HOMOLOGY OF BRAID GROUPS AND THEIR GENERALIZATIONS Vladimir V. Vershinin Abstract. In the paper we give a survey of (co)homologies of braid groups * *and groups connected with them. Among these groups are pure braid groups and general* *ized braid groups. We present explicit formulations of some theorems of V. I. Arnold* *, E. Brieskorn, D. B. Fuks, F. Cohen, V. V. Goryunov and others. The ideas of some proofs * *are outlined. As an application of (co)homologies of braid groups we study the Thom spectra* * of these groups. Introduction The aim of this survey is to give some ideas about (co)homology of braid gro* *ups and their generalizations. It is very well known that braids were rigorously define* *d by E. Artin [Art1] in 1925, although the roots of this notion are seen in the works of A. H* *urwitz ([H], 1891) and R. Fricke and F. Klein ([FK], 1897). Josef Przytycki informed the aut* *hor that he had seen braids in the notebooks of K.-F. Gauss. In his paper [Art2] E. Arti* *n gives the presentation of the braid group which is very well known now. We will denote h* *ere the braid group on n strings by Brn. The group Brn has the generators oei, i = 1; :* *::; n - 1. These generators are subject to the following relations: aeoe ioej = oejoei if |i - j| > 1; oeioei+1oei= oei+1oeioei+1: The cohomologies of the braid groups were studied firstly by V. I. Arnold in* * the work [Arn2], published in 1970. In this paper he discusses the cohomology of the bra* *id groups in a very broad mathematical context and displays connections of this subject w* *ith various mathematical fields. He proves three important theorems about Hi(Brn; Z), namel* *y, the theorems of finiteness, of recurrence and of stabilization (see Theorems 4.1 - * *4.3 below). Also he computes the cohomology groups Hi(Brn; Z) for n 11 and i 9. Cohomolo- gies of pure braid groups were also calculated by V. I. Arnold [Arn1]. These p* *apers of V. I. Arnold had a great influence. His study was continued by D. B. Fuks [F1]* * who calculated the cohomology of the braid groups mod 2. E. Brieskorn [Bri] general* *ized nat- urally the notion of the braid group for any finite Coxeter group W in such a w* *ay that the classical braid group arises when we consider symmetric group as the Coxete* *r group of type An. He proves some analogues of Arnold's results for generalized braid * *groups and _____________ 1991 Mathematics Subject Classification. Primary 20J05, 20F36, 20F55, 55N22. Key words and phrases. Braid group, configuration space, homology, Coxeter g* *roup, generalized braid group, Thom spectrum, Eilenberg-MacLane spectrum. Typeset by AM S-* *TEX 1 2 VLADIMIR V. VERSHININ pure generalized braid groups. Independently of the works of V. I. Arnold and D* *. B. Fuks, the homologies of classical braid groups were studied by Fred Cohen [CF1], [CF2* *], [CLM] by different methods. He computed these homologies with coefficients in Z and * *in Z=p as modules over the Steenrod algebra. The additive structure of these cohomolog* *ies was also computed by V. F. Vainshtein [Vai] who used the methods of D. B. Fuks. Lat* *er these methods were applied by V. V. Goryunov in [G1], [G2] who expressed the cohomolo* *gies of the generalized braid groups of types C and D in terms of classical ones. The cohomologies of the braid groups has the following interesting applicati* *on. The canonical representation of the braid group Brn in the orthogonal group On indu* *ces a map of the corresponding classifying and Thom spaces (details in x6). It was pr* *oved by Fred Cohen [CF3] and Mark Mahowald [Mah1], [Mah2] that the Thom spectrum of the* *se spaces is the Eilenberg-MacLane spectrum of the ordinary homology with coeffici* *ents in Z=2. The paper is organized as follows. In x1 we discuss configuration spaces who* *se funda- mental groups are braid groups. In x2 we give a brief sketch of Coxeter groups * *and study generalized braid groups which Coxeter groups define. The cohomologies of pure* * braid groups are given in x3. Various aspects of (co)homologies of classical braid g* *roups are discussed in x4. Cohomologies of generalized braid groups of types C and D are * *expressed in terms of cohomologies of classical braid groups in x5. The study of the Thom* * spectra of braid groups is carried out in x6. 1. Braid Groups and Configuration Spaces The braid group has a natural interpretation as the fundamental group of the* * config- uration space. For our purposes it will be useful to look at braids from a ver* *y general point of view as it was done by V. Ya. Lin in [Li]. Let Y be a connected top* *ological manifold and W be a finite group acting on Y . A point y 2 Y is called regula* *r if its stabilizer {w 2 W : wy = y} is trivial, i.e. consists only of the unit of the * *group W . The set "Yof all regular points is open. Suppose that it is connected and nonem* *pty. The subspace ORB(Y; W ) of the space of all orbits Orb(Y; W ) consisting of the orb* *its of all regular points is called the space of regular orbits. We have a free action of * *W on "Yand the projection p : "Y! "Y=W = ORB(Y; W ) defines a covering. Let us consider th* *e initial segment of the long exact sequence of this covering: 1 ! ss1(Y"; y0) p*!ss1(ORB(Y; W ); p(y0)) ! W ! 1: The fundamental group ss1(ORB(Y; W ); p(y0)) of the space of regular orbits wil* *l be called the braid group of the action of W on Y and denoted by Br(Y; W ). The fundame* *ntal group ss1(Y"; y0) will be called the pure braid group of the action of W on Y a* *nd denoted by P (Y; W ). The spaces "Yand ORB(Y; W ) are path connected, so the pair of th* *ese groups is defined uniquely up to isomorphism and it is possible not to mention the bas* *e point p0 in the notations. For any space Y the symmetric group m acts on Cartesian powe* *r Y m of Y : w(y1; :::; ym ) = (yw-1(1); :::; yw-1(m)); w 2 m : HOMOLOGY OF BRAID GROUPS AND THEIR GENERALIZATIONS 3 We denote by F (Y; m) the space of m-tuples of pairwise different points in Y : F (Y; m) = {p1; :::; pm ) 2 Y m : pi6= pj fori 6= j}: It is the space of regular points of this action. In the case when the space Y * *is a connected topological manifold M without boundary and dimM 2, the space of regular orbits ORB(Mm ; m ) is open, connected and nonempty. It is called the configuration