arXiv 1010.15072010-10math.ATmath.AG

置换积与胖对角线的基本群与欧拉示性数

Fundamental Group and Euler Characteristic of Permutation Products and Fat Diagonals

Sadok Kallel, Walid Taamallah

备注：: The earlier version of this paper has now been split into two. Title change. The Euler characteristic computation will appear elsewhere. Many new details worked out in this new version, especially pertaining to section 3 on orbit stratifications and to the proof of main Theorem 1.1. Mistakes and mistatements corrected. 18 pages

摘要：

从拓扑和几何的角度研究了置换积及其各种“胖对角”子空间。我们详细描述的稳定剂和轨道分层相关的置换行动，产生一个尖锐的上界，其深度，然后特别注意的对角地层的几何形状。本文给出了具有CW复形同伦型的连通空间$X$的任意置换积的基本群的一个表达式，它是用$π_1(X)$和$H_1(X;\bbz)$表示的.证明了重数不超过$n/2$的$X$上的$n$ -点的位形空间的基本群与$H_1(X;\bbz)$重合.进一步的结果包括在给条件时，脂肪对角子空间的流形可以再次流形。包括各种例子和同调计算。

英文摘要：

Permutation products and their various "fat diagonal" subspaces are studied from the topological and geometric point of view. We describe in detail the stabilizer and orbit stratifications related to the permutation action, producing a sharp upper bound for its depth and then paying particular attention to the geometry of the diagonal stratum. We write down an expression for the fundamental group of any permutation product of a connected space $X$ having the homotopy type of a CW complex in terms of $π_1(X)$ and $H_1(X;\bbz)$. We then prove that the fundamental group of the configuration space of $n$-points on $X$, of which multiplicities do not exceed $n/2$, coincides with $H_1(X;\bbz)$. Further results consist in giving conditions for when fat diagonal subspaces of manifolds can be manifolds again. Various examples and homological calculations are included.

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