Algebraic & Geometric Topology

Cohomology of symplectic groups and Meyer's signature theorem

Dave Benson, Caterina Campagnolo, Andrew Ranicki, and Carmen Rovi

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at msp.org/agt.

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

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 , ) , ) . If we denote by 1 Sp ( 2 g , ) ˜ Sp ( 2 g , ) 1 the pullback of the universal cover of Sp ( 2 g , ) , then by a theorem of Deligne, every finite index subgroup of Sp ( 2 g , ) ˜ contains 2 . As a consequence, a class in the second cohomology of any finite quotient of Sp ( 2 g , ) 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 , ) that contains this information. This quotient 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 2 ̃ 1 , and ̃ appears as a quotient of the metaplectic double cover Mp ( 2 g , ) = Sp ( 2 g , ) ˜ 2 . 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  ̃ is the universal central extension of  . 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
Received: 26 November 2017
Revised: 30 May 2018
Accepted: 16 June 2018
First available in Project Euclid: 18 December 2018

Permanent link to this document
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

Keywords
surface bundles signature modulo 8 signature cocycle Meyer group cohomology symplectic groups

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


Export citation

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