sp* *ace of the manifold M and is denoted by B(M; m). The braid group Br(Mm ; m ) is called* * the braid group of the manifold M on m strings and is denoted by Br(m; M). Analogo* *usly the group P (Mm ; m ) is called the pure braid group of the manifold M on m str* *ings and is denoted by P (m; M). These definitions of braid groups were given by R. * *Fox and L. Neuwirth [FoN]. The classical braid group on m strings Brm and the correspon* *ding pure braid group Pm are obtained in the case when the manifold M is equal to the Eu* *clidean plane R2. Let (qi)i2N be a fixed sequence of distinct points in the manifold M and put* * Qm = {q1; :::; qm }. We use Qm;l= (ql+1; :::; ql+m ) 2 F (M \ Ql; m) as the standard base point of the space F (M \ Ql; m). If k < m we define a pro* *jection proj : F (M \ Ql; m) ! F (M \ Ql; k) by the formula: proj(p1; :::; pm ) = (p1; :::; pk): The following theorems wer* *e proved by E. Fadell and L. Neuwirth [FaN]. Theorem 1.1. The triple proj : F (M \ Ql; m) ! F (M \ Ql; k) is a locally trivi* *al fibre bundle with fibre proj-1 Qk;lhomeomorphic to F (M \ Qk+l; m - k). Considering the sequence of fibrations F (M \ Qm-1 ; 1) ! F (M \ Qm-2 ; 2) ! M \ Qm-2 ; F (M \ Qm-2 ; 2) ! F (M \ Qm-3 ; 3) ! M \ Qm-3 ; ::: ; F (M \ Q1; m - 1) ! F (M; m) ! M E. Fadell and L. Neuwirth proved the following theorem. Theorem 1.2. For any manifold M ssi(F (M \ Q1; m - 1)) = m-1k=1ssi(M \ Qk) for i 2. If ss : F (M; m) ! M admits a section, then proji(F (M; m)) = m-1k=0ssi(M \ Qk); i 2: Corollary 1.1. If M is the Euclidean r-space, then ssi(F (M; m)) = m-1k=0ssi(Sr-1___:::__-Sr-1z______"); i 2: k Corollary 1.2. If M is the Euclidean 2-space, then the space F (R2; m) is the K* *(Pm ; 1)- space and the space B(R2; m) is the K(Brm ; 1)-space. 4 VLADIMIR V. VERSHININ 2. Generalized Braid Groups Let V be a finite dimensional real vector space (dim V = n) with Euclidean s* *tructure. We denote by W a finite subgroup of GL(V ) generated by reflections. We use the te* *rminology and the contents of N. Bourbaki [Bo]. Let M be the set of hyperplanes such that* * W is generated by orthogonal reflections with respect to M 2 M. We suppose that for* * any w 2 W and any hyperplane M 2 M the hyperplane w(M) belongs to M. The space V is divided into cells by hyperplanes of the system M. The cells of the maximal * *dimension (equal to n) are called chambers. The boundary of a chamber A is a subset of a * *union of hyperplanes. These hyperplanes are called the walls of the chamber A: The follo* *wing facts are well known [Bo]. Proposition 2.1. (i) W permutes the chambers of M transitively. (ii) The closure A of a chamber A is the fundamental domain of W acting on V* * . (iii) If x 2 V belongs to A its stabilizer is generated by reflections with * *respect to the walls of A containing x. Also there exists a set I and a one to one correspondence of the elements of* * I with the walls of a chamber A : i 7! Mi(A), which is called a canonical indexation of th* *e walls of the chamber A. Then W is generated by the reflections wi= wi(Mi); i 2 I, satisf* *ying only the following relations (wiwj)mi;j= e; i; j 2 I; where the natural numbers mi;j= mj;iform the Coxeter matrix of W by which the Coxeter graph (W ) of W is constructed. We use the following notations of P. De* *ligne [D]: prod(m; x; y) denotes the product xyxy::: (m factors). The generalized braid gr* *oup Br(W ) of W [Br], [D] is defined as a group with generators {si; i 2 I} and the follow* *ing relations: prod(mi;j; si; sj) = prod(mj;i; sj; si): From this we obtain the presentation of the group W if we add the relations: s2i= e; i 2 I: We will see later in the Theorem 2.1 that this definition of a generalized brai* *d group agrees with our general definition of a braid group of an action of a group W: We deno* *te by oW the canonical map from Br(W ) to W . The classical braids on k strings Brk are * *obtained by this construction if W = Ak = k+1, the symmetric group on k + 1 symbols. In * *this case mi;i+1= 3, and mi;j= 2if j 6= i; i + 1. Now let J1; :::; Js be the sets of vertices of the connected components of t* *he Coxeter graph of W , Wq is the subgroup of W generated by the reflections wi; i 2 Jq. * *Let Vq0 be the subspace of V consisting of vectorsTinvariant under the action of Wq, V* *q is the orthogonal complement of Vq0in V , V0 = 1qs Vq0. Then we get the following f* *acts from the Proposition 5 ([Bo], Chapter V, x3.7). Proposition 2.2. (i) The group W is the direct product of the subgroups Wq (1 * *q s). HOMOLOGY OF BRAID GROUPS AND THEIR GENERALIZATIONS 5 (ii) The vector space V is the direct sum of the orthogonal subspaces V1; :* *::; Vs; V0 invariant under the action of W . If V0 = 0; then the group W acting on V is called essential. In this case ea* *ch chamber is an open simplicial cone. Let us define some order between all the walls Mi;* * 1 i n of a chamber A. The product of reflections wM1 wM2 :::wMn is called the C* *oxeter transformation defined by the ordered chamber A. All the Coxeter transformatio* *ns are conjugate in W and so, all of them have the same characteristic polynomial and * *the same (finite) order. This order is called the Coxeter number of the group W . We de* *note the Coxeter number of the group W by h. Then the characteristic polynomial of a Co* *xeter transformation can be written in the form: Yn 2issm f(t) = (t - exp(_____j_)); j=1 h where m1; m2; :::; mn are the integers such that 0 m1 m2 ::: mn < h: The integers m1; m2; :::; mn are called the exponents of the group W . The classification of irreducible (with connected Coxeter graph) Coxeter gro* *ups is well known (Theorem 1 , Chapter VI, x4 of [Bo]). It consists of the three infinite * *series: A, C and D and groups E6; E7; E8; F4; G2; H3; H4 and I2(p): As the examples there * *are the following Coxeter graphs for An, Cn, Dn and E8: (An) o________o________o________..._________o , 4 (Cn) o________o________o________..._________o , oH H H (Dn) HH o________o________..._________o , o 6 VLADIMIR V. VERSHININ o | | | | (E8) o________o________o________o_______o________o_______o|. The number of vertices in these diagrams is equal to n and the number m over a* *n edge means that mi;j= m for the pair of generators, corresponding to the points conn* *ected by the given edge. Now let us consider theScomplexification VC of V and the complexification * *MC of M 2 M. Let YW = VC - M2M MC . Then we get from (iii) of Proposition 2.1 t* *hat W acts freely on YW . Let XW = YW =W , YW is a covering over XW correspondin* *g to the group W . Let y0 2 A0 be a point in some chamber A0 and x0 is its image in XW .