THOM SPECTRA OF GENERALIZED BRAID GROUPS
Vladimir V. Vershinin
Abstract. It is proved that Thom spectra of generalized braid groups are t*
*he wedges of
suspensions over the EilenbergMacLane spectrum for Z=2. Precise structure*
* of the Thom
spectra of the generalized braid groups of the types C and D is obtained. *
*For the generalized
braid groups of type C the natural pairing analogous to the pairing of the*
* classical braids is
studied. This paring generates the multiplicative structure of the Thom sp*
*ectrum such that
the corresponding bordism theory has the coefficient ring isomorphic to th*
*e polynomial ring
over Z=2 on one generator of dimension one: Z=2[s].
The methods of Algebraic Topology were firstly applied for braid groups by V*
*. I. Arnold
[1]. E. Brieskorn [6] generalized the notion of braid group in connection with*
* Coxeter
groups. It was proved later by Mark Mahowald [18, 19] and Fred Cohen [10] that *
*the Thom
spectrum for the infinite braid group is equivalent to the EilenbergMacLane sp*
*ectrum
K(Z=2). Ralph Cohen [11] established connections between Thom spectra for finit*
*e braid
groups and the BrownGitler spectra. Here we are studying Thom spectra for gene*
*ralized
braid groups and proving that for the infinite generalized braid groups the Tho*
*m spectra
are equivalent to the wedges of suspensions over the EilenbergMacLane spectrum*
* K(Z=2).
For Thom spectra of finite generalized braid groups relations with the BrownGi*
*tler spectra
are considered.
1. Generalized Braid Groups
Let V be a finite dimensional vector space (dim V = n) with euclidean struct*
*ure. We
denote by W a finite subgroup of GL(V ) generated by reflections. We use the te*
*rminology
and the content of N. Bourbaki [4]. 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 and that W acts
totally discontinuously on V . The following facts are well known [4].
Proposition 1. (i) W permutes the chambers of M transitively.
(ii) The closure A of a chamber A is a fundamental domain of W acting on V .
(iii) If x 2 V belongs to A its stabilizer is generated by the reflections w*
*ith respect to
the walls of A containing x.
Also there exists a set I and a one to one correspondence of elements of I w*
*ith the
walls of a chamber A : i 7! Mi(A), which is called a canonical indexation of th*
*e walls of
_____________
1991 Mathematics Subject Classification. Primary 55N22, 20F36, 20F55.
Key words and phrases. Generalized braid group, Thom spectrum, EilenbergMac*
*Lane spectrum.
Typeset by AM S*
*TEX
1
2 VLADIMIR V. VERSHININ
a chamber A. Then W is generated by the reflections wi = wi(Mi); i 2 I, satisfy*
*ing only
the following relations
(wiwj)mi;j= e; i; j 2 I;
where the natural numbers mi;jform the Coxeter matrix of W by which the Coxeter*
* graph
(W ) of W is constructed. We use the following notations of P. Deligne [12]: pr*
*od(m; x; y)
denotes the product xyxy::: (m factors). The generalized braid group Br(W ) of *
*W [12] is
defined as a group with generators from the set I and the following relations:
prod (mi;j; i; j) = prod(mj;i; j; i):
From this we get the presentation of the group W if we add the relations:
w2i= e; i 2 I:
We denote by oW the canonical map from Br(W ) to W . Classical braids on k st*
*rings
Brk are obtained by this construction if W = Ak = k+1, the symmetric group on k*
* + 1
symbols. In this case mi;j= 3 , if i 6= j.
Now let J1; :::; Js be the sets of vertexes 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, l*
*et Vq0be
the subspace of V consisting ofTvectors invariant by the action of Wq, Vq is th*
*e orthogonal
complement of Vq0in V , V0 = 1qs Vq0. Then from the Proposition 5 ([4], Chap*
*ter V,
x3.7) we have the following facts.
Proposition 2. (i) The group W is the direct product of subgroups Wq (1 q s).
(ii) The vector space V is the direct sum of the orthogonal subspaces V1; :*
*::; Vs; V0
invariant by the action of W .
