Homotopy types of fine curve and fine arc complexes

The fine curve complex of a surface is a simplicial complex whose vertices are essential simple closed curves and whose $k$-simplices are collections of $k+1$ disjoint curves. We prove that the fine curve complex is homotopy equivalent to the curve complex. We also prove that the fine arc complex is contractible.

💡 Research Summary

The paper investigates two simplicial complexes associated with an orientable surface (S_{g,b}) of genus (g) with (b) boundary components: the fine curve complex (\mathcal C^{\dagger}(S_{g,b})) and the fine arc complex (\mathcal A^{\dagger}(S_{g,b})). The fine curve complex has as vertices the actual essential simple closed curves on the surface, and a (k)-simplex consists of (k+1) pairwise disjoint curves. In contrast, the classical curve complex (\mathcal C(S_{g,b})) has vertices given by isotopy classes of essential curves, and simplices correspond to collections of isotopy classes that admit disjoint representatives. Because the fine complex records actual curves rather than isotopy classes, it is infinite‑dimensional, whereas the classical complex has dimension (3g+b-3).

The first main result (Theorem 1.1) states that for surfaces with (g\ge1) or (b\ge4) the natural map
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