* * We are in the situation described in x1 in the definition of the braid group of th* *e action of the group W: This braid group is defined as the fundamental group of the space * *of regular orbits of the action of W . In our case ORB(VC ; W ) = XW : So, the generalize* *d braid group is equal to ss1(XW ; x0). For each j 2 I, let `0jbe the homotopy class o* *f paths in YW starting from y0 and ending in wj(y0) which contains a polygon line with su* *ccessive vertices: y0; y0 + iy0; wj(y0) + iy0; wj(y0). The image `j of `0jin XW is a lo* *op with base point x0. Theorem 2.1. (i) The fundamental group ss1(XW ; x0) is generated by the element* *s `j satisfying the following relations: prod (mj;k; `j; `k) = prod(mk;j; `k; `j): (ii) The universal covering of XW is contractible, and so XW is K(ss; 1). This theorem was proved by E. Brieskorn [Bri] for the groups of types Cn; G2* * and I2(p), similarly as E. Fadell and L. Neuwirth [FaN] proved Theorems 1.1, 1.2 and Corol* *lary 1.2. For the types Dn and F4 E. Brieskorn uses this method with small modifications.* * In general case this theorem was proved by P. Deligne [D]. If a group W is the direct product of groups W 0and W 00, then the group Br(* *W ) is the direct product of the groups Br(W 0) and Br(W 00). So, if a group W is the same* * as in the Proposition 2.2, then we have: Br(W ) = Br(W1) x ::: x Br(Ws): There exist pairings for symmetric and braid groups k x l! k+l; : Brk x Brl! Brk+l; which commute with the maps oj : Brj ! j. For the braid group this pairing may * *be constructed by means of adding l extra strings to the initial k. If oe0iare the* * generators of Brk, oe00jare the generators of Brl and oer are the generators of Br(k + l), th* *en the map can be expressed in the form: (oe0i; e) = oei; 1 i k - 1; HOMOLOGY OF BRAID GROUPS AND THEIR GENERALIZATIONS 7 (e; oe00j) = oej+k; 1 j l - 1: In terms of Coxeter graphs it means that we take the vertex corresponding to oe* *k in the Coxeter graph (k+l) andSimbed Brk x Brl into Brk+l in accordance with the inclu* *sion of the (k x l) = (k) (l) into the two components of the graph (k+l) \ oek. This permits us to interpret various imbeddings of products of finite Coxeter g* *roups into a group with greater index. It is true for the corresponding generalized braid* * groups as well. We take away a vertex in a connected Coxeter graph and obtain a graph wh* *ose number of connected components is less than or equal to 3. These components cor* *respond to irreducible Coxeter groups or braid groups whose direct product is the sourc* *e of this mapping. For example, we have the evident pairings: (C; A): Br(Ck) x Br(Al) ! Br(Ck+l+1); (D; A) : Br(Dk) x Br(Al) ! Br(Dk+l+1) for anyk and l; or the pairing (A3; A4; E8) : Br(A3) x Br(A4) ! Br(E8) that corresponds to the forth horizontal vertex of the Coxeter graph of E8. Embeddings of groups (not products) can also be expressed in this language. * *For ex- ample, we have the imbedding ffC : Br(Al-1) ! Br(Cl); and two different imbeddings: ffD : Br(Al-1) ! Br(Dl) in accordance with two different vertices on one end of the Coxeter graph for D* *l. We would like to consider a generalized braid group Br(Ck). We have a relat* *ion in Br(Ck): w1w2w1w2 = w2w1w2w1: Let Br1;n+1be the subgroup of the braid group Brn+1 consisting of all elements * *of Brn+1 with the property that permutations associated with them all leave the number 1* * invariant. It means that the end of the first string is again at the first place. W.-L. Ch* *ow [Ch] found the presentation of this group with generators: oe2; :::; oen; a2; :::; an+1; where oej is the standard generator of the braid group Brn+1 and the elements a* *i are given by the equality ai = oe-11:::oe-1i-2oe2i-1oei-2:::oe1; 2 i n + 1. The e* *lements oe2; :::; oen generate a subgroup in Br1;n+1isomorphic to Brn and the elements a2; :::; an+1 * *generate a normal free subgroup Fn. The following relation is fulfilled in Br1;n+1: oe2a2oe2a2 = a2oe2a2oe2: We define the homomorphism OE : Br(Cn) ! Br1;n+1by the formulae: OE(w1) = a2; OE(wi) = oei; i = 2; :::; n; and obtain the following statement. 8 VLADIMIR V. VERSHININ Proposition 2.3. The map OE defines an isomorphism OE : Br(Cn) ~=Br1;n+1: The claim of this proposition is evident from the geometric point of view. T* *he space XCn can be interpreted as space of n different pairs of points of R2\0, symmetr* *ical with respect to zero [G1, G2]. That is the same as simply the space of n different * *points in R2\0. The group Brn+1 is interpreted as the fundamental group of the space XAn * *of n + 1 different points in R2. If we consider one point (say 0) to be fixed, then we g* *et XCn . For the fundamental group of XAn it means that the first string must have the same * *end as its beginning (equal to zero). We denote by fi the homomorphism from Br(Cn) to Brn defined by the formulae: fi(w1) = e; fi(wi) = oei-1; fori > 1: Then we have fiffC = 1Brn and Br(Cn) is isomorphic to the semidirect product of* * Fn and Brn with the standard braid action of Brn on Fn [Bi]. It is known that the grou* *p Ck is isomorphic to the wreath product of the symmetric group k = Ak-1 with Z=2 : Ck * *~= k o Z=2. The pairing mC : Ck x Cl! Ck+l may be defined using the pairing for the symmetric group k x l! k+l and the wreath product structure. Let w01; :::; w0kbe the generators of Br(Ck) and w001; :::; w00lbe the gener* *ators of Br(Cl). Then it is possible to define the pairing (C; C): (C; C): Br(Ck) x Br(Cl) ! Br(Ck+l) by the formulae: (C; C)(w0i; e) = wi; 1 i k; (C; C)(e; w001) = wk+1:::w2w1w2:::wk+1; (C; C)(e; w00j) = wk+j; 1 j l; This pairing (C; C) was firstly defined in [Ve]. It is easy to check that it is* * associative, that means that the following diagram is commutative: Br(Ck) x Br(Cl) x Br(Cq) -ix1---!Br(Ck+l) x Br(Cq) # 1 x (C; C) # (C; C) Br(Ck) x Br(Cl+q) --i--! Br(Ck+l+q): HOMOLOGY OF BRAID GROUPS AND THEIR GENERALIZATIONS 9 This pairing agrees with the pairing for the Coxeter groups