Arrangements of symmetric products of spaces

Abstract

We study the combinatorics and topology of general arrangements of sub-spaces of the form D + SP n-d (X) in symmetric products SP n (X) where D ∈ SP d (X). Symmetric products SP m (X) : = X m /S m , also known as the spaces of effective "divisors" of order m, together with their companion spaces of divisors/particles, have been studied from many points of view in numerous papers, see [P. Blagojević et al., in: B. Dragović, B. Sazdović (Eds.) Summer School in Modern Mathematical Physics, 2004, math.AT/0408417; S. Kallel, Trans. Amer. Math. Soc. 350 (1998), 1350] for the references. In this paper we approach them from the point of view of geometric combinatorics. Using the topological technique of diagrams of spaces along the lines of [V. Welker et al., J. Reine Angew. Math. 509 (1999), 117; G.M. Ziegler, R.T. Živaljević, Math. Ann. 295 (1993) 527] we calculate the homology of the union and the complement of these arrangements. As an application we include a computation of the homology of the homotopy end space of the open manifold SP n (M g,k ), where M g,k is a Riemann surface of genus g punctured at k points, a problem which was originally motivated by the study of commutative (m + k, m)-groups [K. Trenčevski, D. Dimovski, J. Algebra 240 (2001) 338].