If V0 = 0 then each chamber is an open simplicial cone. The classification o*
*f irreducible
(with connected Coxeter graph) Coxeter groups is well known (Theorem 1, Chapter*
* VI, x4
of [4]). It consists of three infinite series: A, C and D and groups E6; E7; E8*
*; F4; G2; H3; H4
and I2(p):
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 1 that
W acts freely on YW . Let XW = YW =W , YW be a covering of XW corresponding *
*to the
group W . Let y0 2 A0 be a point in some chamber A0 and x0 is its image in XW .*
* For each
j 2 I, let `0jbe a homotopy class of paths in YW starting from y0 and ending i*
*n wj(y0)
which contains a polygon line with successive vertices: y0; y0 + iy0; wj(y0) + *
*iy0; wj(y0).
The image `j of `0jin XW is a loop with a base point x0.
Theorem 1 (E. Brieskorn [6], P. Deligne [12]). (i) The fundamental group ss1(XW*
* ; x0) is
generated by `j satisfying the following relations:
prod (mj;k; `j; `k) = prod(mk;j; `k; `j):
(ii) The universal covering of XW is contractible, so XW is K(ss; 1).
If a group W is a direct product of groups W 0and W 00, then the group Br(W *
*) is a
direct product of groups Br(W 0) and Br(W 00). So in the case of Proposition 2 *
*we have
that Br(W ) = Br(W1) x ::: x Br(Ws):
A good reference for generalized braid groups is the survey of V. Ya. Lin [1*
*7].
THOM SPECTRA OF GENERALIZED BRAID GROUPS 3
2. Pairings
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 braid group this pairing m*
*ay be
constructed as the adding of l more strings to the initial k. If oe0iare genera*
*tors of Brk, oe00j
are generators of Brland oer are generators of Br(k + l), then the map can be *
*expressed
in the form:
(oe0i; e) = oei; 1 i k  1;
(e; oe00j) = oej+1; 1 j l  1:
In terms of Coxeter diagrams it means that we take the vertex, corresponding to*
* oek
and imbedSBrk x Brl into Brk+l in accordance with the inclusion of the (k x l) =
(k) (l) into two components of (k+l) \ oek. This permits us to interpret vari*
*ous
imbeddings of products of finite Coxeter groups into the group of bigger index.*
* This is
true for the corresponding generalized braid groups as well. We take away a ver*
*tex in a
connected Coxeter graph and obtain different connected components (less or equa*
*l than
3), which correspond to irreducible Coxeter groups or braid groups whose direct*
* product
is the source of this mapping. For example, we have 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 pairing
(A3; A4; E8) : Br(A4) x Br(A3) ! Br(E8);
which corresponds to the forth horizontal vertex of the Coxeter graph of E8:
o




(E8) o________o________o________o_______o________o_______o.
Embeddings of groups (not products) can also be expressed in this language. Fo*
*r example,
we have an imbedding
ffC : Br(Al1) ! Br(Cl);
and two different imbeddings:
ffD : Br(Al1) ! Br(Dl)
in accordance with two different vertices on the one end of the Coxeter graph f*
*or Dl:
oH
H H
(Dl) HH o________o________..._________o .
o
4 VLADIMIR V. VERSHININ
We would like to consider generalized braid group Br(Ck). We depict the Coxe*
*ter graph
for Ck in the following way:
4
(Cn) o________o________o________..._________o .
So we have a relation 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 the permutations associated with them all leave the numb*
*er 1
invariant. It means that the end of the first string is again at the first plac*
*e. W.L. Chow
[9] 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 ai*
* =
= oe11:::oe1i2oe2i1oei2:::oe1; 2 i n + 1. The elements oe2; :::; oen gen*
*erate a subgroup in
Br1;n+1isomorphic to Brn and the elements a2; :::; an+1 generate a normal free *
*subgroup
Fn, so that Br1;n+1 is a semidirect product of Brn and Fn. The following rela*
*tion is
fulfilled in Br1;n+1:
oe2a2oe2a2 = a2oe2a2oe2:
So the homomorphism OE : Br(Cn) ! Br1;n+1can be defined by the formulae:
OE(w1) = a2;
OE(wi) = oei; i = 2; :::; n:
We shall use the following statement.
