V. I. Arnold, The cohomology ring of the colored braid group (Russian), Mat. Zametki 5 (  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Tits de type de Coxeter fini. D un point de vue technique, les demonstrations reposent sur les proprietes des ppcm dans un monode. Introduction The (co)homology of Artin s braid groups B n has been computed by methods of di#erential geometry and algebraic topology in the beginning of the 1970 s [3, 4, 29, 16], and the results have then been extended to Artin Tits groups of finite Coxeter type [8, 31, 46] see also [17, 18, 19, 38, 39, 47] A purely algebraic and combinatorial approach was developed by C. Squier in his unpublished PhD thesis of 1980 see [42] relying both on the fact that these ....

V.I. Arnold, The cohomology ring of the colored braid group, Mat. Zametki 5 (1969) 227--231.

....One noteworthy feature is that this result supplies Spin representations arising from the structure of the cohomology algebra for the pure braid group. The theorem also gives the (redundant) statement that Spin representations of the pure braid group give trivial stable vector bundles. Recall [1, 10] that the cohomology ring of the configuration space F (R , n) of n ordered points in R is the quotient of the exterior algebra generated by elements A i,j of degree 1 for 1 n by the ideal generated by the relations A i,j n. The relation A i,j A t,j A i,t A t,j A ....

....# R. Note that M(#) has order 2. Suppose N O(2) commutes with M(#) If the determinant of N is a calculation reveals that N = M(#) If the determinant of N is 1, a similar calculation shows that N = 2 . Thus, for all i, either #(e i ) M(#) or #(e i ) 2 . Now define H : [0, 1] O(2) by the formula H(t, e i ) M(t#) if #(e i ) M(#) 2 if #(e i ) 2 . Notice that the elements appearing as values of H(t, e i ) all commute for any fixed value of t. Thus there is a unique extension of H such that H(t, is a group homomorphism. Consequently, # is ....

V.I. Arnol'd, The cohomology ring of the colored braid group, Math. Notes Academy Sci. USSR 5 (1969), 138--140.

....or Coxeter arrangement (of type An 1 ) is the arrangement An 1 of hyperplanes in Vn 1 R A A A A A A A A A A r Figure 1: The Coxeter hyperplane arrangement A 2 . It is clear that A has r(An 1 ) n regions (called Weyl chambers) and b(An 1 ) 0 bounded regions. Arnold [1] calculated the cohomology ring H (CAn ; C ) In particular, he proved that PoinAn 1 (q) 1 q) 1 2q) 1 (n 1)q) 3.2) In this paper we will study deformations of the arrangement (3.1) which are hyperplane arrangements in Vn 1 R of the following type: x i x j = a ij ; ....

....) c s ; c s 1 ; c r ) are balanced. Equation (8.4) for C is the sum of the equations for C . Thus the statement follows by induction. Remark 8.12 This proposition is an analogue to the well known description of the cohomology ring of the Coxeter arrangement (3. 1) due to Arnold [1]. This cohomology ring is generated by e vw = e (v;w) 1 v w n, subject to relations (8.2) 8.3) and also the following triangle equation: e ab e bc e ab e ac e bc e ac = 0; where 1 a b c n. 9 Truncated Ane Arrangements In this section we study a general class of hyperplane ....

V. I. Arnold, The cohomology ring of colored braid group, Math. Notes 5 (1969), 138-140.

....be the cohomology of the configuration space of the complex line. Given distinct i and j in f1; ng, let ij 2 H (F(C ; n) Z) be the integral cohomology class represented by the closed differential form ij = 1 d(z i Gamma z j ) z i Gamma z j By induction on n, Arnold shows in [1] that the cohomology ring H (F(C ; n) Z) is generated by the classes ij , subject to the relations ij jk jk ki ki ij = 0: 1.13) The action of the group S n on the configuration space F(C ; n) induces an action on the cohomology ring H (F(C ; n) Z) which permutes the ....

