Trisections, intersection forms and the Torelli group

Lambert-Cole, Peter¹

¹School of Mathematics, Georgia Institute of Technology, Atlanta, GA, United States

Algebraic and Geometric Topology, Tome 20 (2020) no. 2, pp. 1015-1040

Résumé

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.

Détail

DOI : 10.2140/agt.2020.20.1015

Classification : 57M27, 57M99
Keywords: 4–manifolds, Torelli group

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