Proposition 3. The map OE defines an isomorphism
OE : Br(Cn) ~=Br1;n+1:
Proof. We define the elements vi 2 Br(Cn); 2 i n + 1; by the formula: vi =
wi1:::w2w1w12:::w1i1. We prove that OE(vi) = ai by induction. For i = 2 w*
*e have
OE(v2) = OE(w1) = a2. Let it be true for j < i + 1. We consider OE(vi+1):
OE(vi+1) = OE(wiviw1i) = oeiaioe1i= oeioe11:::oe1i2oe2i1oei2::*
*:oe1oe1i=
= oe11:::oe1i2oeioe2i1oe1ioei2:::oe1 = oe11:::oe1i2oe1i1oei1*
*oeioe2i1oe1ioei2:::oe1 =
THOM SPECTRA OF GENERALIZED BRAID GROUPS 5
= oe11:::oe1i1oeioei1oeioei1oe1ioei2:::oe1 = oe11:::oe1i1oeioei*
*oei1oeioe1ioei2:::oe1 =
= oe11:::oe1i1oe2ioei1:::oe1 = ai+1
The elements w2; :::; wn; v2; :::; vn+1 can be taken as generators of Br(Cn) wi*
*th the same
relations as for oe2; :::; oen; a2; :::; an+1:
The statement of this proposition is evident from the geometrical point of v*
*iew. The
space XCn can be interpreted as a space of n different pairs of points of R2\0,*
* symmetrical
with respect to zero [16, 17]. That is the same as simply the space of n differ*
*ent points
in R2\0. The group Brn+1 is interpreted as the fundamental group of the space X*
*An of
n + 1 different points in R2. If we consider one point (say 0) to be fixed 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). In this interpretation the first generator of B*
*r(Cn) is the
following braid:
1 2 n + 1
 
 o  
o  
o  
o   
o   
  
"   
"   
  
  
"  
 "  ... 
We denote by fi the homomorphism from Br(Cn) to Brn defined by the formulae:
fi(w1) = e;
fi(wi) = oei1; fori > 1:
Then we have fiff = 1Brn: It is known that the group Ck is isomorphic to a wrea*
*th product
of symmetric group k = Ak1 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 a wreath product structure.
Let zj; j = 1; :::; n; be the following elements in Br(Cn): z1 = w1; zj = wj*
*:::w2w1w2:::wj;
j = 2; :::; n:
6 VLADIMIR V. VERSHININ
Lemma 1. We have the following relations in Br(Cn) between the elements zj and
wi; (i; j = 1; :::; n):
zjwi= wizj; i 6= j; j + 1
zjwj+1zjwj+1 = wj+1zjwj+1zj;
zizj = zjzi:
Proof. We use the induction by the index of zj. For j = 1 the first two relati*
*ons are
the relations between the elements wi and the third follows from the relation b*
*etween the
elements wi. Now let all the relations be true for j < k and consider them for *
*j = k. For
the first one let us suppose that i 6= k  1; k; k + 1. Then we have
zkwi= wkzk1wkwi= wkzk1wiwk = wkwizk1wk = wiwkzk1wk = wizk:
If i = k  1, and k > 2, then we obtain
zkwk1 = wkzk1wkwk1 = wkwk1zk2wk1wkwk1 = wkwk1wkzk2wk1wk =
= wk1wkwk1zk2wk1wk = wk1zk:
If k = 2 and i = 1 then we get z2w1 = w2w1w2w1 = w1w2w1w2 = w1z2: Let us consid*
*er
the second relation. We have:
zkwk+1zkwk+1 = wkzk1wkwk+1wkwkzk1wkwk+1 =
= wkzk1wkwk+1wkzk1w1k+1wk+1wkwk+1 = wkzk1wkwk+1wkzk1w1k+1wkwk+1wk =
= wkzk1wkwk+1wkw1k+1zk1wkwk+1wk = wkzk1wk+1wkwk+1w1k+1zk1wkwk+1wk =
= wkwk+1zk1wkzk1wkwk+1wk = wkwk+1wkzk1wkzk1wk+1wk =
= wk+1wkwk+1zk1wkwk+1zk1wk = wk+1wkzk1wk+1wkwk+1zk1wk =
= wk+1wkzk1wkwk+1wkzk1wk = wk+1zkwk+1zk:
It is sufficient to prove the third relation when i < j. So for j = k we suppos*
*e at first that
i < k  1, then we have
zkzi= wkzk1wkzi= wkzk1ziwk = wkzizk1wk = ziwkzk1wk = zizk:
If i = k  1 then we obtain
zkzk1 = wkzk1wkzk1 = (by the second relation)= zk1wkzk1wk = zk1zk:
Lemma is proved.