....Then (i) real C ffl (X; U ; F) Gammad;0] is isomorphic to j ffl j (ii) real cone C ffl (X; U ; F) F Delta Delta [ Gammad Gamma1;0] is isomorphic to i ffl i F . Proof. Note that (ii) is implied by (i) and the five lemma: in the diagram 0 F i ffl i F j ffl j F [1] 0 0 F cone C ffl (X; U ; F) Gamma F C ffl (X; U ; F) 1] 0 the top row is exact by axiom (iii) for exact Mackey 2 functors, and the bottom row is obviously exact. We now prove (i) by induction on d: for d = 0, C ffl (X; U ; F) j ffl j F , and the proposition is a ....

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V.I. Arnold, The cohomology ring of the colored braid group, Mat. Zametki 5 (1969), 227--231.

.... proof of the existence of sign twists in the action of Sn on the homology of a product of n copies of a poset P: These sign twists predict, for example, the exterior algebra structure of the cohomology of the complement of the complexified hyperplane arrangement of type An Gamma1 (see Arnold [A]) This action is of added interest because in many of the recently studied examples of hyperplane arrangements, and more generally subspace arrangements (see [Bj3] BjWe] and [SWe1] lower intervals in the intersection lattice Ln of the arrangement (where n is the dimension of the ambient ....

V.I. Arnold, The cohomology ring of the colored braid group, Math. Notes 5 (1969), 138-140.

....cones (times a copy of R) Hence, the cohomology of M n;2 is free with fi n;2 ) n as its only non vanishing Betti number. Furthermore, M n;2 is known as the pure braid space . It is an Eilenberg MacLane space with the pure braid group as fundamental group [FaN62] FoN62] Arnol d [Arn69] showed that the cohomology of M n;2 is free and he also computed its Betti numbers. See [OT92] for more information about the k = 2 case. The interest in obtaining information about the cohomology of M n;k , also for k 3, arose in connection with a problem from computer science in ....

....groups What can be said about the fundamental group of M n;3 Is M n;3 a K( 1) space 7. 3 Algebra structure of the cohomology rings Can the multiplicative structure of the free graded modules H n;k ) and H n;k ) be described This was done in the k = 2 case by Arnol d [Arn69] see also Orlik and Solomon [OS80] and [OT92] 7.4 CW complexes Is there any geometrically motivated CW complex C, such that C and M n;k are homotopy equivalent and C has cells only in dimensions t Delta (k Gamma 2) for 0 t b k c 7.5 Recursions for Betti numbers Let b(n; k) ....

V.I. Arnol'd. The cohomology ring of the colored braid group. Math. Notes 5, 138--140 (1969).

....Braid Arrangement The classical braid arrangement A n in complex n space is given by the thick diagonals H ij : z i = z j for 1 i j n. The braid arrangement, also known as the complexified Coxeter arrangement of type A, is a well studied object (see for example Fox Neuwirth [7] Arnol d [1], Brieskorn [4] and Lehrer Solomon [8] Its name is derived from the fact that by a result of Fox Neuwirth [7] the complement MAn is the classifying space of the pure braid group on n strings. We enlarge the central (i.e. all hyperplanes pass through the origin) arrangement A n by some ....

.... Then = S jB1 j Theta Delta Delta Delta Theta S jB f Gamma1 j Theta S e1 [S 1 ] Theta Delta Delta Delta Theta S en [S n ] The S n character on e H n Gammat f Gamma2 ( 0; l 1 ; l t ) is given by G r jB1 j Delta Delta Delta r jB f Gamma1 j Delta sgn [ 1 ] Delta 1 e2 [ 2 ] Delta Delta Delta Proof. The assertion follows immediately from Proposition 4.2 and the [16, Theorem 1.1] We are grateful to Richard Stanley for pointing out that the characteristic polynomial (see [14] of Pi n can be easily computed using a result about ....

