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Cluster algebras and triangulated surfaces. Part I: Cluster complexes

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arxiv math/0608367 v3 pith:WLMF5POV submitted 2006-08-15 math.RA math.COmath.GT

Sergey Fomin , Michael Shapiro , Dylan Thurston This is my paper · ORCID
classification math.RAmath.COmath.GT
keywords clusteralgebrascomplexsurfacesalgebraassociatedbasicbordered
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We establish basic properties of cluster algebras associated with oriented bordered surfaces with marked points. In particular, we show that the underlying cluster complex of such a cluster algebra does not depend on the choice of coefficients, describe this complex explicitly in terms of "tagged triangulations" of the surface, and determine its homotopy type and its growth rate.

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Cited by 3 Pith papers

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  2. From green mutation to $\mathrm{X}$-evolution: flows and foliations on cluster complexes

    math.RT 2025-01 conditional novelty 7.0 of 10

    Introduces a piecewise-linear flow on cluster complexes whose leaves generalize green mutation, proving these complexes are spheres for Dynkin quivers and contractible for Euclidean quivers.

  3. Topological model for derived category associated to sphere with four binaries

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