The Geometry and Fundamental Group of Permutation Products and Fat Diagonals

Abstract

AbstractPermutation products and their various “fat diagonal” subspaces are studied from the topological and geometric points 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 exhibit an expression for the fundamental group of any permutation product of a connected space Xhaving the homotopy type of a CW complex in terms of π1(X) and H1(X;Z). We then prove that the fundamental group of the configuration space of n-points on X, of whichmultiplicities do not exceed n2, coincides with H1(X; Z). Further results consist in giving conditions for when fat diagonal subspaces of manifolds can be manifolds again. Various examples and homological calculations are included.