In the geometric interpretation zj is the following braid:
THOM SPECTRA OF GENERALIZED BRAID GROUPS 7
1 2 j j + 1 n + 1
    
    
    
   
   
    
H     
H     
H    
 H   
  HH   
    
  ... HH  ... 
Let w01; :::; w0kbe the generators of Br(Ck) and w001; :::; w00lare the generat*
*ors of Br(Cl).
Then lemma 1 allows us to define a pairing (C; C)= C :
(C; C): Br(Ck) x Br(Cl) ! Br(Ck+l)
by the formulae:
(C; C)(w0i) = wi; 1 i k;
(C; C)(w001) = wk+1:::w2w1w2:::wk+1;
(C; C)(w00j) = wk+j; 1 j l:
It is easy to check that this pairing is associative, what means that the follo*
*wing diagram
is commutative:
Br(Ck) x Br(Cl) x Br(Cq) Cx1!Br(Ck+l) x Br(Cq)
# 1 x C # C
Br(Ck) x Br(Cl+q) C! Br(Ck+l+q):
It agrees with the pairing for the Coxeter groups mC : Ck x Cl ! Ck+l+1, so we *
*have a
commutative diagram
Br(Ck) x Br(Cl) oCxoC!Ck x Cl
# 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 ff :
Brk x Brl ffxff!Br(Ck) x Br(Cl)
# # C
Brk+l ff! Br(Ck+l):
8 VLADIMIR V. VERSHININ
But there is no analogous commutativity for fi : Br(Ck) ! Brk. To see this let *
*k = 2, then
m(fi x fi)(e; w001) = m(e; e) = e and fi(e; w001) = fi(z3) = fi(w3w2w1w2w3) = o*
*e3oe22oe3 6= e.
So the homomorphism fi does not agree with the pairings.
Geometrically the pairing for the braids of the series C can be described in*
* the following
way. We map R2\0 (with k different points) diffeomorphically onto open disk of*
* radius
k+1=2 without zero Dk+1=2\0 in such a way that the points with coordinates (1; *
*0); :::; (k; 0)
map 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 coordinates (1; 0); :::; (l; 0) ma*
*p onto the
points (k + 1; 0); :::; (k + l; 0). This map
R2\0 x R2\0 ! R2\0
is equivalent to the map of configuration spaces, described by Viktor Vassiliev*
* [21]:
X(k) x X(l) ! X(k + l)
where the space X can be presented in the form X = Y x R. Then for the fundamen*
*tal
groups we obtain our pairing. Considering the generalized braid groups of the t*
*ype C as
the subgroups of the ordinary braid groups our pairing can be described as putt*
*ing 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 a semidirect pro*
*duct of
Brk with Z=2 ::: Z=2 ( k copies) where Brk acts on Z=2 ::: Z=2 by permutati*
*ons.
We denote by s1 the element (a; e; :::; e) 2 Z=2 ::: Z=2; where a is a genera*
*tor of Z=2,
and we denote by s2; :::; sk the standard generators of Brk. Then we have a rel*
*ation:
s1s2s1s2 = s2s1s2s1:
We define a homomorphism
fl : Br(Ck) ! Brk o Z=2
by the formula
fl(wi) = si:
This homomorphism does not agree with the pairings (C and the pairing determine*
*d by
the wreath product structure).
Now we would like to consider the direct limits of finite Coxeter groups. We*
* denote by
W the category whose objects are finite Coxeter groups and morphisms are the in*
*clusions
W 07! W , corresponding to inclusions of Coxeter graphs 0 7! . We call by a ch*
*ain 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 some *
*natural
number NE (for a subgroup W 0of W we consider 0 as a subgraph of ).
Definition 1. 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 series A, C or D with canonical i*
*nclusions
as morphisms we obtain A1 , C1 or D1 as the corresponding limit Coxeter group*
*s.
THOM SPECTRA OF GENERALIZED BRAID GROUPS 9
Proposition 4. (i) The limit Coxeter group W1 is isomorphic to a direct produc*
*t 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.
