Crossed homomorphisms and low dimensional representations of mapping class groups of surfaces.

We continue the study of low dimensional linear representations of mapping class groups of surfaces initiated by Franks–Handel [Proc. Amer. Math. So. 141 (2013), pp. 2951–2962] and Korkmaz [ Low-dimensional linear representations of mapping class groups , preprint, arXiv:1104.4816v2 (2011)]. We consider (2g+1)-dimensional complex linear representations of the pure mapping class groups of compact orientable surfaces of genus g. We give a complete classification of such representations for g \geq 7 up to conjugation, in terms of certain twisted 1-cohomology groups of the mapping class groups. A new ingredient is to use the computation of a related twisted 1-cohomology group by Morita [Ann. Inst. Fourier (Grenoble) 39 (1989), pp. 777–810]. The classification result implies in particular that there are no irreducible linear representations of dimension 2g+1 for g \geq 7, which marks a contrast with the case g=2. [ABSTRACT FROM AUTHOR]