空间（有限有理型）$X$在Quillen和Sullivan代数模型下的HR $_*(X,G)$.当$G$是约化的时，我们还计算了表示同调的$G$ -不变部分HR $_*(X,G)^G$，并研究了HR $_*(X,G)^G$作为分次交换代数不含局部有限型的问题.这个问题与B. Feigin和P. Hanlon在20世纪80年代提出的，并由S.费舍尔岛Grojnowski和C. 2008年的Teleman。将强Macdonald猜想用拓扑术语重新表述，给出了HR $_*(X,G)^G$是复约化群$G$的分次对称代数的空间$X$的一个简单刻画.

英文摘要：

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 $G$.

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