mC : Ck x Cl! Ck+l; so we have a commutative diagram Br(Ck) x Br(Cl) -oCxoC---!Ck x Cl # (C; C) # mC Br(Ck+l) - oC---! Ck+l: It also agrees with the pairing Br(Ck) x Brl ! Br(Ck+l) through the canonical i* *nclu- sion Brl ! Br(Cl). It is also easy to check the commutativity of the diagram f* *or the homomorphism ffC : Brk x Brl ffCxffC-----!Br(Ck) x Br(Cl) # # (C; C) Brk+l --ffC--! Br(Ck+l): But there is no analogous commutativity for fi : Br(Ck) ! Brk and (C; C). To se* *e this let k = 2; then (fi; fi)(e; w001) = (e; e) = e and fi(C; C)(e; w001) = fi(w3w2w* *1w2w3) = oe2oe21oe2 6= e. So the homomorphism fi does not agree with the pairings and (* *C; C). On the level of configuration spaces the pairing (C; C) for the braids of th* *e series C can be described in the following way. We map R2\0 (with k different points) diffeo* *morphically onto the open disk of radius k + 1=2 without zero Dk+1=2\0 in such a way that t* *he points with coordinates (1; 0); :::; (k; 0) go onto themselves and we map R2\0 (with l* * different points) diffeomorphically onto R2\Dk+1=2 in such a way that the points with coo* *rdinates (1; 0); :::; (l; 0) go onto the points (k + 1; 0); :::; (k + l; 0). This map (2.1) R2\0 x R2\0 ! R2\0 is a particular case of the map of configuration spaces, which was described by* * Viktor Vassiliev [Vas, p. 25]: X(k) x X(l) ! X(k + l) where the space X(k) in our notations is equal to B(X; k) and the space X can b* *e presented in the form X = Y x R for some other space Y . The map (2.1) generates the pair* *ing of fundamental groups of configuration spaces which coincides with (C; C): Conside* *ring the generalized braid groups of type C as the subgroups of the ordinary braid group* *s the pairing 1 can be described as putting k + 1 strings of the first group instead * *of the zero string of the second group. Let us consider the group Brk o Z=2 which can be viewed as semi-direct produ* *ct of Brk with Z=2 ::: Z=2 (k copies) where Brk acts on Z=2 ::: Z=2 by permutations. * *We denote by s1 the element (a; e; :::; e) 2 Z=2 ::: Z=2; where a is a generator* * of Z=2, and 10 VLADIMIR V. VERSHININ we denote the standard generators of Brk by s2; :::; sk in this context. Then w* *e have the relation: s1s2s1s2 = s2s1s2s1: We define the homomorphism fl : Br(Ck) ! Brk o Z=2 by the formula fl(wi) = si: This homomorphism does not agree with the pairings (i and the pairing determine* *d by the wreath product structure). Now we will consider the direct limits of finite Coxeter groups. We denote * *by W the category whose objects are finite Coxeter groups and morphisms are the incl* *usions W 07! W corresponding to inclusions of Coxeter graphs 0 7! . We call a chain a subcategory E of W which is a well ordered countable set and such that the tota* *l number of connected components of Coxeter graphs of the elements of E is bounded by so* *me natural number NE (for a subgroup W 0of W we consider 0 as a subgraph of ). We call by a limit Coxeter group W1 such an infinite group that there exists a chain E for * *which W1 is equal to the direct limit of E. If we take as E the groups from one of the s* *eries A, C or D with canonical inclusions as morphisms we obtain A1 , C1 or D1 as the corre* *sponding limit Coxeter groups. Proposition 2.4. The limit Coxeter group W1 is isomorphic to a direct product * *of a finite number (greater or equal than one) of groups of type A1 , C1 or D1 and* * of a finite number of finite Coxeter groups. The proof follows from the fact that W1 must be infinite and its Coxeter gr* *aph is to have finitely many components. Pairings described above generate the pairings of the limit Coxeter groups a* *nd the corresponding braid groups, for example (C; A): Br(C1 ) x Br(A1 ) ! Br(C1 ); (C; C): Br(C1 ) x Br(C1 ) ! Br(C1 ); (D; A) : Br(D1 ) x Br(A1 ) ! Br(D1 ): For the general limit Coxeter group W1 we may have several different pairings * *with Br(A1 ) = Br1 depending on the copy of one of the infinite groups of types A1 * *, C1 or D1 for which this pairing is taken (W; A) : Br(W1 ) x Br(A1 ) ! Br(W1 ): HOMOLOGY OF BRAID GROUPS AND THEIR GENERALIZATIONS 11 3. Cohomology of Pure Braid Groups Cohomologies of pure braid groups were first calculated by V. I. Arnold in [* *Arn1] using the Serre spectral sequence. We consider a somewhat more general case of config* *uration space for Rn [O], [CT]. Let us consider F (Rn; 2). The map OE : Sn-1 ! F (Rn; 2); described by OE(x) = (x; -x), is a 2-equivariant homotopy equivalence. Define b* *y A the generator of Hn-1 (F (Rn; 2); Z) which maps to the standard generator of Hn-1 (* *Sn-1 ; Z) by OE*. For i and j, such that 1 i; j m; i 6= j, specify ssi;j: F (Rn; m) ! F* * (Rn; 2) by ssi;j(p1; :::; pm ) = (pi; pj). Let Ai;j= ss*i;j(A) 2 Hn-1 (F (Rn; m); Z): It follows that Ai;j= (-1)nAj;iand A2i;j= 0. For w 2 m we have an action w(Ai;* *j) = Aw-1(i);w-1(j), since ssi;jw = ssw-1(i);w-1(j): Note also that under restrictio* *n to F (Rn \ Qk; m - k) ~=ss-1 (Qk) F (Rn; m) the classes Ai;jwith 1 i; j k go to zero since in this case the map ssi;jis c* *onstant on ss-1 (Qk). Considering the Serre spectral sequence we have the following theore* *m. Theorem 3.1. The cohomology group H*(F (Rn \Qk; m-k); Z) is the free Abelian gr* *oup with generators Ai1;j1Ai2;j2:::Ais;js; where k < j1 < j2 < ::: < js m and ir < jr for r = 1; :::s: The multiplicative structure and the m -algebra structure of H*(F (Rn; m); Z* *) are given by the following theorem which is proved using the 3-action on H*(F (Rn; 3); Z). Theorem 3.2. The cohomology ring H*(F (Rn; m); Z) is multiplicatively generated* * by the square-zero elements Ai;j2 Hn-1 (F (Rn; m); Z); 1 i < j m; subject to the only relations Ai;kAj;k= Ai;jAj;k- Ai;jAi;kfori < j < k: Q m-1 The Poincare series for F (Rn; m) is j=1(1 + jtn-1): Remark. In the case of R2 = C the cohomology classes Aj;kcan be interpreted as * *the classes of cohomology of differential forms !j;k= _1_2ssidzj_-_dzkz: j - zk The cohomologies of pure generalized braid groups were computed by E. Briesk* *orn [Bri] using ideas of V. I. Arnold for classical case. Let V be a finite-dimensional c* *omplex vector space and Hj 2 V; j 2 I be the finite family of complex affine hyperplanes give* *n by linear forms lj. E. Brieskorn proves the following theorem. 