## Algebraic & Geometric Topology

### Cohomology of symplectic groups and Meyer's signature theorem

#### Abstract

Meyer showed that the signature of a closed oriented surface bundle over a surface is a multiple of $4$, and can be computed using an element of $H 2 ( Sp ( 2 g , ℤ ) , ℤ )$. If we denote by $1 → ℤ → Sp ( 2 g , ℤ ) ˜ → Sp ( 2 g , ℤ ) → 1$ the pullback of the universal cover of $Sp ( 2 g , ℝ )$, then by a theorem of Deligne, every finite index subgroup of $Sp ( 2 g , ℤ ) ˜$ contains $2 ℤ$. As a consequence, a class in the second cohomology of any finite quotient of $Sp ( 2 g , ℤ )$ can at most enable us to compute the signature of a surface bundle modulo $8$. We show that this is in fact possible and investigate the smallest quotient of $Sp ( 2 g , ℤ )$ that contains this information. This quotient $ℌ$ is a nonsplit extension of $Sp ( 2 g , 2 )$ by an elementary abelian group of order $2 2 g + 1$. There is a central extension $1 → ℤ ∕ 2 → ℌ ̃ → ℌ → 1$, and $ℌ ̃$ appears as a quotient of the metaplectic double cover $Mp ( 2 g , ℤ ) = Sp ( 2 g , ℤ ) ˜ ∕ 2 ℤ$. It is an extension of $Sp ( 2 g , 2 )$ by an almost extraspecial group of order $2 2 g + 2$, and has a faithful irreducible complex representation of dimension $2 g$. Provided $g ⩾ 4$, the extension $ℌ ̃$ is the universal central extension of $ℌ$. Putting all this together, in Section 4 we provide a recipe for computing the signature modulo $8$, and indicate some consequences.

#### Article information

Source
Algebr. Geom. Topol., Volume 18, Number 7 (2018), 4069-4091.

Dates
Revised: 30 May 2018
Accepted: 16 June 2018
First available in Project Euclid: 18 December 2018

https://projecteuclid.org/euclid.agt/1545102064

Digital Object Identifier
doi:10.2140/agt.2018.18.4069

Mathematical Reviews number (MathSciNet)
MR3892239

Zentralblatt MATH identifier
07006385

Subjects
Primary: 20J06: Cohomology of groups
Secondary: 20C33: Representations of finite groups of Lie type 55R10: Fiber bundles

#### Citation

Benson, Dave; Campagnolo, Caterina; Ranicki, Andrew; Rovi, Carmen. Cohomology of symplectic groups and Meyer's signature theorem. Algebr. Geom. Topol. 18 (2018), no. 7, 4069--4091. doi:10.2140/agt.2018.18.4069. https://projecteuclid.org/euclid.agt/1545102064

