Abstract
Meyer showed that the signature of a closed oriented surface bundle over a surface is a multiple of , and can be computed using an element of . If we denote by the pullback of the universal cover of , then by a theorem of Deligne, every finite index subgroup of contains . As a consequence, a class in the second cohomology of any finite quotient of can at most enable us to compute the signature of a surface bundle modulo . We show that this is in fact possible and investigate the smallest quotient of that contains this information. This quotient is a nonsplit extension of by an elementary abelian group of order . There is a central extension , and appears as a quotient of the metaplectic double cover . It is an extension of by an almost extraspecial group of order , and has a faithful irreducible complex representation of dimension . Provided , the extension is the universal central extension of . Putting all this together, in Section 4 we provide a recipe for computing the signature modulo , and indicate some consequences.
Citation
Dave Benson. Caterina Campagnolo. Andrew Ranicki. Carmen Rovi. "Cohomology of symplectic groups and Meyer's signature theorem." Algebr. Geom. Topol. 18 (7) 4069 - 4091, 2018. https://doi.org/10.2140/agt.2018.18.4069
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