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V.I. Arnol'd, The cohomology ring of the colored braid group, Math. Notes 5 (1969), 138--140.

....commutative quotient of E n , since they satisfy the relations (2.2) and (2.3) but not (2. 1) It would be interesting to understand the connections between the algebra E n and the Orlik Solomon algebra [22] that corresponds to the Coxeter hyperplane arrangement (of type A) Recall that Arnold [1] described this algebra as the quotient of the exterior algebra of the vector space spanned by the [ij] by the ideal generated by the left hand sides of (2.6) 7 5 Dunkl elements The Dunkl elements j , for j = 1; n, in the quadratic algebra E n are defined by j = Gamma 1i j ....

.... Gamma [15] 12] 13] 14] 15] Gamma[13] 14] 25] 15] 13] Gamma [14] 25] 15] 13] 14] 15] 12] 13] 14] 15] Gamma[25] 15] 13] 14] 15] 13] 14] 15] 13] 14] 15] 13] 14] i i 1] Delta Delta Delta [i m] i 1] i 2] Delta Delta Delta [i i Gamma 1] 0 ; 7. 5) where [1 0] should be interpreted as [1 m] Proof We present the proof in a manner similar to the proof of Lemma 7.2 above. Induction on m. Base of induction (m = 3) is [12] 13] 23] 21] 31] 32] 0 ; which is immediate from (2.4) 2.6) Again, we will explain the general induction argument by ....

V. I. Arnold, The cohomology ring of the colored braid group, Math. Notes 5 (1969), 138--140.

....always denote dimension of an arrangement. The braid arrangement or Coxeter arrangement (of type A n Gamma1 ) is the arrangement A n Gamma1 of hyperplanes in V n Gamma1 ae R It is clear that A has r(A n Gamma1 ) n regions (called Weyl chambers) and b(A n Gamma1 ) 0 bounded regions. Arnold [1] calculated the cohomology ring H (CAn ; C ) In particular, he proved that Poin An Gamma1 (q) 1 q) 1 2q) Delta Delta Delta (1 (n Gamma 1)q) 3.2) In this paper we will study deformations of the arrangement (3.1) which are hyperplane arrangements in V n Gamma1 ae R of the ....

....C (c s ; c s 1 ; c r ) are balanced. Equation (8.4) for C is the sum of the equations for C . Thus the statement follows by induction. Remark 8.12 This proposition is an analogue to the well known description of the cohomology ring of the Coxeter arrangement (3. 1) due to Arnold [1]. This cohomology ring is generated by e vw = e (v;w) 1 v w n, subject to relations (8.2) 8.3) and also the following triangle equation: e ab e bc Gamma e ab e ac e bc e ac = 0; where 1 a b c n. 9 Truncated affine arrangements In this section we study a general class of ....

V. I. Arnold, The cohomology ring of colored braid group, Math. Notes 5 (1969), 138--140.

....representations of finite groups on the rational cohomology of the complement of an arrangement of linear subspaces. The connection between arrangements of linear subspaces and configuration spaces was first suggested by Bjorner [Bj4, 8. 5] who was himself inspired by the work of Arnol d [Ar1] Ar2] Ar3] and Vassiliev [Va] Although the relation between hyperplane arrangements and configuration spaces had been established a long time ago by Fadell and Neuwirth [Fa Ne] the idea of using subspace arrangements as a unifying approach in this context seems to be recent. In this paper ....

....we consider the following examples : A) Let M be the set of all n = k q tuples (x 1 ; xn ) of points x i 2 R such that there is no subset E of f1; ng of cardinality k satisfying x t = x s for all t; s 2 E. In particular M is the pure braid space (see [Fa Ne] Ar1] for k = d = 2. See [Co La Ma] for k = 2 and d 2: For general k the space M is first mentioned in [Co Lu] in connection with a generalisation of the Borsuk Ulam Theorem. The cohomology of the spaces M was determined in [Bj We] for d = 1; 2 (and implicitly for general d) We study ....

