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Permutation products and their various "fat diagonal" subspaces are studiedfrom the topological and geometric point of view. We describe in detail thestabilizer and orbit stratifications related to the permutation action,producing a sharp upper bound for its depth and then paying particularattention to the geometry of the diagonal stratum. We write down an expressionfor the fundamental group of any permutation product of a connected space $X$having the homotopy type of a CW complex in terms of $\pi_1(X)$ and$H_1(X;\bbz)$. We then prove that the fundamental group of the configurationspace 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 forwhen fat diagonal subspaces of manifolds can be manifolds again. Variousexamples and homological calculations are included.

Fundamental Group and Euler Characteristic of Permutation Products and Fat Diagonals