12 VLADIMIR V. VERSHININ Theorem 3.3. The cohomology classes, corresponding to holomorphic differential * *forms !j = _1_2ssidlj_l j generate the cohomology ring H*(V \ [j2IHj; Z): Moreover, this ring is isomorph* *ic to Z-subalgebra generated by forms !j in the algebra of meromorphic forms on V: The cohomologies of generalized pure braid groups are described by the follo* *wing the- orem. Theorem 3.4. (i)The cohomology group of the pure braid group P (W ) with coeffi* *cients in the ring of the integer numbers Hk(P (W ); Z) is a free abelian group, and i* *ts rank is equal to the number of elements w 2 W of the length l(w) = k, where l is the length c* *onsidered with respect to the system of generators consisting of all reflections of the g* *roup W . (ii) The Poincare series for H*(P (W ); Z) is Yn (1 + mjt); j=1 where mj are the exponents of the group W . (iii) The multiplicative structure of H*(P (W ); Z) coincides with the struc* *ture of algebra, generated by 1-forms described in the previous theorem. 4. Homology of Braid Groups The cohomologies of the classical braid groups were first studied by V. I. A* *rnold in the article [Arn2]. To investigate H*(Brn; Z) he interprets the space K(Brn; 1) ~=* *B(R2; n) as the space of complex polynomials of degree n without multiple roots with the* * first coefficient equal to 1: (4.1) Pn(t) = tn + z1tn-1 + ::: + zn-1t + zn: More precisely let us consider the "Vieta map" from Cn which we denote at this * *place by Cn()to Cn which we denote by Cn(z)to distinguish between the domain and the ima* *ge: (4.2) p : Cn()! Cn(z): It maps a point (1; :::; n) 2 Cn()to the polynomial Pn(t) = tn + z1tn-1 + ::: +* * zn-1t + zn, which has the roots 1; :::; n (with multiplicity counted). The space Cn(z)is i* *nterpreted as a space of orbits Orb(Cn; n) of the canonical action of symmetric group n on* * Cn by taking values of symmetrical polynomials in 1; :::; n. The standard basis here * *consists of the basic symmetric polynomials: X zk() = (-1)k i1:::ik; 1 k n: i1<::: 0 and 0 < t < (n - 1)(p - 1): F. Cohen calculates the homology of B(Rn; m) with coefficients in Z=2 (as in* * the The- orems 4.4 and 4.5) and with coefficients in Z=p; p > 2. Theorem 4.8. The homology of the infinite braid group with coefficients in Z=p;* * p > 2 as a Hopf algebra is isomorphic to the tenzor product of exterior and polynomia* *l algebras: E(a1; :::; ai; :::) Z=p[b1; :::; bj; :::]; i = 1; 2; :::; j = 1* *; 2; :::; deg ai= 2pi-1- 1; deg bj = 2pj - 2; with the coproduct given by the formulae: (ai) = 1 ai+ ai 1; (bj) = 1 bj + bj 1: Theorem 4.9. The canonical inclusion Brn ! Br1 induces a monomorphism in homol- ogy with coefficients in Z=p; p > 2. Its image is the subcoalgebra of the tenzo* *r product E(a1; :::aj; :::) Z=p[b1; :::; br; :::] with Z=p-basis consisting of monomials X X affl11:::afflllbk11:::bkss; where ffli= 0; 1; and 2( fflipi-1+ * * kjpj) n: i j F. Cohen describes also the action of the Steenrod algebra A [CF3]. We woul* *d like to consider his results in the language of homology and coaction of the dual * *of the Steenrod algebra A*. We remind, that A* as an algebra for p = 2 is isomorphic * *to the polynomial algebra Z=2[1; :::; k; :::]; deg k = 2k - 1; and for p > 2 it is iso* *morphic to the tenzor product of exterior and polynomial algebras E(o0; o1; :::; ol; :::) Z=p* *[1; :::; k; :::]; deg k = 2(pk - 1); deg ol= 2pl- 1. The coproduct is given by the formulae X pi X pi (k) = j i; (ol) = ol 1 + j oi: i+j=k i+j=l 16 VLADIMIR V. VERSHININ Theorem 4.10. The coaction of the dual of the Steenrod algebra A* on the homolo* *gy of the braid groups Brm , 1 m 1 is given for p = 2 by the formula: ae1 a ; if j = 1; (aj) = j 1 aj + 1 a2j-1 if j 2; and for p > 2 by the formulae ae1 a ; ifj = 1; (aj) = j 1 aj + o0 bj-1; ifj 2; ae1 b ; ifj = 1; (bj) = j p 1 bj - 1 bj-1; ifj 2: Let 20S2 be the connected component of the trivial loop in the double loop s* *pace 2S2: Graeme Segal [Se] established the following connection between the infinite bra* *id group and iterated loop spaces. Theorem 4.11. There exists a map K(Br1 ; 1) ! 20S2; that induces an isomorphism* * in homology for any group of coefficients G (with trivial action of Br1 on G): H*(Br1 ; G) ~=H*(20S2; G): The classical Hopf fibration S3 ! S2 induces isomorphism in homology H*(2S3;* * G) ~= H*(20S2; G): This allows us to consider the spaces 2S3 and 20S2 as a plus-const* *ruction for K(Br1 ; 1): We can also use the computations of homologies of nSn+1 by E. D* *yer and R. Lashof [DL] and by Fred Cohen [CLM, p. 227]. It follows from these calcu* *lations that H*(nSn+1 ; Z=p) is isomorphic to the polynomial algebra over Z=p on genera* *tors Qk11:::Qkn-1n-1a1; where Qi are the Araki-Kudo-Dyer-Lashof operations, which ac* *t in homol- ogy of iterated loop spaces [CLM] and a1 is the image of the generator of H1(S1* *; Z=p) by the map H*(S1; Z=p) ! H*(nSn+1 ; Z=p); induced by the canonical inclusion S1 ! nSn+* *1 : This gives us another proof of the Theorems 4.4 and 4.8. The generators aj; j >* * 1; can be considered as Qj-11a1 and bj as fiaj+1: The coaction of the dual of the Stee* *nrod alge- bra, described in Theorem 4.10 may be obtained from the Theorem 4.11 with the h* *elp of Nishida relations [CLM] which relate the action of Steenrod operations and Arak* *i-Kudo- Dyer-Lashof operations. F. Cohen uses the Bockstein spectral sequence to calculate the integral homo* *logy of the braid groups [CLM]. Theorem 4.12. The p-torsion in H*(Brn; Z); n 1 is all of order p: The p-torsio* *n of H*(Br1 ; Z) in degrees strictly greater than one is isomorphic to the following: (i) If p = 2, to the polynomial algebra generated by a1 and a2j; j > 1: (ii) If p > 2, to the tenzor product of exterior algebra generated by a1; an* *d polynomial algebra generated by bj: HOMOLOGY OF BRAID GROUPS AND THEIR GENERALIZATIONS 17 Theorem 4.13. The canonical inclusion Brn ! Br1 induces a monomorphism in ho- mology with coefficients in Z. Its 2-torsion image in degrees strictly greater * *than 1 has the Z=2-basis consisting of monomials X ak11:::akll; such, thatki 0 mod 2 fori > 1 and ki2i n; i and its p-torsion image, p > 2; in degrees strictly greater than 1 has the Z=p-* *basis consisting of monomials X affl11bk11:::bkss; where ffl1 = 0; 1; and 2(ffl1 + kjpj) n: j The methods of