On the pure virtual braid group P V 3
aa r X i v : . [ m a t h . G T ] J un ON THE PURE VIRTUAL BRAID GROUP
P V V. G.BARDAKOV, R. MIKHAILOV, V. V. VERSHININ, AND J. WU
Abstract.
In this article, we investigate various properties of thepure virtual braid group
P V . From its canonical presentation, weobtain a free product decomposition of P V . As a consequence, weshow that P V is residually torsion free nilpotent, which impliesthat the set of finite type invariants in the sense of Goussarov-Polyak-Viro is complete for virtual pure braids with three strands.Moreover we prove that the presentation of P V is aspherical. Fi-nally we determine the cohomology ring and the associated gradedLie algebra of P V . Introduction
Virtual knots were introduced by L. Kauffman [14] and studied bymany authors. One of the motivations lies in the theory of Gaussdiagrams and Gauss codes of knots [14], [19]. Namely, for any knot di-agram it is possible to construct its Gauss diagram and form its Gausscode. The problem is that not every Gauss diagram (or Gauss code)corresponds to some knot. To escape this difficulty virtual knots wereintroduced. Strictly virtual knots are those whose Gauss diagram is nota Gauss diagram of any classical knot. Many notions from the classi-cal knot theory are generalized to virtual knots, such as fundamentalgroup, rack, quandle, Kauffman and Jones polynomials. M. Goussarov,M. Polyak and O. Viro [9] proved that the analogues of the upper andthe lower presentations of the classical fundamental group of a knotgive two different groups for virtual knot. Theorem of M. Goussarov,
Mathematics Subject Classification.
Primary 20F38; Secondary 20F36,57M.
Key words and phrases.
Pure virtual braid group, Lie algebra, presentation.This work was started during the stay of the four authors in Oberwolfach in theframework of the pro gramme RiP during May 18th - June 7th, 2008. The authorswould like to thank the Mathematical Institute of Oberwolfach for its hospitality.The work was continued during the visit of the third author in the Institute forMathematical Sciences of the National University of Singapore during December 4- 19, 2008. He would like to express his gratitude to the IMS .The last author is partially supported by the Academic Research Fund of theNational University of Singapore R-146-000-101-112.
M. Polyak and O. Viro [9] says that two virtually equivalent classicalknots are classically equivalent. This theorem means that inclusion ofusual knots in the universe of virtual knots does not spoil the classi-cal theory. On the other hand virtual knots are appeared to be useful:combinatorial formulas for Vassiliev invariants were obtained by meansof virtual knot theory.Virtual braid groups
V B n were introduced in [14] and [21]. SeichiKamada [12] proved that any virtual link can be described as the clo-sure of a virtual braid, which is unique up to certain basic moves. Thisis analogous to the Alexander and Markov theorems for classical braidsand links. So, the same way as in classical case virtual braids can beused in the study of virtual links.In the virtual braid group two types of crossings are allowed: 1) asusual braids, or 2) as an intersection of lines on the plane. This groupis given by the following set of generators: { ζ i , σ i , i = 1 , , ..., n − } and relations: ζ i = 1 ,ζ i ζ j = ζ j ζ i , if | i − j | > ,ζ i ζ i +1 ζ i = ζ i +1 ζ i ζ i +1 . The symmetric group relations ( σ i σ j = σ j σ i , if | i − j | > ,σ i σ i +1 σ i = σ i +1 σ i σ i +1 . The braid group relations ( σ i ζ j = ζ j σ i , if | i − j | > ,ζ i ζ i +1 σ i = σ i +1 ζ i ζ i +1 . The mixed relationsThe generator σ i corresponds to the canonical generator of the braidgroup Br n . The generators ζ i correspond to the intersection of lines.As for the classical braid groups there exists the canonical epimor-phism to the symmetric group V B n → Σ n with the kernel called thepure virtual braid group P V n . So we have a short exact sequence1 → P V n → V B n → Σ n → . Define the following elements in
