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There are various complexes associated to a surface using the curves and arcs e.g. Curve complex, Arc complex, curve arc complex and so on (for a collection of such objects see This). Now to understand the propeties of Mapping class group it is necessary to understand the connectedness of various subcomplexes of these.

Here are some methods to prove such connectivity arguments.

1) Hatcher flow.(Triangulation of surfaces.)

2) Using Morse theory.(A presentation for the MCG of closed orientable surfaces.)

3) Showing the related complexes are homemorphc to product of Teichmuller space with a cell.("Natural triangulations associated to a surface" by B.H. Bowditch, D.B.A. Epstein , and Decorated teichmuller space of punctured surfaces.)

4) Combinatorial arguments like Lickorish ("A finite set of generators for the homotopy gruop of 2 manifolds" by W.B.R. Lickorish ).

5) Considering all possibilities of the given complex and using explicit description.(The second homology group of the MCG of an orientable surfaces.).

I want to know other methods to prove results of these type. Any reference or idea will be extremely helpfull.

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