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Trisections, intersection forms and the Torelli group

Peter Lambert-Cole

Algebraic & Geometric Topology 20 (2020) 1015–1040

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.

4–manifolds, Torelli group
Mathematical Subject Classification 2010
Primary: 57M27, 57M99
Received: 26 March 2019
Revised: 24 July 2019
Accepted: 9 August 2019
Published: 23 April 2020
Peter Lambert-Cole
School of Mathematics
Georgia Institute of Technology
Atlanta, GA
United States