Trisections, intersection forms and the Torelli group

Abstract

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 $S^3$ 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.