Fundamental Group and Euler Characteristic of Permutation Products and Fat Diagonals

Details Fundamental Group and Euler Characteristic of Permutation Products and Fat Diagonals Fundamental Group and Euler Characteristic of Permutation Products and Fat Diagonals

2010-10-07

arxiv.org/abs/1010.1507v1

Mathematics

Algebraic Topology

24 pages

Scientific paper

Permutation products and their various "fat diagonal" subspaces are studied from the topological and geometric point of view. We first write down an expression for the fundamental group of any permutation product of a connected space $X$, having the homotopy type of a simplicial complex, in terms of $\pi_1(X)$ and $H_1(X;{\mathbb Z})$. 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 $H_1(X;{\mathbb Z})$. Useful additivity properties for the Euler characteristic are then spelled out and used to give explicit formulae for the Euler characteristics of various fat diagonals. Several examples and calculations are included.

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