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Revista Colombiana de Matemáticas
Print version ISSN 0034-7426
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 → Dr0 where E is a complex surface with boundary, Dr0 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 fg:Eg → D10 having genus g regular fiber and violating this conjecture.
Keywords : Milnor number; monodromy; right handed Dehn twist; morsification.