Algebraic & Geometric Topology

Vortices and a TQFT for Lefschetz fibrations on 4–manifolds

Michael Usher

Full-text: Open access

Abstract

Adapting a construction of D Salamon involving the U(1) vortex equations, we explore the properties of a Floer theory for 3–manifolds that fiber over S1 which exhibits several parallels with monopole Floer homology, and in all likelihood coincides with it. The theory fits into a restricted analogue of a TQFT in which the cobordisms are required to be equipped with Lefschetz fibrations, and has connections to the dynamics of surface symplectomorphisms.

Article information

Source
Algebr. Geom. Topol., Volume 6, Number 4 (2006), 1677-1743.

Dates
Received: 10 July 2006
Accepted: 29 August 2006
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513796603

Digital Object Identifier
doi:10.2140/agt.2006.6.1677

Mathematical Reviews number (MathSciNet)
MR2263047

Zentralblatt MATH identifier
1131.57031

Subjects
Primary: 57R57: Applications of global analysis to structures on manifolds, Donaldson and Seiberg-Witten invariants [See also 58-XX]
Secondary: 57R56: Topological quantum field theories 53D40: Floer homology and cohomology, symplectic aspects

Keywords
Lefschetz fibration Floer homology symmetric product TQFT

Citation

Usher, Michael. Vortices and a TQFT for Lefschetz fibrations on 4–manifolds. Algebr. Geom. Topol. 6 (2006), no. 4, 1677--1743. doi:10.2140/agt.2006.6.1677. https://projecteuclid.org/euclid.agt/1513796603