D. B. Fuks were applied by F. V. Vainshtein [Vai] for calcula* *tion of the cohomologies of the braid groups with coefficients in Z=p and Z. As a result he* * obtained a complete information about the additive structure of these cohomologies and a* *bout the action of the Bockstein homomorphism. We call the Coxeter representation of the symmetric group n the representati* *on n ! GLn(Z); corresponding to the permutations ofPthe basic vectors in Zn: The restriction t* *o the hyper- plane in Zn; given by the formula xi = 0, is called the reduced Coxeter repre* *sentation. These representations define the structures of n-modules on Zn and Zn-1: With t* *he help of the canonical map Brn ! n Zn and Zn-1 become modules over Brn: We denote the* *se modules over Brn by Kn and K"n: The following theorem was proved by V. V. Vassi* *liev [Vas]. Theorem 4.14. The cohomologies of the braid group with coefficients in the Coxe* *ter representation and reduced Coxeter representation are given by the formulae: n-1M Hq(Brn; Kn) = Hq-i(Brn-1-i; Z); n 2; i=0 n-1M Hq(Brn; "Kn) = Hq-i(Brn-i; Z); n 2; i=1 where we put formally Br0 = {e}, the group consisting of a single element. Proof. We consider the first formula now. The module Kn is isomorphic to the m* *odule CoindBrnBr1;nZ, coinduced from the trivial module Z over Br1;n, where the subgr* *oup Br1;n was defined in x2. So by Shapiro's Lemma [Bro] we have: H*(Brn; Kn) ~=H*(Br1;n;* * Z): We prove the first isomorphism by induction. For n = 2 Br1;2is equal to Z and * *so, its cohomologies are the same as of the circle. The formula is true. Let n > 2. * *Consider the homomorphism fi : Br1;n! Brn-1, defined in x2 and the Serre-Hochschild spec* *tral sequence for fi. As it was also described in x2 Kerfi is the free subgroup Fn-* *1 of Brn 18 VLADIMIR V. VERSHININ generated by braids a2; :::; an. The initial term of the Serre-Hochschild spect* *ral sequence Ep;q2is isomorphic to: Hp(Brn-1; Hq(Fn-1; Z)). There are only two first nonzero* * lines in this spectral sequence, because H0(Fn-1; Z) ~=Z; H1(Fn-1; Z) ~=Zn-1; Hq(Fn-1; Z) ~=0 forq > 1: We remind that Br1;nis isomorphic to the semidirect product of Fn-1 and Brn-1 w* *ith the standard braid action of Brn-1 on Fn-1. This action on H0(Fn-1; Z) ~=Z generate* *s the trivial ZBrn-1-module structure and on H1(Fn-1; Z) ~=Zn-1 it generates the stru* *cture of module Kn-1. This ends the induction step. The isomorphism for the reduced C* *oxeter representation follows from the exact sequence 0 ! "Kn! Kn ! Z ! 0. 5. Cohomology of Generalized Braid Groups The methods of D. B. Fuks and V. F. Vainshtein were applied by V. V. Goryuno* *v in [G1], [G2] to calculations of the cohomologies of the generalized braid groups * *of types C and D. The configuration space XCn for the braid groups of type C was described* * in x2 as the space of n different pairs of points of C\0, symmetric with respect to z* *ero what is the same as simply the space of n different points in C\0. The configuration sp* *ace XDn for the braid groups of type D can be described in the following way. Let us co* *nsider the geometrically distinct pairs of points in C; symmetric with respect to zero. Th* *e degenerate case when the pair consists of one point, equal to zero, is included. Then we s* *uppose that each nondegenerate pair is considered with different signs (plus or minus) of p* *oints. The involution acts on nondegenerate pairs by changing the signs and is identical * *on the degenerate pair. We call the two unordered sets of n distinct pairs ofPpoints {* *p1; :::; pn} and {q1; :::; qn} equivalent if qi = ffli(pi), ffli = 0; 1, such that i=ni=1f* *fli is even. The space XDn is the factor space of n-tuples of geometrically different pairs in C with * *respect to the equivalence relation just described. The space XAn-1 = B(C; n) in x4 was interpreted as the space of the polynomi* *als over the complex numbers of degree n without multiple roots with the first coefficie* *nt equal to 1 (4.1). Analogously it is possible to interpret the space XCn as the space of * *polynomials over C of the form (4.1) without multiple and zero roots. Let n be one of the * *spaces XCn or XDn and *nis its one point compactification. The same way as D. B. Fuk* *s, V. V. Goryunov uses the Poincare duality Hk(n; Z) ~=H"2n-k(*n; Z) for the calculations of the cohomology of the generalized braid groups of types* * C and D. He constructs the cellular subdivision of the space *nand proves the following * *theorems. Theorem 5.1. The cohomologies of the infinite generalized braid groups of types* * C and D with coefficients in Z are expressed in terms of the cohomologies of classica* *l ones in the following way: Mq Hq(Br(C1 ); Z) = Hq-i(Br1 ; Z); i=0 HOMOLOGY OF BRAID GROUPS AND THEIR GENERALIZATIONS 19 M1 Hq(Br(D1 ); Z) = Hq(Br1 ; Z) [ Hq-2i-3(Br1 ; Z=2)]: i=0 We denote by fln the canonical inclusion fln : Brn-1 ! Brn; and by flqnthe map induced by fln in cohomologies: flqn: Hq(Brn; Z) ! Hq(Brn-1; Z): Kerflqndenotes as usual the kernel of this map. Theorem 5.2. The cohomologies of the finite generalized braid groups of types C* * and D with coefficients in Z are expressed in terms of the cohomologies of classical * *ones in the following way: Mn Hq(Br(Cn); Z) = Hq-i(Brn-i; Z); n 2; i=0 M1 M1 Hq(Br(Dn); Z) = Hq(Brn; Z) [ Kerflq-2in-2i [ Hq-2j-3(Brn-2j-3; Z=2)]; n* * 3; i=0 j=0 where we put formally Br0 = {e}, the group consisting of a single element. The formula for the cohomologies of Br(Cn) was proved in x4 (Theorem 4.14), * *because Hq(Br(Cn); Z) = Hq(Br1;n+1; Z) = Hq(Brn; "Kn): Corollary 5.1 (Theorem of stabilization). With the increasing of n the cohomolo* *gy groups of the generalized braid groups of types C and D stabilize: Hi(Br(Cn); Z) = Hi(Br(C2i-2); Z) ifn 2i - 2; Hi(Br(Dn); Z) = Hi(Br(D2i-1); Z) ifn 2i - 1: The analogues of G. Segal's theorem about the plus-construction for classify* *ing space of infinite braid group (Theorem 4.11) were discovered by D. B. Fuks [F2]: the* * plus- construction of K(Br(C1 ); 1) is equal to 2S3 x S2 and the plus-construction of* * the space K(Br(D1 ); 1) is equal to 2S3 x F , where F is a homotopy fibre of a map * *of degree 2 from S3 to S3. 