 More tersely, the map Homeo(T"n") → SHE(T"n") is 1-connected (isomorphic on path-components, onto fundamental group).This is a "homeomorphism reduces to homotopy reduces to algebra" result.Thus the short exact sequence of the mapping class group splits (an identification of the torus as the quotient of R"n" gives a splitting, via the linear maps, as above)