Moduli spaces of tropical curves of higher genus with marked points and homotopy colimits

Details Moduli spaces of tropical curves of higher genus with marked points and homotopy colimits Moduli spaces of tropical curves of higher genus with marked points and homotopy colimits

2008-09-25

arxiv.org/abs/0809.4367v1

Mathematics

Algebraic Topology

Scientific paper

The main characters of this paper are the moduli spaces $TM_{g,n}$ of rational tropical curves of genus $g$ with $n$ marked points, with $g\geq 2$. We reduce the study of the homotopy type of these spaces to the analysis of compact spaces $X_{g,n}$, which in turn possess natural representations as a homotopy colimits of diagrams of topological spaces over combinatorially defined generalized simplicial complexes $\Delta_g$, with the latter being interesting on their own right. We use these homotopy colimit representations to describe a CW complex decomposition for each $X_{g,n}$. 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 $X_{2,0}$, $X_{2,1}$, $X_{2,2}$, and $X_{3,0}$, as well as a partial understanding of other cases, resulting in several open questions and in further conjectures.

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