P V n λ i,i +1 = ρ i σ − i , λ i +1 ,i = ρ i λ i,i +1 ρ i = σ − i ρ i , i = 1 , , . . . , n − ,λ ij = ρ j − ρ j − . . . ρ i +1 λ i,i +1 ρ i +1 . . . ρ j − ρ j − ,λ ji = ρ j − ρ j − . . . ρ i +1 λ i +1 ,i ρ i +1 . . . ρ j − ρ j − , ≤ i < j − ≤ n − . N THE PURE VIRTUAL BRAID GROUP
P V These elements belong to
V P n . It is shown in [2] that the group P V n ( n ≥
3) admits a presentation with the generators λ ij , ≤ i = j ≤ n, and the following relations: λ ij λ kl = λ kl λ ij (1.1) λ ki λ kj λ ij = λ ij λ kj λ ki , (1.2)where distinct letters stand for distinct indices.Like the usual pure braid groups, groups P V n admit a semi-directproduct decompositions [2]: for n ≥ , the n -th virtual pure braidgroup can be decomposed as P V n = V ∗ n − ⋊ P V n − , where V ∗ n − is a free subgroup of P V n − .As it happens in Mathematics the virtual braid groups appearedin another context under the other name “ n -th quasitriangular group QTr n ” in the work of L. Bartholdi, B. Enriquez, P. Etingof and E.Rain [5] as groups associated to the Yang-Baxter equations. Reallyconsider the equation (1.2) for k = 1, i = 2, j = 3, we get the quantumYang-Baxter equation λ λ λ = λ λ λ . Also the authors of introduce quadratic Lie algebras qtr n and map themonto the associated graded algebras of the Malcev Lie algebras of thegroups QTr n . Among the other results of [5], the complete descrip-tion of integral homology groups of P V n is given: the homology group H r ( P V n , Z ) is free abelian of rank equal to (cid:0) n − r (cid:1) n !( n − r )! (the number ofunordered partitions of n in r ordered parts).2. Residually Nilpotence of
P V The group
P V admits the following presentation:(2.1) h λ , λ , λ , λ , λ , λ | λ λ λ λ − λ − λ − = 1 λ λ λ λ − λ − λ − = 1 , λ λ λ λ − λ − λ − = 1 ,λ λ λ λ − λ − λ − = 1 , λ λ λ λ − λ − λ − = 1 ,λ λ λ λ − λ − λ − = 1 i . Recall that(2.2)
P V = V ∗ ⋋ V , where V = h λ , λ i is a free 2-generated group and V ∗ is the normalclosure of group V = h λ , λ , λ , λ i in P V . The group V is a freeof rank 4. V. G.BARDAKOV, R. MIKHAILOV, V. V. VERSHININ, AND J. WU
Proposition 2.1.
There exists a group G , such that there is the fol-lowing free product decomposition: P V ∼ = G ∗ Z .Proof. We can represent generators λ ij as virtual braids (see [2]) anduse the operations of doubling of string we construct new generators of P V from generators of V .We have a = λ λ , b = λ λ ,b = λ λ , a = λ λ , and it is easy to check that P V = h a , a , b , b , λ , λ i . We can write the old generators as words in new generators: λ = λ − b , λ = b λ − , λ = λ − a , λ = a λ − . The relations from the presentation (1.2) of
P V in new generators havethe following form a b = b a , b a = b λ λ ,b a − = b ( λ λ ) − , a b = a λ λ ,a b − = a ( λ λ ) − , a b = b a , where y x = x − y x. If we define c = λ λ , c = λ , then P V = h a , a , b , b , c , c | [ a , b ] = [ a , b ] = 1 ,b c = b a , a c = a b , b c = b a b , a c = a b a i , where [ x, y ] as usual denotes the commutator of the elements x and y : [ x, y ] = x − y − xy . Hence P V = G ∗ h c i , where the group G can be presented as having generators a , a , b , b , c and the followingrelations(2.3) [ a , b ] = [ a , b ] = 1 , (2.4) b c = b a , a c = a b , b c = b a b , a c = a b a and G = Q ⋋ h c i , where Q is a subgroup of G with the set ofgenerators a , b , a , b and with infinite set of relations(2.5) [ a i , b i ] c k = 1 , i = 1 , , k ∈ Z , rewritten in generators a , b , a , b using relations (2.4). (cid:3) The main result of this section is the following:
N THE PURE VIRTUAL BRAID GROUP
P V Theorem 2.2.
The pure virtual braid group
P V is residually torsionfree nilpotent. Before proof of this theorem let us make a short deviation and ob-serve that the semi-direct product decomposition (2.2) of
P V does notimply the residual nilpotence of P V immediately.2.1. Remarks of Non-residually Nilpotent Groups.
Let G and G be residually nilpotent groups and let G = G ⋋ G be a semi-directproduct of these groups. The following question naturally arises: when G is residually nilpotent? As was proved in [10] if G acts trivially onthe abelianization of G then the answer is positive.We consider the case when G and G are free. