Topology of the diffeomorphism group

[electroweak]

I would like to study the path components (isotopy classes) of the diffeomorphism group of some compact Riemann surface. To make sense of path connectedness, I require a notion of continuity; hence, I require a notion of an open set of diffeomorphisms. What sort of topology should I put on the diffeomorphism group?

 

[Office_Shredder]

http://en.wikipedia.org/wiki/Diffeomorphism#Topology

 

[quasar987]

The classical reference is

Augustin Bangaya, The Structure of Classical Diffeomorphism Groups