Proof. It follows from the fact that W1 must be infinite and its Coxeter graph*
* is to have
finite number of components.
Pairings described above generate the pairings of limit Coxeter groups and c*
*orrespond
ing 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 general limit Coxeter group W1 we have different pairings with Br(A1 ) = B*
*r1
depending on the copy of one of the infinite groups of types A1 , C1 or D1 fo*
*r which this
pairing is taken
(W; A) : Br(W1 ) x Br(A1 ) ! Br(W1 ):
3. Construction of Thom Spectra for Generalized Braid Groups
From the construction of a finite Coxeter group W we have the inclusion int*
*o the
orthogonal group O(n) acting in V :
W : W ! O(n);
which can be involved into the following commutative diagram:
Br(W ) oW! W !
flfl fl
fl flfl
Br(W1) x . .x.Br(Ws) o1x...xos!W1 x . .x.Ws!
! O(n)
x?
?
! O(n1) x . .x.O(ns) x O(n0):
This commutative diagram generates the commutative diagram of classifying space*
*s:
BBr(W ) BoW! BW !
flfl fl
fl flfl
BBr(W1) x . .x.BBr(Ws) Bo1x...xBos!BW1 x . .x.BWs!
10 VLADIMIR V. VERSHININ
 W! BO(n)
x?
?
1x...xs!BO(n
1) x . .x.BO(ns) x BO(n0):
This commutative diagram generates in its turn the commutative diagram of Th*
*om
spectra:
MBr(W ) MoW! MW !
flfl fl
fl flfl
MBr(W1) ^ . .^.MBr(Ws) ^ Sn0 Mo1^...^Mos^1!MW1 ^ . .^.MWs^Sn0*
*!
MW! MO(n)
x?
?
M1^...^Ms^!MO(n
1) ^ . .^.MO(ns) ^ MO(n0);
where is the inclusion of the sphere Sn0 into the Thom space: Sn0! MO(n0). T*
*he
composition of the maps oW and W classifies the bundle
YW xW Rn ! XW ;
which has its Thom space YW nW Bn=Sn where Bn is a unit ball and n denotes th*
*e half
smash product: A n B = A x B=A x b0, b0 2 B is the base point. For the series C*
* it is
equivalent to YCn nCn S1 (n), where S1 (n)denotes the nfold smash product of S*
*1 on which
Cn acts by permutations between copies of S1 and by complex conjugation on eac*
*h S1.
For 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 S*
*1, but
according to of 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 = S1 (RP 1), MBr2 = S1 (RP 2) and the
map Mo is the canonical inclusion.
4. Computation of Thom Spectra for Limit Groups
Using the procedure described above we get Thom spectra MW1 and MBr(W1 ) fo*
*r a
limit Coxeter group and corresponding infinite braid group. Pairings of Coxeter*
* and braid
groups generate the pairings of Thom spaces and spectra (which we shall denote *
*by the
same symbol ). For W1 = A1 it was proved by Mark Mahowald [18, 19] and Fred
Cohen [10] that MBr1 is multiplicatively isomorphic to the EilenbergMacLane s*
*pectrum
K(Z=2). The pairing described earlier induce on MW1 module structures over M1 *
*. So
MW1 has at least one module structure over M1 and the same way MBr(W1 ) has
THOM SPECTRA OF GENERALIZED BRAID GROUPS 11
at least one module structure over MBr1 . Let j : S0 ! MBr1 be the unit map of*
* the
spectrum MBr1 . The composition of 1 ^ j 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 EilenbergMacLane spectra. The spaces XW are connected, so ss0(MBr(W1 )) = *
*Z=2.
Shaun Bullet studied in [8] 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 t*
*hese
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. The spectrum M1 is equivalent to the wedge of EilenbergMacLa*
*ne
spectra K(Z=2). Being a module over M1 a Thom spectrum MW1 for the limit Coxeter
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 proved the following theorem.
Theorem 2. The Thom spectra MBr(W1 ) and MW1 for the limit Coxeter groups are
equivalent to the wedges of EilenbergMacLane spectrum K(Z*
*=2),
ss0(MBr(W1 )) = Z=2.
