I am currently studying mapping class groups, and would like to know why one uses orientation-preserving homeomorphisms in their definition. I read in 'A Primer on Mapping Class Groups' by Benson and Farb that there are all sorts of replacements one can make, e.g. diffeomorphisms instead of homeomorphisms, isotopy instead of homotopy, but one always uses orientation-preserving maps. Is there a special reason one does that?
Maybe this hasn't to do with only mapping class groups either. Is there a general reason why one would only consider orientation-preserving maps?