Algebraic & Geometric Topology
- Algebr. Geom. Topol.
- Volume 20, Number 2 (2020), 1015-1040.
Trisections, intersection forms and the Torelli group
We apply mapping class group techniques and trisections to study intersection forms of smooth –manifolds. Johnson defined a well-known homomorphism from the Torelli group of a compact surface. Morita later showed that every homology –sphere can be obtained from the standard Heegaard decomposition of by regluing according to a map in the kernel of this homomorphism. We prove an analogous result for trisections of –manifolds. Specifically, if and admit handle decompositions without – or –handles and have isomorphic intersection forms, then a trisection of can be obtained from a trisection of by cutting and regluing by an element of the Johnson kernel. We also describe how invariants of homology –spheres can be applied, via this result, to obstruct intersection forms of smooth –manifolds. As an application, we use the Casson invariant to recover Rohlin’s theorem on the signature of spin –manifolds.
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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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. https://projecteuclid.org/euclid.agt/1588212081