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

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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 ( 2 g + 1 ) (2g+1) -dimensional complex linear representations of the pure mapping class groups of compact orientable surfaces of genus g g . We give a complete classification of such representations for g ≥ 7 g \geq 7 up to conjugation, in terms of certain twisted 1 1 -cohomology groups of the mapping class groups. A new ingredient is to use the computation of a related twisted 1 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 2 g + 1 2g+1 for g ≥ 7 g \geq 7 , which marks a contrast with the case g = 2 g=2 .