Arnol'd, V.I.: The cohomology ring of the colored braid group. Math. Notes 5, 138--140 (1969)

....that allows one to compute these subvarieties is their relation to the cohomology support loci of rank one local systems on the complement X : P n C. One can de ne this loci as k (X) f 2 Hom( 1 = 1 ; C This presentation for braid arrangements had already been found by V.I. Arnold [2]. Libgober [30] and E.Hironaka [26] have studied the relation between k (X) and the characteristic variety Char k (X) The main result in this direction states that there is a bijection between the closed points of Char k (X) and k (X) cf. Proposition 1.13. The space k (X) has been studied ....

V. I. Arnold `The cohomology ring of the colored braid group', Math. Notes of the Academy of Sci. of the USSR, 5 (1969), 138-140.

....Braid Arrangement The classical braid arrangement A n in complex n space is given by the thick diagonals H ij : z i = z j for 1 # i j#n. The braid arrangement, also known as the complexified Coxeter arrangement of type A, is a well studied object (see for example Fox Neuwirth [7] Arnol d [1], Brieskorn [4] and Lehrer Solomon [8] Its name is derived from the fact that by a result of Fox Neuwirth [7] the complement MAn is the classifying space of the pure braid group on n strings. We enlarge the central (i.e. all hyperplanes pass through the origin) arrangement A n by some ....

V.I. Arnol'd, The cohomology ring of the colored braid group,Math.Notes5 (1969), 138--140.

....des invariants topologiques du couple (P 2 ; R) Nous commen cons par l anneau de cohomologie de l espace X = P 2 nR a coecients dans C , not e H (X; C ) Cet invariant a et e classiquement etudi e dans le cas d arrangements d hyperplans dans P r (ou C r 1 ) par V.I. Arnold [2], H. Brieskorn [4] et P. Orlik L. Solomon [11] Notons que le cas r = 2 est un cas particulier du n otre. Nous introduisons quelques notations pour pouvoir d ecrire l invariant H (X; C ) Soit d i le degr e de la composante irr eductible C i et soit i : d log(C i =C d i 0 ) une 1 forme ....

....is to study topological invariants of the pair (P 2 ; R) A rst type of invariant is the cohomology ring H (X; C ) of the complement X = P 2 nR with coecients in C . This invariant has been classically studied in the case of hyperplane arrangements in P r (or C r ) by V.I. Arnold [2], H.Brieskorn [4] P.Orlik and L.Solomon [11] Note that the case r = 2 is a particular case of ours. To describe this invariant we rst x a resolution : X P 2 of the divisor (R) in P 2 so that R : R) red is a divisor with normal crossings. Then we consider the sheaves A k ....

V. I. Arnold `The cohomology ring of the colored braid group', Math. Notes of the Academy of Sci. of the USSR, 5 (1969), 138-140.

....The braid arrangement or Coxeter arrangement (of type A n 1 ) is the arrangement A n 1 of hyperplanes in V n 1 R n given by x i x =0, 1#i j#n (3.1) see Fig. 1 for an example. It is clear that A n 1 has r(A n 1 ) n regions (called Weyl chambers) and b(A n 1 ) 0 bounded regions. Arnold [1] calculated the cohomology ring H (C A n 1 ; C) In particular, he proved that Poin A n 1 (q) 1 q) 1 2q) 1 (n 1) q) 3.2) In this paper we will study deformations of the arrangement (3.1) which are hyperplane arrangements in V n 1 R n of the type x i x j =a (1) ij , a (m ij ) ....

....s , c s 1 , c r ) are balanced. Equation (8.4) for C is the sum of the equations for C and C . Thus the statement follows by induction. K Remark 8.12. This proposition is an analogue to the well known description of the cohomology ring of the Coxeter arrangement (3. 1) due to Arnold [1]. This cohomology ring is generated by e vw =e (v, w) 1#v w#n, subject to relations (8.2) 8.3) and also the triangle equation: e ab e bc e ab e ac e bc e ac =0, where 1#a b c#n. 9. TRUNCATED AFFINE ARRANGEMENTS In this section we study a general class of hyperplane arrangements which ....

