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Torsions and intersection forms of 4-manifolds from trisection diagrams

Part of: Low-dimensional topology PL-topology

Published online by Cambridge University Press:  14 December 2020

Vincent Florens  and
Delphine Moussard
Vincent Florens*
Affiliation:
Université de Pau, Pau, Laboratoire LMAP, France e-mail: [email protected]
Delphine Moussard
Affiliation:
Université d’Aix-Marseille, I2M, Marseille, France
*
e-mail: [email protected]
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Abstract

Gay and Kirby introduced trisections, which describe any closed, oriented, smooth 4-manifold X as a union of three 4-dimensional handlebodies. A trisection is encoded in a diagram, namely three collections of curves in a closed oriented surface $\Sigma $ , guiding the gluing of the handlebodies. Any morphism $\varphi $ from $\pi _1(X)$ to a finitely generated free abelian group induces a morphism on $\pi _1(\Sigma )$ . We express the twisted homology and Reidemeister torsion of $(X;\varphi )$ in terms of the first homology of $(\Sigma ;\varphi )$ and the three subspaces generated by the collections of curves. We also express the intersection form of $(X;\varphi )$ in terms of the intersection form of $(\Sigma ;\varphi )$ .

Keywords

4-manifold trisectionsintersection formsReidemeister torsion

MSC classification

Primary: 57Q10: Simple homotopy type, Whitehead torsion, Reidemeister-Franz torsion, etc.
Secondary: 57M99: None of the above, but in this section
Type
Article
Information
Canadian Journal of Mathematics , Volume 74 , Issue 2 , April 2022 , pp. 527 - 549
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Copyright
© Canadian Mathematical Society 2020

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Footnotes

The first author was partially supported by the ANR Project LISA 17-CE40-0023-01. While working on the contents of this paper, the second author has been supported by a Postdoctoral Fellowship of the Japan Society for the Promotion of Science. She is grateful to Tomotada Ohtsuki and the Research Institute for Mathematical Sciences for their support. She is now supported by the Région Bourgogne Franche-Comté project ITIQ–3D. She thanks Gwénaël Massuyeau and the Institut de Mathématiques de Bourgogne for their support.

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