Homotopy types of fine curve and fine arc complexes

TL;DR

This paper demonstrates that the fine curve complex shares the same homotopy type as the classical curve complex, and establishes that the fine arc complex is contractible, advancing understanding of surface topology.

Contribution

It proves the homotopy equivalence between the fine and classical curve complexes and shows the contractibility of the fine arc complex, providing new insights into their topological properties.

Findings

Fine curve complex is homotopy equivalent to the curve complex.

Fine arc complex is contractible.

Results deepen understanding of surface topology and complex structures.

Abstract

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.