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In this paper, we calculate the p-torsion of the Farrell cohomology for lowgenus pure mapping class groups with punctures, where p is an odd prime. Here,`low genus' means g=1,2,3; and `pure mapping class groups with punctures' meansthe mapping class groups with any number of punctures, where the punctures arenot allowed to be permuted. These calculations use our previous results aboutthe periodicity of pure mapping class groups with punctures, as well as othercohomological tools. The low genus cases are interesting because we know thatthe high genus cases can be reduced to the low genus ones. Also, thecohomological properties of the mapping class groups without punctures areclosely related to our cases.

Farrell cohomology of low genus pure mapping class groups with punctures