Volume 25, issue 6 (2021)

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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

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 An and Dn, 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.

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singularity theory, mapping class groups, low dimensional topology
Mathematical Subject Classification
Primary: 14D05, 57R45
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
Pablo Portilla Cuadrado
Departamento de Matemáticas Básicas
Nick Salter
Department of Mathematics
Columbia University
New York, NY
United States