6. Thom Spectra for Coxeter and Braid Groups From the definition of a finite Coxeter group W we have the inclusion into t* *he orthogonal group O(n) acting in th n-dimensional real vector space V with Euclidean struct* *ure: (6.1) W : W ! O(n); 20 VLADIMIR V. VERSHININ which can be involved into the following commutative diagram: Br(W ) --oW--! W ----! flfl fl fl flfl oW1x...xoWs Br(W1) x . .x.Br(Ws) --------! W1 x . .x.Ws ----! ----! O(n) x? ? ----! O(n1) x . .x.O(ns) x O(n0): This commutative diagram generates the commutative diagram of the classifying s* *paces: BBr(W ) --BoW--! BW ----! flfl fl fl flfl BoW1x...xBoWs BBr(W1) x . .x.BBr(Ws) -----------! BW1 x . .x.BWs ----! - W---! BO(n) x? ? W1x...xWs ---------! BO(n1) x . .x.BO(ns) x BO(n0): This commutative diagram generates in its turn the commutative diagram of th* *e Thom spaces: MBr(W ) -MoW---! MW ----! flfl fl fl flfl MoW1^...^MoWs^1 MBr(W1) ^ . .^.MBr(Ws) ^ Sn0 -------------! MW1 ^ . .^.MWs ^ Sn0 ----! -MW---! MO(n) x? ? MW1^...^MWs^ -------------! MO(n1) ^ . .^.MO(ns) ^ MO(n0); where is the inclusion of the sphere Sn0 into the Thom space: Sn0! MO(n0). We * *remind the definition of a Thom space (see [St], for examle) and a spectrum [Ad], [Sw]* *. For a vector bundle : E ! B with a Riemannian metric we define the corresponding dis* *c and spherical bunles ED ! B and ES ! B. The Thom space M() of the bundle is defined as the factor space ED =ES, what is the same as the one-point compactification * *of the space HOMOLOGY OF BRAID GROUPS AND THEIR GENERALIZATIONS 21 E. By definition, a spectrum X is a sequence of cellular spaces Xn, n 2 Z with * *baise-point provided with structure maps ffln : SXn ! Xn+1; n 2 Z; where S denotes the suspension of a space. The composition of the maps oW and W classifies the bundle YW xW Rn ! XW ; where the space YW was defined in x2. The Thom space of this bundle is equiva* *lent to YW nW Bn=Sn, where Bn is the unit ball and n denotes the half smash produc* *t: A n B = A x B=A x b0, b0 2 B is the base point. For the series C it is equival* *ent to YCn nCn S1 (n), where S1 (n)denotes the n-fold smash product of S1 on which Cn * * acts by permutations between copies of S1 and by complex conjugation on each S1. Fo* *r the series D the Thom space is equivalent to YCn nDn S1 (n), where Dn also acts on * *S1 (n)by permutations between copies of S1 and by complex conjugation on each S1, but ac* *cording to the description of the group Dn the number of conjugations must be even. If* * the Coxeter graph of a group consists of one point (A1 = 2 = Z=2), then Br2 = Z and* * we have B2 = RP 1, BBr2 = S1 and M2 = S-1 (RP 1) (S-1 denotes the inverse of the suspension functor S, which is invertable in the category of spectra), MBr2 = S* *-1 (RP 2) and the map Mo is the canonical inclusion. Using the procedure described above we get the Thom spectra MW1 and MBr(W1 ) for a limit Coxeter group and the corresponding infinite braid group. This gene* *ral situation of Thom spectra for Coxeter groups and generalized braid groups was considered * *in [Ve]. The pairings of Coxeter and braids groups generate the pairings of Thom spac* *es and spectra (which we shall denote by the same symbol ). It means in particular th* *at the Thom spectra for the classical braid groups and generalized braid groups of typ* *e C are multiplicative. There is a beautiful identification of the Thom spectrum for th* *e classical braid group made by Mark Mahowald [Mah1, Mah2] and Fred Cohen [CF3]. Theorem 6.1. The Thom spectrum of the braid group MBr1 is multiplicatively iso* *mor- phic to the Eilenberg-MacLane spectrum K(Z=2). At first Mark Mahowald studied the following situation. Consider the map S1 ! BO; realizing the generator of ss1(BO) ~=Z=2: Apply the functor 2S2 to this map and* * consider the composition j : 2S3 ! 2S2(BO) ! BO; where the second map is the retraction, arising from the infinite loop structur* *e of BO. It was proved by Mark Mahowald [Mah1, Mah2] that the Thom spectrum of j is equival* *ent to K(Z=2). Then Fred Cohen [CF3] considered the composition BBr1 ! 2S3 ! BO; 22 VLADIMIR V. VERSHININ where the first map is that of G. Segal from the Theorem 4.11 and the second on* *e is j. He proves that this composition is homotopic to B1 from (6.1) and that the Thom s* *pectrum of this composition MBr1 is multiplicatively isomorphic to the Eilenberg-MacLa* *ne spec- trum K(Z=2). The geometric aspects of the corresponding bordism theories were d* *iscussed by Fred Cohen [CF3] and Brian Sanderson [Sa]. The pairings described at the end of x2 induce on MW1 for any limit Coxeter* * group W1 at least one module structure over M1 . The same way MBr(W1 ) has at least * *one module structure over MBr1 . Let : S0 ! MBr1 be the unit map of the spectrum MBr1 . The composition of 1 ^ and : MBr(W1 ) ^ S0 ! MBr(W1 ) ^ MBr1 ! MBr(W1 ) is equal to the identity map of MBr(W1 ). This follows from the fact that the c* *omposition: Wk = Wk x A0 ! Wk x Al! Wk+l+1 is equal to the inclusion Wk ! Wk+l+1. The same is true for MBr(Wk). Hence the spectrum MBr(W1 ) is a direct summand in MBr(W1 ) ^ K(Z=2) and it is itself a w* *edge of Eilenberg-MacLane spectra. The spaces XW are connected, so ss0(MBr(W1 )) = * *Z=2. Analogously we prove that the spectrum M1 is equivalent to the wedge of Eilenb* *erg- MacLane spectra K(Z=2) being the module over MBr1 . Shaun Bullet studied in [Bu] Thom spectra and corresponding bordism theories* * for the following groups: 1 ; 1 o Z=2 = C1 ; Br1 o Z=2. It was proved by him that* * these bordism theories are multiplicative and that M*; M( o Z=2)* and M(Br o Z=2)* are polynomial algebras over Z=2. He also proved that the canonical map induces the* * injective multiplicative morphism of cobordism theories: M*( ) ! M( o Z=2)*( ); such that the composition M*( ) ! M( o Z=2)*( )! MO*( ); and the map M(Br o Z=2)*( )! MO*( ) are surjective. Being a module over M1 the Thom spectrum MW1 for a limit Coxe* *ter group W1 becomes a module over K(Z=2) as well. So MW1 is also a wedge of Eile* *nberg- MacLane spectra K(Z=2). As a result we have the following theorem. Theorem 6.2. The Thom spectra MBr(W1 ) and MW1 for limit Coxeter groups are equivalent to the wedges of suspensions over the Eilenberg-MacLane spectrum K(Z* *=2), ss0(MBr(W1 )) = Z=2. From the cofibre sequence S1 ! S1 ! RP 2! : :;: where the first map is multiplication by 2 we obtain HOMOLOGY OF BRAID GROUPS