Algebraic & Geometric Topology

A function on the homology of 3–manifolds

Vladimir Turaev

Full-text: Open access

Abstract

In analogy with the Thurston norm, we define for an orientable 3–manifold M a numerical function on H2(M;). This function measures the minimal complexity of folded surfaces representing a given homology class. A similar function is defined on the torsion subgroup of H1(M;). These functions are estimated from below in terms of abelian torsions of M.

Article information

Source
Algebr. Geom. Topol., Volume 7, Number 1 (2007), 135-156.

Dates
Received: 21 November 2006
Accepted: 11 January 2007
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513796662

Digital Object Identifier
doi:10.2140/agt.2007.7.135

Mathematical Reviews number (MathSciNet)
MR2308939

Zentralblatt MATH identifier
1145.57011

Subjects
Primary: 57M27: Invariants of knots and 3-manifolds
Secondary: 57Q10: Simple homotopy type, Whitehead torsion, Reidemeister-Franz torsion, etc. [See also 19B28]

Keywords
genus Thurston norm torsion

Citation

Turaev, Vladimir. A function on the homology of 3–manifolds. Algebr. Geom. Topol. 7 (2007), no. 1, 135--156. doi:10.2140/agt.2007.7.135. https://projecteuclid.org/euclid.agt/1513796662


Export citation

References

  • F Deloup, On abelian quantum invariants of links in 3-manifolds, Math. Ann. 319 (2001) 759–795
  • F Deloup, G Massuyeau, Reidemeister-Turaev torsion modulo one of rational homology three-spheres, Geom. Topol. 7 (2003) 773–787
  • S Friedl, Reidemeister torsion, the Thurston norm and Harvey's invariants
  • P B Kronheimer, T S Mrowka, Scalar curvature and the Thurston norm, Math. Res. Lett. 4 (1997) 931–937
  • E Looijenga, J Wahl, Quadratic functions and smoothing surface singularities, Topology 25 (1986) 261–291
  • C T McMullen, The Alexander polynomial of a 3-manifold and the Thurston norm on cohomology, Ann. Sci. École Norm. Sup. $(4)$ 35 (2002) 153–171
  • L I Nicolaescu, The Reidemeister torsion of 3-manifolds, de Gruyter Studies in Mathematics 30, Walter de Gruyter & Co., Berlin (2003)
  • P Ozsváth, Z Szabó, Holomorphic disks and genus bounds, Geom. Topol. 8 (2004) 311–334
  • W P Thurston, A norm for the homology of $3$-manifolds, Mem. Amer. Math. Soc. 59 (1986) i–vi and 99–130
  • V Turaev, Introduction to combinatorial torsions, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel (2001) Notes taken by Felix Schlenk
  • V Turaev, Torsions of $3$-dimensional manifolds, Progress in Mathematics 208, Birkhäuser Verlag, Basel (2002)