2 S.YU.OREVKOV est conuguee a la tresse, construite de cette facon. Une tresse est dite algebrique si elle est une tresse locale d'une fonction multiv

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1 REALIZABILITY OF A BRAID MONODROMY BY AN ALGEBRAIC FUNCTION IN A DISK S.Yu.Orevkov Abstract. A braid is called algebraic if it is conugated to the local braid of an algebraic function at a singular point. It is shown that any homomorphism of a free group into a braid group which takes each generator to an algebraic braid, can be realized as the braid monodromy of an algebraic function in a disk. Realisabilite d'une monodromie des tresses par une fonction algebrique dans le disque. Resume. Une tresse appelee algebrique si elle est conuguee a la tresse locale d'une fonction algebrique en un point singulier. On demontre que tout homomorphisme du groupe libre dans le groupe des tresses, qui envoie chaque generateur sur une tresse algebrique, peut ^etre realise comme monodromie des tresses d'une fonction algebrique dans le disque. Version francaise abregee x. Monodromie des tresses. Soit p(x; y) 2 C[z; w] un polyn^ome sans facteurs multiples de deux variables complexes. Supposons que p soit unitaire par rapport a w, c.-a-d. p = w n + a (z)w n + + a n (z). Alors, pour chaque domaine D C, le polyn^ome p denit une fonction multivoque F (z) = fw p(z; w) = 0g. Cette fonction a n valeurs en un point generique, et moins que n valeurs dans un ensemble ni S = S(F ) = fz ; : : : ; z s g. Le fait que p soit unitaire implique que F n'a pas des p^oles. Notons V n C[w] la variete des polyn^omes unitaires de degre n et soit n V n l'hypersurface du discriminant, constituee des polyn^omes qui ont des racines multiples. Evidemment, V n ' C n. On considere le groupe des tresses a n brins B n = (V n n ; p 0 ), ou p 0 est un polyn^ome xe. Soit ~p : C! V n l'application denie par ~p(z 0 ) = p(z 0 ; ). Par denition de S, le polyn^ome ~p(z) 62 n si z 62 S. Fixons un point de base z 0 2 D S et un chemin dans V n n entre ~p(z 0 ) et p 0. On peut denir l'homomorphisme ~p : (D S)! (V n n ) = B n, appele monodromie des tresses de la fonction algebrique F (z). Etant donne un lacet : [0; ]! D S, la classe de conugaison de la tresse ~p ([]) ne depend que de F et. Considerons un germe en z = 0 de fonction n-valuee w = F (z) denie par un polyn^ome unitaire de w dont les coecients dependent analytiquement de z. Pour un " assez petit, la tresse associee au chemin t 7! "e it, t 2 [0; 2] ne depend pas de " a conugaison pres. On dit qu'une tresse est une tresse locale de F en z = 0 si elle I am grateful to University Bordeaux-I for nancial support and hospitality Typeset by AMS-TEX

2 2 S.YU.OREVKOV est conuguee a la tresse, construite de cette facon. Une tresse est dite algebrique si elle est une tresse locale d'une fonction multivoque en un point singulier. Soient S 0 = f; : : : ; sg et D 0 C un disque contenant S 0. Fixons une systeme de generateurs ; : : : ; s du (D 0 S 0 ) tel que i soit represente par un petit cercle autour du point. Proposition. Soient b ; : : : ; b s 2 B n des tresses algebriques et soit D C un disque. Alors, il existent un dieomorphisme ' : (D 0 ; S 0 )! (D; S) et une fonction algebrique w = F (z) denie par un polyn^ome p(z; w), unitaire par rapport a w, tels que S(F ) = S et la monodromie des tresses ~p envoie les generateurs ' ( ); : : :, ' ( s ) de (D S) sur b ; : : : ; b s. Exemple. Toute tresse conuguee a un generateur standard de B n, est algebrique (elle est une tresse locale de F (z) = f p z; 3; 4; : : : ; ng). Donc, si on choisi les i tels soit librement homotope a Q, la proposition donne le theoreme de L. Rudolph [] qui dit que toute tresse quasi-positive peut ^etre realisee par une fonction algebrique dans un disque (une tresse est dite quasi-positive si elle est un produit de tresses conuguees aux generateurs standards du groupe des tresses). La proposition est une consequence immediate de la proposition 2. x2. Lacets analytiques dans C n. Soit une hypersurface algebrique dans C n. Etant donne un voisinage U C de 0 et une application analytique f : U! C n telle que f(u 0) C n, on dit que la classe d'homotopie libre locale de f en 0 est la classe de conugaison dans (C n ), qui correspond au lacet [0; 2]! C n, t 7! f("e it ) pour ". Soient (D 0 ; S 0 ) et ; : : : ; s comme ci-dessus. Proposition 2. Soit U C un voisinage de 0. Pour chaque = ; : : : ; s, soit f : (U; 0)! (C n ; ), une application analytique telle que f (U) C n et soit b 2 (C n ) un representant arbitraire de la classe d'homotopie libre locale de f en 0. Soit D C un disque. Alors, il existe un dieomorphisme ' : (D 0 ; S 0 )! (D; S) et une application polyn^omiale p : D! C n tels que p () \ D = S et p (' ( )) = b, = ; : : : ; s. L'auteur tient a remercier M. Delzant pour avoir corrige le francais. x. Braid monodromy. Let p(z; w) 2 C[z; w] be a reduced (i.e. without multiple factors) polynomial in two complex variables. Suppose that p is monic with respect to w, i.e. p = w n + a (z)w n + + a n (z). Then in any domain D C, the polynomial p denes a multi-valued function F (z) = fw p(z; w) = 0g. This function takes n values in a generic point of D and takes less than n values in a nite set S = S(F ) = fz ; : : : ; z s g. The fact that p is monic means that F has no poles. Denote by V n C[w] the variety of all monic polynomials of degree n, and let n V n be the discriminant hypersurface consisting of polynomials with multiple roots. Clearly that V n ' C n. We dene the braid group on n strings as B n = (V n n ; p 0 ), where p 0 is a xed polynomial.

