Vanishing cycles, plane curve singularities and framed mapping class groups

Portilla Cuadrado, Pablo¹ ; Salter, Nick²

¹ Departamento de Matemáticas Básicas, CIMAT, Guanajuato, Mexico
² Department of Mathematics, Columbia University, New York, NY, United States

Geometry & topology, Tome 25 (2021) no. 6, pp. 3179-3228.

Résumé

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 $7$.

Détail

DOI : 10.2140/gt.2021.25.3179

Keywords: singularity theory, mapping class groups, low dimensional topology

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