Milnor bration and bred links at

Milnor bration and bred links at innity Arnaud Bodin January 29, 1999 Introduction Let f : C 2 ! C be a polynomial function. By denition c 2 C is a regular value at innity if there exists a disc D centred at c and a compact set C of C 2 such that the map f : f 1 (D) n C ! D is a locally trivial bration. There are only a nite number of critical (or irregular) values at innity. For c 2 C and a suciently large real number R, the link at innity K c = f 1 (c) \ S 3 R is well-dened. In this paper we sketch the proof of the following theorem which gives a characterization of bred multilinks at innity. Theorem. A multilink K 0 = f 1 (0) \ S 3 R is bred if and only if all the values c 6= 0 are regular at innity. We rst obtain theorem 1, a version of this theorem was proved by A. Nemethi and A. Zaharia in [NZ] (with \semitame as a hypothesis). Here we give a new proof using resolution of singularities at innity.

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