Abstract

CADAVID, CARLOS A.  and  VELEZ, JUAN D.. Towards a new interpretation of Milnor's number. Rev.colomb.mat. [online]. 2008, vol.42, n.2, pp.153-166. ISSN 0034-7426.

The Milnor number is a fundamental invariant of the biholomorphism type of the singularity of the germ of a holomorphic function f defined on an open neighborhood W of 0 ∈ Cⁿ, and such that 0 is the only critical point of f in W. The present article describes a conjecture that would provide an interpretation of this invariant, in the case n=2, as a sharp lower bound for the number of factors in any factorization in terms of right-handed Dehn twists of the monodromy around the singular fiber of f. Also, towards the end of the paper, an analogue conjecture for proper holomorphic maps f:E → D_r⁰ where E is a complex surface with boundary, D_r⁰ is {z ∈ C: |z| < r }, and f has f^-1(0) as its unique singular fiber and all other fibers are closed and connected 2-manifolds of (necessarily the same) genus g ≥ 0, is briefly described. The latter conjecture has been proved recently by the authors in the case when the regular fiber of f has genus 1 ([3]), and in ([5]), that author provides for each g ≥ 2 an f_g:E_g → D₁⁰ having genus g regular fiber and violating this conjecture.

Keywords : Milnor number; monodromy; right handed Dehn twist; morsification.

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