Complex of curves

In mathematics , the complex of curves of a surface is an essential aid for investigating the mapping class group of the surface .

definition

An abstract simplicial complex is associated with a surface . It is given by the following data. ${\displaystyle S}$ ${\displaystyle {\mathcal {C}}(S)}$

0-Simplices: Every isotope class of essential simple closed curves in corresponds to a corner in . ${\displaystyle S}$ ${\displaystyle {\mathcal {C}}(S)}$

1- Simplizes: Two corners in are connected by an edge if the corresponding isotopic classes of curves apply to the number of intersections . ${\displaystyle {\mathcal {C}}(S)}$ ${\displaystyle a,b}$ ${\displaystyle i(a,b)=0}$

k-Simplizes: Corners create a k-Simplex if and only if they are connected in pairs by edges. is therefore a flag complex . ${\displaystyle k+1}$ ${\displaystyle {\mathcal {C}}(S)}$

properties

• The complex of curves is empty for . For the complex of curves is a countable set of 0-simplices. ${\displaystyle S^{2},S_{0,1},S_{0,2},S_{0,3}}$ ${\displaystyle T^{2},S_{1,1},S_{0,4}}$
• For is contiguous . ${\displaystyle 3g+n\geq 5}$ ${\displaystyle {\mathcal {C}}(S_{g,n})}$
• The complex of curves is a Gromov hyperbolic space . Except for , it has an infinite diameter . ${\displaystyle S^{2},S_{0,1},S_{0,2},S_{0,3}}$

Applications

• From the connection of the curve complex it follows that the mapping class group is finitely generated .
• Two simplices in determine a Heegaard decomposition of a 3-manifold . The decomposition is reducible if and only if the two simplices have a common corner. The decomposition is weakly reducible if the two simplices are connected by an edge. ${\displaystyle {\mathcal {C}}(S)}$

literature

• Benson Farb, Dan Margalit: A primer on mapping class groups. Princeton Mathematical Series, 49th Princeton University Press, Princeton, NJ, 2012. ISBN 978-0-691-14794-9 online (pdf)
• Nikolai Ivanov: Mapping class groups. Handbook of geometric topology, 523-633, North-Holland, Amsterdam, 2002.

Web links

• Saul Schleimer: Notes on the complex of curves