AND THEIR GENERALIZATIONS 23 Corollary 6.1. If the Coxeter graph of W1 contains an isolated vertex, W1 = W* * 0x (Z=2), then MBr(W1 ) = MBr(W 0) ^ SMBr(W 0), where S, denotes a suspension over* * a spectrum. Now let us consider the Thom spectra for the groups C1 and D1 . We would li* *ke to know the number of summands K(Z=2) in each dimension for these spectra. This me* *ans to know modules ss*(MBr(C1 )) = MBr(C1 )* and ss*(MBr(D1 )) = MBr(D1 )*. We use the knowledge of cohomology of the braid groups of the type C and D (Theore* *m 5.1) and then the Thom isomorphism. Theorem 6.3. The Thom spectra MBr(C1 ) and MBr(D1 ) are equivalent to the fol- lowing wedges of the Eilenberg-MacLane spectra 1_ MBr(C1 ) = SiK(Z=2); i=0 _1 MBr(D1 ) = K(Z=2) _ [ S2+iK(Z=2)]: i=0 The pairing defined for the braid groups of type C induces a multiplicative * *structure (probably not commutative) for the theory MBr(C1 )*( ). So MBr(C1 )* has a ring* * struc- ture which we would like to consider. We take the circle S1 with its standard i* *mbedding in Rn+1. Its normal bundle is trivial, so the corresponding classifying map n : S1 ! BO(n) is homotopic to zero. Now we take a fibration fn : BCn ! BO(n) homotopic to the canonical map and analogously a fibration : BBr(Cn) ! BCn; so that the composition fn = f0n: BBr(Cn) ! BO(n) is a fibration homotopic to the canonical map from BBr(Cn) to BO(n). We have H1(BBr(Cn); Z) = Br(Cn)=[Br(Cn); Br(Cn)] = Z Z; H1(B(Cn); Z) = Br(Cn)=[Cn; Cn] = Z=2 Z=2; and the map H1( ) is the canonical projection. We consider a map g0 : S1 ! BBr(* *Cn), such that in homology the generator of H1(S1; Z) maps by H1(g0) to some generat* *or v of H1(BBr(Cn); Z) and such that the composition f0ng0: S1 ! BBr(Cn) ! BO(n) 24 VLADIMIR V. VERSHININ is homotopic to zero. We take g : S1 ! BBr(Cn) as a map homotopic to g0 and suc* *h that f0ng = n. The map g defines a (BBr(Cn); f0n)-structure on S1, and the map g de* *fines a (BCn; fn)-structure on S1 [St]. Let w0 2 H1(BBr(Cn); Z) be the element dual t* *o v 2 H1(BBr(Cn); Z) and w is the reduction mod 2 of w0. By our construction the char* *acteristic number of S1 with (BBr(Cn); f0n)-structure which corresponds to w is nonzero el* *ement of Z=2. So the bordism class of S1 may be considered as a generator of MBr(C1 )1 a* *nd its reduction from BBr(Cn) to B(Cn) is a nonzero element of (MC1 )1. The ring (MC1 * *)* is a free algebra over Z=2. So we have the following theorem. Theorem 6.4. The coefficient ring MBr(C1 )* of the bordism theory corresponding* * to the braid group of type C is a polynomial algebra from one generator s in dimen* *sion 1: MBr(C)* ~=Z=2[s]: Corollary 6.2. The image of the ring MBr(C1 )* in the unoriented cobordism ring* * is equal to zero in positive dimensions. Remark. In the unoriented cobordism ring MO2 = Z=2; MO3 = 0. So the canonical m* *ap to unoriented cobordism for the bordism groups of the braids of type D MBr(D1 )* ! MO*; is neither monomorphism nor epimorphism. Let us consider Thom spectra, corresponding to braid groups of finite Coxete* *r groups. We have seen that these spectra are smash products of spectra for irreducible C* *oxeter groups. Thom spectra MBrk were studied by E. Brown and F. Peterson [BP] and Ral* *ph Cohen [CR]. Let B(l) denote the Brown-Gitler spectrum [BG]. E. Brown and F. Pet* *erson [BP] proved the following theorem. Theorem 6.5. The Thom spectrum MBrk is 2-equivalent to the Brown-Gitler spectrum B([k=2]), where [a] denotes the integer part of a. Corollary 6.3. If a morphism tn : MBrn ! K(Z=2) represents the generator of cohomologies of MBrn as a module over the Steenrod * *algebra and X is any CW complex, then the corresponding morphism of generalized homology theories (MBrn)q(X) ! Hq(X; Z=p) is surjective for q 2[n=2] + 1. For odd primes Ralph Cohen [CR] proved the following theorem. HOMOLOGY OF BRAID GROUPS AND THEIR GENERALIZATIONS 25 Theorem 6.6. For p > 2 MBrkp; (k 6 0 mod p) is homotopy p-equivalent to the (p-* *2)k- fold suspension over the Brown-Gitler spectrum S(p-2)kB([k=p]; p). If a morphism sk : MBrkp ! K(Z=p; (p - 2)k) represents the generator of cohomologies of MBrkp as module over the Steenrod a* *lgebra and X is any CW complex, then the corresponding morphism of generalized homology theories MBr(Ckp)q+(p-2)k(X) ! Hq(X; Z=p) is surjective for ae 2p([k=p] + 1) - 1;if k 6 0 mod p; q 2k - 1; if k 0 mod p: Let p be the mod p -algebra described in [BCKQRS]. So 2 is the graded Z=2-al* *gebra generated by the elements i of degree i for i 0, which are subject to certain * *relations. If p is odd, p is the graded Z=p-algebra generated by the elements i-1 of degree 2* *i(p-1)-1 for i 1 , and the elements i-1 of degree 2i(p - 1) for i 0, which are also su* *bject to certain relations. Let Jk be the left ideal of p generated by 0; :::; k-1 if p * *= 2 and by 0; :::; k-1; -1; :::; k-1 for p odd. Then from the results of the papers [CR], * *[BCKQRS] we obtain the following facts. Corollary 6.4. The 2-localization of the homotopy group ssq(MBrn) is isomorphic* * to (2=J[n=2])q for q < 2[n=2]. The p-localization of the homotopy group ssq(MBr(Ck* *p)); k 6 0 mod p is isomorphic to (p=J[k=p])q-(p-2)kfor q < p(2[k=p] + k + 2) - 2(k + 1). We denote by tW the Thom class of the spectrum MBr(W ): tW : MBr(W ) ! K(Z=2): Let ffC : MBrn ! MBr(Cn) be the map induced by the imbeddings of Coxeter graphs which are described in x2. The composition: MBrn ! MBr(Cn) ! MO(n) ! MO ! K(Z=2); where the last map is the Thom class of MO is equal to the Thom class of MBrn. * *The analogous compositions for the series D and E: MBrn ! MBr(Dn) ! MO(n) ! MO ! K(Z=2); MBrn ! MBr(En) ! MO(n) ! MO ! K(Z=2); n = 6; 7; 8 are equal to the Thom class of MBrn. So we get that the homomorphisms induced * *in cohomology: H*(MBr(Cn); Z=2) ! H*(MBrn); Z=2); H*(MBr(Dn); Z=2) ! H*(MBrn); Z=2); H*(MBr(En); Z=2) ! H*(MBrn); Z=2); n = 6; 7; 8; are epimorphisms. Using the Corollary 6.3 we obtain the following theorem [Ve]. 26 VLADIMIR V. VERSHININ Theorem 6.7. If X is any CW complex then the maps for bordism theories MBr(Cn)** *( ), MBr(Dn)*( ) and MBr(En)*( ), induced by the Thom class t : MBr(Cn)q(X) ! Hq(X; Z=2); MBr(Dn)q(X) ! Hq(X; Z=2); MBr(En)q(X) ! Hq(X; Z=2); n = 6; 7; 8; are epimorphisms for q 2[n=2] + 1. 7. 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