COLLOQUIUM MATHEMATICUM VOL. 89 2001 NO. 1 ESTIMATES FOR HOMOLOGICAL DIMENSION OF CONFIGURATION SPACES OF GRAPHS BY JACEK ŚWIĄTKOWSKI (Wrocław and Warszawa) Abstract. We show that the homological dimension of a configuration space of a graph Γ is estimated from above by the number b of vertices in Γ whose valence is greater than 2. We show that this estimate is optimal for the n-point configuration space of Γ if n ≥ 2b. 0. Introduction. Let Γ be a finite graph and n a natural number. The marked n-point configuration space of Γ is a subspace  C n Γ in the nth cartesian power of Γ defined by  C n Γ := {(x 1 ,...,x n ) ∈ Γ n : x i = x j for i = j }. Consider the natural free action of the symmetric group S n on the space  C n Γ defined by σ(x 1 ,...,x n )=(x σ(1) ,...,x σ(n) ) and put C n Γ :=  C n Γ/S n . The space C n Γ is called the (unmarked) n-point configuration space of Γ . This paper reports on partial progress towards understanding the ho- mology of configuration spaces of graphs, or even more generally of compact polyhedra. For another recent result in that direction, see [G]. We call a vertex v of Γ branched if it is adjacent to at least three edges. We denote by b = b(Γ ) the number of branched vertices in Γ . The main result of this paper is the following. 0.1. Theorem. Let Γ be a finite graph and n a natural number. (1) There exists a cube complex K n Γ of dimension min(b(Γ ),n) which embeds as a deformation retract into the configuration space C n Γ . (2) The fundamental group π 1 (C n Γ ) contains a subgroup isomorphic to the free abelian group Z k with k = min(b(Γ ), [n/2]), where [x] denotes the integer part of x. 2000 Mathematics Subject Classification : Primary 55M10; Secondary 20J05, 51F99. The author was supported by the Polish State Committee for Scientific Research (KBN) grant 2 P03A 023 14. [69]