Vanishing cycles, plane curve singularities and framed mapping class groups

Portilla Cuadrado, Pablo; Salter, Nick

doi:10.2140/gt.2021.25.3179

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Vanishing cycles, plane curve singularities and framed mapping class groups

Pablo Portilla Cuadrado and Nick Salter

Geometry & Topology 25 (2021) 3179–3228

DOI: 10.2140/gt.2021.25.3179

Abstract

Let $f$ be an isolated plane curve singularity with Milnor fiber of genus at least $5$ . For all such $f$ , we give an intrinsic description of the geometric monodromy group that does not invoke the notion of the versal unfolding space, and an easy criterion to decide if a given simple closed curve in the Milnor fiber is a vanishing cycle or not. With the lone exception of singularities of type $A_{n}$ and $D_{n}$ , we find that both are determined completely by a canonical framing of the Milnor fiber induced by the Hamiltonian vector field associated to $f$ . As a corollary we answer a question of Sullivan concerning the injectivity of monodromy groups for all singularities having Milnor fiber of genus at least $7$ .

Keywords

singularity theory, mapping class groups, low dimensional topology

Mathematical Subject Classification

Primary: 14D05, 57R45

References

Publication

Received: 15 May 2020

Revised: 9 September 2020

Accepted: 22 October 2020

Published: 30 November 2021

Proposed: András I Stipsicz

Seconded: Dan Abramovich, Benson Farb

Authors

Pablo Portilla Cuadrado
Departamento de Matemáticas Básicas CIMAT Guanajuato Mexico

Nick Salter
Department of Mathematics Columbia University New York, NY United States

Volume 25, issue 6 (2021)