Moduli spaces of metric graphs of genus 1 with marks on vertices

Details Moduli spaces of metric graphs of genus 1 with marks on vertices Moduli spaces of metric graphs of genus 1 with marks on vertices

2008-09-25

arxiv.org/abs/0809.4364v1

Mathematics

Algebraic Topology

Topology and its Applications, In Press, Corrected Proof, Available online 3 August 2008

Scientific paper

In this paper we study homotopy type of certain moduli spaces of metric graphs. More precisely, we show that the spaces $MG_{1,n}^v$, which parametrize the isometry classes of metric graphs of genus 1 with $n$ marks on vertices are homotopy equivalent to the spaces $TM_{1,n}$, which are the moduli spaces of tropical curves of genus 1 with $n$ marked points. Our proof proceeds by providing a sequence of explicit homotopies, with key role played by the so-called scanning homotopy. We conjecture that our result generalizes to the case of arbitrary genus.

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