Osaka Journal of Mathematics

On diagrams of simplified trisections and mapping class groups

Kenta Hayano

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A simplified trisection is a trisection map on a 4--manifold such that, in its critical value set, there is no double point and cusps only appear in triples on innermost fold circles. We give a necessary and sufficient condition for a 3--tuple of systems of simple closed curves in a surface to be a diagram of a simplified trisection in terms of mapping class groups. As an application of this criterion, we show that trisections of spun 4--manifolds due to Meier are diffeomorphic (as trisections) to simplified ones. Baykur and Saeki recently gave an algorithmic construction of a simplified trisection from a directed broken Lefschetz fibration. We also give an algorithm to obtain a diagram of a simplified trisection derived from their construction.

Article information

Osaka J. Math., Volume 57, Number 1 (2020), 17-37.

First available in Project Euclid: 15 January 2020

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Mathematical Reviews number (MathSciNet)

Primary: 57R45: Singularities of differentiable mappings
Secondary: 20F38: Other groups related to topology or analysis 57M99: None of the above, but in this section 57R65: Surgery and handlebodies


Hayano, Kenta. On diagrams of simplified trisections and mapping class groups. Osaka J. Math. 57 (2020), no. 1, 17--37.

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