3 REALIZABILITY OF A BRAID MONODROMY 3 Let ~p : C! V n be the mapping dened by ~p(z 0 ) = p(z 0 ; ). By the denition of S, we have ~p(z) 62 n for z 62 S. Chose a base point z 0 2 D S and a path in V n n between ~p(z 0 ) and p 0. Then we can dene the homomorphism ~p : (D S)! (V n n ) = B n which is called the braid monodromy of the algebraic function F (z). (We do not speak of braid monodromy of a curve, because it depends on the choice of coordinates). The braid monodromy is dened by F up to an inner automorphism of B n and it determines (D C; graph(f )) up to an ambient isotopy of D C preserving the proection onto D. Given a closed path : [0; ]! D S, the conugacy class of the braid ~p ([]) depends only on F and and does not depend on the choice of base points and paths between them. Consider a germ at z = 0 of an n-valued function w = F (z) dened by a monic reduced polynomial in w with coecients analytically depending on z. For " suciently small, the braid associated with the path t 7! "e it, t 2 [0; 2] does not depend on " up to conugacy. A braid is called a local braid of F at 0 if it is conugated to a braid which appears in this way. A braid is called algebraic if it is a local braid of a multi-valued function at a singular point. L. Rudolph [] has proved that if is a simple closed positively oriented path then the braid ~p ([]) is quasipositive (equivalent to a product of braids conugated to standard generators of the braid group) and that any quasipositive braid can be realized as the boundary braid of some algebraic function on a disk without poles. We generalize the Rudolph's theorem showing (Proposition below) that any homomorphism (D S)! B n which takes small loops around points of S to algebraic braids, can be realized as the braid monodromy of an algebraic function without poles in a disk. Let S 0 = f; : : : ; sg and let D 0 C be a disk containing S 0. Fix a system of generators ; : : : ; s of (D 0 S 0 ) such that is represented by a small positively oriented loop around the point. Proposition. Let b ; : : : ; b s 2 B n be any algebraic braids and D C a disk. Then there exist a dieomorphism ' : (D 0 ; S 0 )! (D; S) and an algebraic function w = F (z) on D dened by a monic in w polynomial p(z; w) such that S(F ) = S and the braid monodromy ~p takes the generators ' ( ); : : :, ' ( s ) of (D S) to b ; : : : ; b s. Example. Any braid conugated to a standard generator of B n, is algebraic (it is a local braid of F (z) = f p z; 3; 4; :::; ng). Thus, if we chose 's so is freely homotopic to Q then Proposition gives the Rudolph's theorem mentioned above. Remark. Proposition remains true if we omit in it (and in the denition of algebraic braids) the condition that the polynomials in w are monic. Proposition as well as its non-monic version are direct consequences of Proposition 2 below. x2. Analytic loops in C n. Let be any algebraic hypersurface in C n given by = 0. If U C is a neighborhood of 0 and f : U! C n is an analytic mapping such that f(u 0) C n then we dene the local free homotopy class of f at 0 as the conugacy class in (C n ) corresponding to the loop [0; 2]! C n, t 7! f("e it ) for a suciently small ". Let (D 0 ; S 0 ) and ; : : : ; s be as above.

