Algebraic & Geometric Topology

Trisections, intersection forms and the Torelli group

Peter Lambert-Cole

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We apply mapping class group techniques and trisections to study intersection forms of smooth 4–manifolds. Johnson defined a well-known homomorphism from the Torelli group of a compact surface. Morita later showed that every homology 3–sphere can be obtained from the standard Heegaard decomposition of S3 by regluing according to a map in the kernel of this homomorphism. We prove an analogous result for trisections of 4–manifolds. Specifically, if X and Y admit handle decompositions without 1– or 3–handles and have isomorphic intersection forms, then a trisection of Y can be obtained from a trisection of X by cutting and regluing by an element of the Johnson kernel. We also describe how invariants of homology 3–spheres can be applied, via this result, to obstruct intersection forms of smooth 4–manifolds. As an application, we use the Casson invariant to recover Rohlin’s theorem on the signature of spin 4–manifolds.

Article information

Algebr. Geom. Topol., Volume 20, Number 2 (2020), 1015-1040.

Received: 26 March 2019
Revised: 24 July 2019
Accepted: 9 August 2019
First available in Project Euclid: 30 April 2020

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M27: Invariants of knots and 3-manifolds 57M99: None of the above, but in this section

4–manifolds Torelli group


Lambert-Cole, Peter. Trisections, intersection forms and the Torelli group. Algebr. Geom. Topol. 20 (2020), no. 2, 1015--1040. doi:10.2140/agt.2020.20.1015.

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  • J S Birman, R Craggs, The $\mu $–invariant of $3$–manifolds and certain structural properties of the group of homeomorphisms of a closed, oriented $2$–manifold, Trans. Amer. Math. Soc. 237 (1978) 283–309
  • S K Donaldson, An application of gauge theory to four-dimensional topology, J. Differential Geom. 18 (1983) 279–315
  • B Farb, D Margalit, A primer on mapping class groups, Princeton Mathematical Series 49, Princeton Univ. Press (2012)
  • P Feller, M Klug, T Schirmer, D Zemke, Calculating the homology and intersection form of a $4$–manifold from a trisection diagram, Proc. Natl. Acad. Sci. USA 115 (2018) 10869–10874
  • M Freedman, R Kirby, A geometric proof of Rochlin's theorem, from “Algebraic and geometric topology, II” (R J Milgram, editor), Proc. Sympos. Pure Math. XXXII, Amer. Math. Soc., Providence, RI (1978) 85–97
  • D Gay, R Kirby, Trisecting $4$–manifolds, Geom. Topol. 20 (2016) 3097–3132
  • D Johnson, Quadratic forms and the Birman–Craggs homomorphisms, Trans. Amer. Math. Soc. 261 (1980) 235–254
  • D Johnson, The structure of the Torelli group, II: A characterization of the group generated by twists on bounding curves, Topology 24 (1985) 113–126
  • D Johnson, The structure of the Torelli group, III: The abelianization of $\mathscr{T}$, Topology 24 (1985) 127–144
  • M A Kervaire, J W Milnor, On $2$–spheres in $4$–manifolds, Proc. Nat. Acad. Sci. U.S.A. 47 (1961) 1651–1657
  • R C Kirby, The topology of $4$–manifolds, Lecture Notes in Mathematics 1374, Springer (1989)
  • R Kirby, P Melvin, The $3$–manifold invariants of Witten and Reshetikhin–Turaev for $\mathrm{sl}(2,\mathbb{C})$, Invent. Math. 105 (1991) 473–545
  • R C Kirby, M G Scharlemann, Eight faces of the Poincaré homology $3$–sphere, from “Geometric topology” (J C Cantrell, editor), Academic, New York (1979) 113–146
  • F Laudenbach, V Poénaru, A note on $4$–dimensional handlebodies, Bull. Soc. Math. France 100 (1972) 337–344
  • H B Lawson, Jr, M-L Michelsohn, Spin geometry, Princeton Mathematical Series 38, Princeton Univ. Press (1989)
  • S Morita, Casson's invariant for homology $3$–spheres and characteristic classes of surface bundles, I, Topology 28 (1989) 305–323
  • S Morita, On the structure of the Torelli group and the Casson invariant, Topology 30 (1991) 603–621
  • V A Rohlin, New results in the theory of four-dimensional manifolds, Doklady Akad. Nauk SSSR 84 (1952) 221–224 In Russian
  • N Saveliev, Invariants for homology $3$–spheres, Encyclopaedia of Mathematical Sciences 140, Springer (2002)
  • F Waldhausen, Heegaard–Zerlegungen der $3$–Sphäre, Topology 7 (1968) 195–203