#### References

• H Behr, Explizite Präsentation von Chevalley-gruppen über Z, Math. Z. 141 (1975) 235–241
• D J Benson, Theta functions and a presentation of $2^{1+(2g+1)}\mathsf{Sp}(2g,2)$, preprint (2017) \setbox0\makeatletter\@url https://homepages.abdn.ac.uk/d.j.benson/pages/html/archive/benson.html {\unhbox0
• S Bouc, N Mazza, The Dade group of (almost) extraspecial $p$–groups, J. Pure Appl. Algebra 192 (2004) 21–51
• J F Carlson, J Thévenaz, Torsion endo-trivial modules, Algebr. Represent. Theory 3 (2000) 303–335
• J H Conway, R T Curtis, S P Norton, R A Parker, R A Wilson, Atlas of finite groups: maximal subgroups and ordinary characters for simple groups, Oxford Univ. Press, Eynsham (1985)
• P Deligne, Extensions centrales non résiduellement finies de groupes arithmétiques, C. R. Acad. Sci. Paris Sér. A-B 287 (1978) A203–A208
• U Dempwolff, Extensions of elementary abelian groups of order $2\sp{2n}$ by $S\sb{2n}(2)$ and the degree $2$–cohomology of $S\sb{2n}(2)$, Illinois J. Math. 18 (1974) 451–468
• P Diaconis, Threads through group theory, from “Character theory of finite groups” (M L Lewis, G Navarro, D S Passman, T R Wolf, editors), Contemp. Math. 524, Amer. Math. Soc., Providence, RI (2010) 33–47
• C J Earle, J Eells, A fibre bundle description of Teichmüller theory, J. Differential Geometry 3 (1969) 19–43
• H Endo, A construction of surface bundles over surfaces with non-zero signature, Osaka J. Math. 35 (1998) 915–930
• L Funar, W Pitsch, Finite quotients of symplectic groups vs mapping class groups, preprint (2016) \setbox0\makeatletter\@url http://www.maths.ed.ac.uk/~v1ranick/papers/funarpitsch1.pdf {\unhbox0
• S P Glasby, On the faithful representations, of degree $2^n$, of certain extensions of $2$–groups by orthogonal and symplectic groups, J. Austral. Math. Soc. Ser. A 58 (1995) 232–247
• T Gocho, The topological invariant of three-manifolds based on the $\mathrm{U}(1)$ gauge theory, Proc. Japan Acad. Ser. A Math. Sci. 66 (1990) 237–239
• T Gocho, The topological invariant of three-manifolds based on the $\mathrm{U}(1)$ gauge theory, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 39 (1992) 169–184
• D Gorenstein, Finite groups, Harper & Row, New York (1968)
• R L Griess, Jr, Automorphisms of extra special groups and nonvanishing degree $2$ cohomology, Pacific J. Math. 48 (1973) 403–422
• P Hall, G Higman, On the $p$–length of $p$–soluble groups and reduction theorems for Burnside's problem, Proc. London Math. Soc. 6 (1956) 1–42
• I Hambleton, A Korzeniewski, A Ranicki, The signature of a fibre bundle is multiplicative mod $4$, Geom. Topol. 11 (2007) 251–314
• M-E Hamstrom, Homotopy groups of the space of homeomorphisms on a $2$–manifold, Illinois J. Math. 10 (1966) 563–573
• G Hiss, Die adjungierten Darstellungen der Chevalley–Gruppen, Arch. Math. (Basel) 42 (1984) 408–416
• B Huppert, Endliche Gruppen, I, Grundl. Math. Wissen. 134, Springer (1967)
• J-i Igusa, On the graded ring of theta-constants, Amer. J. Math. 86 (1964) 219–246
• A Korzeniewski, On the signature of fibre bundles and absolute Whitehead torsion, PhD thesis, University of Edinburgh (2005) \setbox0\makeatletter\@url http://hdl.handle.net/1842/12106 {\unhbox0
• T Y Lam, T Smith, On the Clifford–Littlewood–Eckmann groups: a new look at periodicity mod $8$, Rocky Mountain J. Math. 19 (1989) 749–786
• R Luke, W K Mason, The space of homeomorphisms on a compact two-manifold is an absolute neighborhood retract, Trans. Amer. Math. Soc. 164 (1972) 275–285
• W Meyer, Die Signatur von Flächenbündeln, Math. Ann. 201 (1973) 239–264
• G Nebe, E M Rains, N J A Sloane, The invariants of the Clifford groups, Des. Codes Cryptogr. 24 (2001) 99–121
• M Newman, J R Smart, Symplectic modulary groups, Acta Arith. 9 (1964) 83–89
• A Putman, The Picard group of the moduli space of curves with level structures, Duke Math. J. 161 (2012) 623–674
• D Quillen, The mod $2$ cohomology rings of extra-special $2$–groups and the spinor groups, Math. Ann. 194 (1971) 197–212
• C Rovi, The nonmultiplicativity of the signature modulo $8$ of a fibre bundle is an Arf–Kervaire invariant, Algebr. Geom. Topol. 18 (2018) 1281–1322
• B Runge, On Siegel modular forms, I, J. Reine Angew. Math. 436 (1993) 57–85
• B Runge, On Siegel modular forms, II, Nagoya Math. J. 138 (1995) 179–197
• B Runge, Codes and Siegel modular forms, Discrete Math. 148 (1996) 175–204
• M Sato, The abelianization of the level $d$ mapping class group, J. Topol. 3 (2010) 847–882
• P Schmid, On the automorphism group of extraspecial $2$–groups, J. Algebra 234 (2000) 492–506
• R Stancu, Almost all generalized extraspecial $p$–groups are resistant, J. Algebra 249 (2002) 120–126
• M R Stein, Generators, relations and coverings of Chevalley groups over commutative rings, Amer. J. Math. 93 (1971) 965–1004
• M R Stein, The Schur multipliers of $\mathrm{Sp}\sb{6}({\mathbb Z})$, $\mathrm {Spin}\sb{8}({\mathbb Z})$, $\mathrm{Spin}\sb{7}({\mathbb Z})$, and $F\sb{4}({\mathbb Z})$, Math. Ann. 215 (1975) 165–172
• R Steinberg, Generators, relations and coverings of algebraic groups, II, J. Algebra 71 (1981) 527–543
• K Tsushima, On a decomposition of Bruhat type for a certain finite group, Tsukuba J. Math. 27 (2003) 307–317