V. I. Arnold, The cohomology ring of a colored braid group, Math. Notes 5 (1969), 138#140.

....Coxeter arrangement of type A n Gamma1 is the arrangement of hyperplanes x i Gamma x j = 0; 1 i j n: 1.1.1) The regions of this arrangement, n in number, correspond different ways of ordering the sequence x 1 ; x n . The cohomology ring of the complement was calculated by Arnold [1]. In particular, he showed that its Poincar e polynomial, which is the generating function for the Betti numbers, is equal to (1 q) 1 2q) Delta Delta Delta (1 (n Gamma 1)q) In this chapter we study a more general class of arrangements which can be viewed as deformations of the arrangement ....

....) is the arrangement of hyperplanes in V n explicitly given by x i Gamma x j = 0; 1 i j n: 1.2.4) 4 or Braid arrangement 13 To compute the number of regions of this arrangement is not much harder than to compute the order of the symmetric group S n both these numbers are n . Arnold [1] calculated the cohomology ring H (CAn ; C ) see Corollary 1.2.14) In particular, he demonstrated that the characteristic polynomial of A n is equal to An (q) q Gamma 1) q Gamma 2) Delta Delta Delta (q Gamma n 1) 1.2.5) Brieskorn [9] generalized Arnold s result to the case ....

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V. I. Arnold, The cohomology ring of colored braid group, Math. Notes 5 (1969), 138--140.

....also describes factored arrangements in this interval. These results use the compact notation of signed graphs introduced by Zaslavsky. 1. Introduction The topology of the complement of an arrangement of hyperplanes started to receive a significant amount of attention with the work of Arnold [Ar], and of Fox, Fadell and Neuwirth [FaN] FoN] on the braid arrangements A n in the 1960 s. A portion of the work since then (by Saito [Sa] Terao [Te1 2] Stanley [St1 2] Falk and Randell [FR] Jambu and Paris [JP] has been aimed at identifying properties of the braid arrangement and other ....

Arnold, V.I.: The cohomology ring of the colored braid group. Matematicheskie Zametki, 5, 227-231 (1969): Mathematical Notes, 5, 138-40 (1969)

.... proof of the existence of sign twists in the action of Sn on the homology of a product of n copies of a poset P: These sign twists predict, for example, the exterior algebra structure of the cohomology of the complement of the complexified hyperplane arrangement of type An Gamma1 (see Arnold [A]) This action is of added interest because in many of the recently studied examples of hyperplane arrangements, and more generally subspace arrangements (see [Bj3] BjWe] and [SWe1] lower intervals in the intersection lattice Ln of the arrangement (where n is the dimension of the ambient ....

V.I. Arnold, The cohomology ring of the colored braid group, Math. Notes 5 (1969), 138-140.

....n distinct ordered points in C . We have 1 (M) PB n . Let ij = ji = 1 2 p Gamma1 d log (z i Gamma z j ) Then, f ij ; 1 i j ng represents a basis for H 1 (M) and the only kind of relations among wedges of these closed 1 forms are ij j j i i ij = 0: See [A]. Let fC ij = C ji ; 1 i j ng be a set of formal non commutative variables. It is not hard to check that in order to make the connection (3.1) Omega = X 1i jn ij C ij flat, we should assume (3.2) ae [C ij ; C k ] 0; i; j; k; are distinct [C ij ; C i C j ] 0: 10 XIAO SONG ....

V.I. Arnold, "The cohomology ring of the colored braid group". Math. Notes of Acad. of Sci. of USSR, 5, (1969), pp. 138--140.

