#### Volume 25, issue 6 (2021)

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Vanishing cycles, plane curve singularities and framed mapping class groups

Geometry & Topology 25 (2021) 3179–3228
##### Abstract

Let $f$ be an isolated plane curve singularity with Milnor fiber of genus at least $5$. For all such $f\phantom{\rule{-0.17em}{0ex}}$, 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 $f\phantom{\rule{-0.17em}{0ex}}$. 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