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2020 Trisections, intersection forms and the Torelli group
Peter Lambert-Cole
Algebr. Geom. Topol. 20(2): 1015-1040 (2020). DOI: 10.2140/agt.2020.20.1015

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 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.

Citation

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Peter Lambert-Cole. "Trisections, intersection forms and the Torelli group." Algebr. Geom. Topol. 20 (2) 1015 - 1040, 2020. https://doi.org/10.2140/agt.2020.20.1015

Information

Received: 26 March 2019; Revised: 24 July 2019; Accepted: 9 August 2019; Published: 2020
First available in Project Euclid: 30 April 2020

zbMATH: 07195383
MathSciNet: MR4092318
Digital Object Identifier: 10.2140/agt.2020.20.1015

Subjects:
Primary: 57M27 , 57M99

Keywords: 4–manifolds , Torelli group

Rights: Copyright © 2020 Mathematical Sciences Publishers

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