....arrangement or Coxeter arrangement (of type A n Gamma1 ) is the arrangement A n Gamma1 of hyperplanes in V n Gamma1 ae R n given by x i Gamma x j = 0; 1 i j n: 3.1) It is clear that A has r(A n Gamma1 ) n regions (called Weyl chambers) and b(A n Gamma1 ) 0 bounded regions. Arnold [1] calculated the cohomology ring H (CAn ; C ) In particular, he proved that Poin An Gamma1 (q) 1 q) 1 2q) Delta Delta Delta (1 (n Gamma 1)q) 3.2) In this paper we will study deformations of the arrangement (3.1) which are hyperplane arrangements in V n Gamma1 ae R n of the ....

....; c s 1 ; c r ) are balanced. Equation (8.4) for C is the sum of the equations for C 0 and C 00 . Thus the statement follows by induction. Remark 8.12 This proposition is an analogue to the well known description of the cohomology ring of the Coxeter arrangement (3. 1) due to Arnold [1]. This cohomology ring is generated by e vw = e (v;w) 1 v w n, subject to relations (8.2) 8.3) and also the following triangle equation: e ab e bc Gamma e ab e ac e bc e ac = 0; where 1 a b c n. 9 Truncated affine arrangements In this section we study a general class of ....

V. I. Arnold, The cohomology ring of colored braid group, Math. Notes 5 (1969), 138--140.

....of the configuration space of the complex line. Given distinct j and k in f1; ng, let jk 2 H 1 (F(C ; n) Z) be the integral cohomology class represented by the closed differential form Omega jk = 1 2 i d(z j Gamma z k ) z j Gamma z k : By induction on n, Arnold shows in [1] that the cohomology ring H ffl (F(C ; n) Z) is generated by the classes jk , subject to the relations jk = kj and ij jk jk ki ki ij = 0: 1.13) The action of the group S n on the configuration space F(C ; n) induces an action on the cohomology ring H ffl (F(C ; n) Z) ....

....[ Gammad;0] is isomorphic to j ffl j ffl F ; ii) real Gamma cone Gamma C ffl (X; U ; F) F Delta Delta 2 ObD(X) Gammad Gamma1;0] is isomorphic to i ffl i ffl F . Proof. Note that (ii) is implied by (i) and the five lemma: in the diagram 0 F i ffl i ffl F j ffl j ffl F[1] 0 0 F cone Gamma C ffl (X; U ; F) Gamma F Delta C ffl (X; U ; F) 1] 0 w w u w u w u w w w the top row is exact by axiom (iii) for exact Mackey 2 functors, and the bottom row is obviously exact. We now prove (i) by induction on d: for d = 0, C ffl (X; U ; F) j ffl j ffl F ....

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V.I. Arnold, The cohomology ring of the colored braid group, Mat. Zametki 5 (1969), 227--231.

....ff A = A. In this case, the lattice X is geometric and the codimension of its each element can be recovered from the ordering as the rank of this element. There are two main results describing the structure of the ring R = H (C(A) ZZ) in this case. The first one is the Arnold Brieskorn theorem [1, 4] stating that the algebra of differential forms generated by the closed forms 1 2 i dff A ff A is isomorphic to R under the de Rham homomorphism. The second one is the Orlik Solomon theorem proving that R depends only on X. Moreover the theorem gives a presentation of R by generators and ....

V. I. Arnold, The cohomology ring of the colored braid group, Mat. Zametki 5 (1969), 227-231 (Math. Notes 5 (1969), 138-140).

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V. I. Arnold, The cohomology ring of the colored braid group (Russian), Mat. Zametki 5 (

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V. I. Arnold, The cohomology ring of the colored braid group, Math. Notes 5 (1969), 138--140.

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V.I. Arnold, The cohomology ring of the colored braid group, Mat. Zametki 5 (1969) 227--231.

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Arnol'd, V.I.: The cohomology ring of the colored braid group, Math. Notes 5(1969), 138--140.

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