From the cofibre sequence
S1 ! S1 ! RP 2! : :;:
where the first map is a multiplication by 2 we obtain
Corollary 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
12 VLADIMIR V. VERSHININ
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 and then th*
*e Thom
isomorphism. These cohomologies with coefficients in Z were computed by V. Gori*
*unov
[15, 16]. Namely there are expressions for the cohomologies of generalized brai*
*ds in terms
of classical ones: 1
M
Hq(Br(C1 ); Z) = Hqi(Br1 ; Z);
i=0
M1
Hq(Br(D1 ); Z) = Hq(Br1 ; Z) [ Hq2i3(Br1 ; Z=2)]:
i=0
The formula for the cohomologies of Br(C1 ) may be also proved using the propos*
*ition
3 and the fact that the cohomologies of Br1;n+1 are isomorphic to the cohomolog*
*ies of
Brn+1 with coefficients in the Coxeter representation Xn+1 [10], [21].
H*(Br1;n+1; Z) ~=H*(Brn+1; Xn+1):
Representation Xn+1 is defined by the composition
Brn+1 ! n+1 ! Aut Zn+1 ;
n+1 acts on the basis of Zn+1 by permutations.
Theorem 3. The Thom spectra MBr(C1 ) and MBr(D1 ) are equivalent to the followi*
*ng
wedges of the EilenbergMacLane 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
structure which we would like to consider. We take a circle S1 with its standar*
*d imbedding
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;
THOM SPECTRA OF GENERALIZED BRAID GROUPS 13
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)
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 [20]. 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 bordism class of S1 may be considered as a generator of MBr(C1 )1 an*
*d 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 proved the following theorem.
Theorem 4. The coefficient ring MBr(C1 )* of the bordism theory corresponding t*
*o the
braid group of type C is a polynomial algebra from one generator s in dimension*
* 1:
MBr(C)* ~=Z=2[s]:
It is possible to prove theorem 4 by studing the Hopf algebra structure on t*
*he cohomolo
gies of Br(C1 ) as it was done by D. B. Fuks [13] for the ordinary braid group.*
* One more
way of proving theorems 3 and 4 is to use the results of D. B. Fuks [14] that t*
*he "quilleni
sation" of K(Br(C1 ); 1) is equal to 2S3 x S2 nd the "quillenisation" of K(Br(D*
*1 ); 1)
is equal to 2S3 x F , where F is a homotopy fibre of a map of degree 2 from S3 *
*to S3.
Corollary 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.
14 VLADIMIR V. VERSHININ
5. Thom Spectra of Groups of Finite Type
Let us consider Thom spectra, corresponding to braid groups of finite Coxete*
*r groups.
We have seen that these spectra are smash products of the spectra for irreducib*
*le Coxeter
groups. Thom spectra MBrk were studied by E. Brown and F. Peterson [7] and Ral*
*ph
Cohen [11]. In particular, it was proved, that MBrk is 2equivalent to the Brow*
*nGitler
spectrum B([k=2]).
We denote by tW the Thom class of the spectrum MBr(W ):
tW : MBr(W ) ! K(Z=2):
Let MBrn ! MBr(Cn) be the map induced by the imbeddings of Coxeter graphs de
scribed earlier. 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 results of Ralph Cohen [11] we obtain the following*
* theorem.
Theorem 5. 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, where [a] denotes the integer part of a.
6. Acknowledgements
The author is thankful to the Universitat Autonoma de Barcelona where the re*
*search of
this paper was started during the visit in 19931994. Special thanks are to the*
* topologists
for the hospitality and the stimulating atmosphere in the Departament de Matema*
*tiques
and in the Centre de Recerca Matematica: to Professors Manuel Castellet, Irene *
*Llerena,
Jaume Aguade, Carles Broto and Carlos Casacuberta, to Professor N. H. V. Hung a*
*lso for
the information on the cohomology of groups. The author is grateful to the Univ*
*ersite de
Nantes where this research was finished. I thank Professor Fred Cohen for the d*
*iscussions
which started in Stockholm, 1993, and were continued by email and Professor A.*
* Z. Ananin
for conversations on the material of this paper in Italy at the begining of 199*
*4.
THOM SPECTRA OF GENERALIZED BRAID GROUPS 15
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Sobolev Institute of Mathematics, Novosibirsk, 630090, Russia
Email address: versh@math.nsc.ru