4 4 S.YU.OREVKOV Proposition 2. Let U C be some neighborhood of 0. For = ; : : : ; s, let f : (U; 0)! (C n ; ) be an analytic mapping such that f (U 0) C n, and let b 2 (C n ) be any representative of the local free homotopy class of f at 0. Let D C be a disk. Then there exist a dieomorphism ' : (D 0 ; S 0 )! (D; S) and a polynomial mapping p : D! C n such that p () \ D = S and for the induced mapping (D 0 S 0 )! (D S)! (C n ) one has p (' ( )) = b, = ; : : : s. The rest of the paper is devoted to the proof of Proposition 2. Introduce the following notation. For paths k : [a k ; b k ]! X, k = ; 2 we denote by 2 their product and by k the inverse in the sense of the multiplication in the fundamental gruppoid of X ( 2 is dened only when 2 (a 2 ) = (b )). The class of a loop in (X) we denote by []. Thus, [ ] = [] and [] = [][]. In other aspects we treat paths as functions, for instance, if : [a; b]! C is a path and c; d 2 C then c+d is ust t 7! c(t)+d. We use the same notation for a segment [a; b] C and for a path which parameterizes it. Denote the path [0; 2]! C, t 7! e it by and let = [0;]. For = ; 2; ::: and 0 < " < = 2 dene the path ;" as ( +"), where = [0; "] ( + " ) [ + "; 2 "] (2 + " ) [ + "; "]. ;" : Suppose that a (non-closed) path f : ([0; a]; S 0 )! (C n ; ) is analytic near S 0. For 0 < < a and " suciently small the element [f ;" ] 2 (C n ) does not depend on " and we denote it by [[f]]. Lemma. Let f 0 ; f : ([0; a]; S 0 )! (C n ; ) be two paths which are analytic near S 0. Let N = + ord x= ( f 0 ). Suppose that (i) f 0 (0) = f (0), (ii) N -ets of f 0 and f at x coincide, = ; :::; s and (iii) f (x) f 0 (x) < dist(f 0 (x); ) for any x 2 [0; a] n S 0. Then [[f ]] = [[f 0 ]]. Proof. Continue f 0 ; f into some complex neighborhoods of points ; 2; ::: Let f u = f 0 + u(f f 0 ). For any we have (f 0 (x)) = (x ) N + (x ) N g (x) and f f 0 = (x ) N h (x) where g and h are analytic near x =. Hence, (f u (x)) = (x ) N + (x ) N H (x; u) where H is analytic near the line x = in C 2. Hence, there exists > 0 such that for any, H is continuous in V ; = fx 2 C : x < g [0; ]. Let M = max max V; H =. Then for " < min(; =M) we have a homotopy f u ;" in C n. Without loss of generality we may assume that = [ ;" ] for some ". Otherwise, for each we replace with [ ;" ] and b with W (b ; :::; b s ) where W is the word such that [ ;" ] = W ( ; :::; s ). Lemma 2. In the hypothesis of Proposition 2 there exists a path f : ([0; a]; S 0 )! (C n ; ), analytic near S 0, such that [[f]] = b for any = ; :::; s. Proof. Let " be so small that b = c (f ") c for some c 's. If x < ", we put f(x) = f (x). Dene f on [0; "] as a reparameterization of c and on [ + "; + "] for as a reparameterization of (f " ) c c +.

5 REALIZABILITY OF A BRAID MONODROMY 5 Lemma 3. Let N be an integer and S = fx ; :::; x s g [a; b] R. Let f and be real continuous functions on [a; b] such that f is analytic near S and is piecewise algebraic near S. Suppose that 0 on [a; b] and (x) > 0 for x 62 S. Then there exists a polynomial p such that for x 62 S one has f(x) p(x) < (x) and the N -ets of f and p at x coincide for any = ; : : : ; s. Proof. Denote Q by 2 Q the order of zero of at x and let 2 2Z, > max. Put q = (x x ). Then q = is continuous on [a; b]. Denote by M its maximum and let q = q =M. Then on [a; b] we have 0 q. Put N = max(n; ). Let p be a polynomial whose N -ets at x ; : : : ; x s are the same as those of f and put g = (f p )=q. The function g is continuous and hence, by Weierstrass approximation theorem, there exists a polynomial p 2 such that g p 2 <. Then for p = p + p 2 q and x 2 [a; b] S we have f p= f p=q = g p 2 <. Proof of Proposition 2. Let f be as in Lemma 2. Put a = s + = 2, (x) = dist(f(x); ) and N = + max ord x= f. Applying Lemma 3 to Re and Im of each coordinate, we can approximate f by a polynomial mapping p : [0; a]! C n with the same N-ets at S 0, so that p f < on [0; a] S 0. Hence, [[p ]] = [[f]] = b by Lemmas, 2. Chose " < dist([0; a]; p () S 0) and let D be the "-neighborhood of [0; a] in C. Let ' : D! D be a polynomial approximation of a conformal isomorphism D! D and let t : (' (D); S 0 )! (D 0 ; S 0 ) be a homotopy such that 0 = id and is a dieomorphism. To complete the proof, put p = p ', ' = ' and note that p (' ( )) = [p ;" ] = [p ;" ] = [[p ]] = b. REFERENCES. L.Rudolph, Algebraic functions and closed braids, Topology 22 (983), System Studies Institute, Russ. Acad. Sci. Avtozavodskaya 23, Moscow, Russia Laboratoire de Mathematiques Pures de Bordeaux, Univ. Bordeaux I, 35 crs. de la Liberation, Talence, France address: orevkov@math.u-bordeaux.fr