Export citation

References

  • D Auroux, S K Donaldson, L Katzarkov, Singular Lefschetz pencils, Geom. Topol. 9 (2005) 1043–1114
  • I G Avramidi, Green functions of higher-order differential operators, J. Math. Phys. 39 (1998) 2889–2909
  • M Berger, P Gauduchon, E Mazet, Le spectre d'une variété riemannienne, Lecture Notes in Mathematics, Vol. 194, Springer, Berlin (1971)
  • W Dicks, J Llibre, Orientation-preserving self-homeomorphisms of the surface of genus two have points of period at most two, Proc. Amer. Math. Soc. 124 (1996) 1583–1591
  • S K Donaldson, Lefschetz pencils on symplectic manifolds, J. Differential Geom. 53 (1999) 205–236
  • S K Donaldson, Floer homology groups in Yang-Mills theory, Cambridge Tracts in Mathematics 147, Cambridge University Press, Cambridge (2002)
  • S Donaldson, I Smith, Lefschetz pencils and the canonical class for symplectic four-manifolds, Topology 42 (2003) 743–785
  • C J Earle, J Eells, The diffeomorphism group of a compact Riemann surface, Bull. Amer. Math. Soc. 73 (1967) 557–559
  • A Floer, Morse theory for Lagrangian intersections, J. Differential Geom. 28 (1988) 513–547
  • A Floer, H Hofer, Coherent orientations for periodic orbit problems in symplectic geometry, Math. Z. 212 (1993) 13–38
  • O García-Prada, A direct existence proof for the vortex equations over a compact Riemann surface, Bull. London Math. Soc. 26 (1994) 88–96
  • P B Gilkey, Invariance theory, the heat equation, and the Atiyah-Singer index theorem, second edition, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL (1995)
  • R E Gompf, A I Stipsicz, $4$-manifolds and Kirby calculus, Graduate Studies in Mathematics 20, American Mathematical Society, Providence, RI (1999)
  • M Hutchings, Reidemeister torsion in generalized Morse theory, Forum Math. 14 (2002) 209–244
  • M Hutchings, M Sullivan, The periodic Floer homology of a Dehn twist, Algebr. Geom. Topol. 5 (2005) 301–354
  • M Hutchings, M Sullivan, Rounding corners of polygons and the embedded contact homology of $T\sp 3$, Geom. Topol. 10 (2006) 169–266
  • M Hutchings, M Thaddeus, Periodic Floer homology, in preparation
  • A Jaffe, C Taubes, Vortices and monopoles: Structure of static gauge theories
  • P B Kronheimer, T S Mrowka, Floer homology for Seiberg–Witten monopoles, book in preparation
  • P B Kronheimer, T S Mrowka, P Ozsváth, Z Szabó, Monopoles and lens space surgeries
  • Y-J Lee, Heegaard splittings and Seiberg-Witten monopoles, from: “Geometry and topology of manifolds”, Fields Inst. Commun. 47, Amer. Math. Soc., Providence, RI (2005) 173–202
  • Y-J Lee, Reidemeister torsion in Floer-Novikov theory and counting pseudo-holomorphic tori. I, J. Symplectic Geom. 3 (2005) 221–311
  • Y-J Lee, Reidemeister torsion in Floer-Novikov theory and counting pseudo-holomorphic tori. II, J. Symplectic Geom. 3 (2005) 385–480
  • G Liu, G Tian, Floer homology and Arnold conjecture, J. Differential Geom. 49 (1998) 1–74
  • G Liu, G Tian, On the equivalence of multiplicative structures in Floer homology and quantum homology, Acta Math. Sin. $($Engl. Ser.$)$ 15 (1999) 53–80
  • I G Macdonald, Symmetric products of an algebraic curve, Topology 1 (1962) 319–343
  • M Marcolli, B-L Wang, Equivariant Seiberg-Witten Floer homology, Comm. Anal. Geom. 9 (2001) 451–639
  • D McDuff, D Salamon, Introduction to symplectic topology, second edition, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York (1998)
  • J Moser, On the volume elements on a manifold, Trans. Amer. Math. Soc. 120 (1965) 286–294
  • V Muñoz, B-L Wang, Seiberg–Witten Floer homology of a surface times a circle
  • J Nielsen, Fixpunktfrie Afbildninger, Mat. Tidsskr. B (1942) 25–41 Reproduced in an English translation by J Stillwell in “Jakob Nielsen: Collected Papers”, vol. 2, Birkhäuser, Boston, 1986, 221–232
  • P Ozsváth, Z Szabó, Holomorphic disks and knot invariants, Adv. Math. 186 (2004) 58–116
  • P Ozsváth, Z Szabó, Holomorphic triangle invariants and the topology of symplectic four-manifolds, Duke Math. J. 121 (2004) 1–34
  • T Perutz, Lagrangian matching invariants for fibred four–manifolds. I
  • T Perutz, Surface–fibrations, four–manifolds, and symplectic Floer homology, PhD thesis, Imperial College (2005)
  • P Petersen, Riemannian geometry, second edition, Graduate Texts in Mathematics 171, Springer, New York (2006)
  • S Piunikhin, D Salamon, M Schwarz, Symplectic Floer-Donaldson theory and quantum cohomology, from: “Contact and symplectic geometry (Cambridge, 1994)”, (C B Thomas, editor), Publ. Newton Inst. 8, Cambridge Univ. Press, Cambridge (1996) 171–200
  • D Salamon, Lectures on Floer homology, from: “Symplectic geometry and topology (Park City, UT, 1997)”, IAS/Park City Math. Ser. 7, Amer. Math. Soc., Providence, RI (1999) 143–229
  • D A Salamon, Seiberg-Witten invariants of mapping tori, symplectic fixed points, and Lefschetz numbers, from: “Proceedings of 6th Gökova Geometry-Topology Conference”, volume 23 (1999) 117–143
  • M Schwarz, Cohomology operations from $S^1$–cobordisms in Floer homology, ETH thesis (1995)
  • P Seidel, Symplectic Floer homology and the mapping class group, Pacific J. Math. 206 (2002) 219–229
  • I Smith, Serre-Taubes duality for pseudoholomorphic curves, Topology 42 (2003) 931–979
  • A I Stipsicz, On the number of vanishing cycles in Lefschetz fibrations, Math. Res. Lett. 6 (1999) 449–456
  • C H Taubes, ${\rm SW}\Rightarrow{\rm Gr}$: from the Seiberg-Witten equations to pseudo-holomorphic curves, J. Amer. Math. Soc. 9 (1996) 845–918
  • C H Taubes, Seiberg Witten and Gromov invariants for symplectic $4$-manifolds, First International Press Lecture Series 2, International Press, Somerville, MA (2000)
  • M Usher, The Gromov invariant and the Donaldson-Smith standard surface count, Geom. Topol. 8 (2004) 565–610
  • C Viterbo, The cup-product on the Thom-Smale-Witten complex, and Floer cohomology, from: “The Floer memorial volume”, Progr. Math. 133, Birkhäuser, Basel (1995) 609–625