Trisections, intersection forms and the Torelli group (Q2175887)

Trisections of closed 4-manifolds are analogous to Heegaard splittings of 3-manifolds; specifically, a trisection of a closed 4-manifold \(X\) is a decomposition of \(X\) into three 4-dimensional handlebodies \(\sharp (S^1 \times B^3)\) such that the intersection of each two of these 4-dimensional handlebodies is a 3-dimensional handlebody \(\sharp (S^1 \times B^2\)), and the intersection of all three of them is a closed orientable surface. \textit{S. Morita} showed [Topology 28, No. 3, 305--323 (1989; Zbl 0684.57008); ibid. 30, No. 4, 603--621 (1991; Zbl 0747.57010)] that every integral homology 3-sphere can be obtained by cutting the standard Heegaard splitting of the 3-sphere along the Heegaard surface and regluing by an element of the Torelli group (the subgroup of the mapping class group acting trivially on homology) which lies in the kernel of the Johnson homomorphism (the subgroup of the Torelli group generated by seperating twists: \textit{D. Johnson} [Topology 24, 127--144 (1985; Zbl 0571.57010)]). The main result of the present paper is a 4-dimensional analogue of Morita's result. ``Specifically, if \(X\) and \(Y\) admit handle decompositions without 1- and 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.''