In this survey paper, we give a complete list of known results on the first and the second homology groups of surface mapping class groups. Some known results on higher (co) homology are also mentioned.
KORKMAZ, MUSTAFA (2002) "Low-dimensional homology groups of mapping class groups: a survey," Turkish Journal of Mathematics: Vol. 26: No. 1, Article 8. Available at: https://journals.tubitak.gov.tr/math/vol26/iss1/8