Moduli Spaces of Tropical Curves of Higher Genus with Marked Points and Homotopy Colimits

The main characters of this paper are the moduli spaces TMg,n of rational tropical curves of genus g with n marked points, with g ≥ 2. We reduce the study of the homotopy type of these spaces to the analysis of compact spaces Xg,n, which in turn possess natural representations as a homotopy colimits of diagrams of topological spaces over combinatorially defined generalized simplicial complexes ∆g, with the latter being interesting on their own right. We use these homotopy colimit representations to describe a CW complex decomposition for each Xg,n. Furthermore, we use these developments, coupled with some standard tools for working with homotopy colimits, to perform an in-depth analysis of special cases of genus 2 and 3, gaining a complete understanding of the moduli spaces X2,0, X2,1, X2,2, and X3,0, as well as a partial understanding of other cases, resulting in several open questions and in further conjectures. 1. Moduli spaces of tropical curves Tropical geometry is a fairly recent new field within the broader context of algebraic geometry. During the time which elapsed since its inception, tropical geometry has already developed its language and its methods, and has furthermore found numerous applications; we refer the interested reader to [St02, Chapter 9], and more recently to [DFS07, DY07, Mi06], for both applications and the general background information. Some spaces arising in tropical geometry are of interest from the point of view of algebraic topology as well. Often these have natural definitions and fit well in more general structures. One such instance is furnished by the moduli spaces of rational tropical curves of genus g with n marked points TMg,n, which were introduced by Mikhalkin in [Mi07], see also [Ko08a] for a purely topological definition. Prior to this work, only the case of genus 1 has been studied systematically, see [Ko08a, Ko08b], where, e.g., the homology groups with coefficients in Z2 were computed for this family of spaces. In this paper, we present an in-depth analysis of the moduli spaces of rational tropical curves of higher genus. As a first step, we complement the known shrinking bridges strong deformation retraction, leading from TMg,n to TM b g,n as described in [Ko08a, Section 3], by a further new simplification. On the intuitive level, that newly discovered strong deformation retraction increases the edge lengths proportionally, to reach the length of the longest one, stopping when the edges which are strictly shorter than the longest one form a forest. This process is described in detail in Section 2, where 2000 Mathematics Subject Classification. Primary: 57xx, secondary 14Mxx, 55xx.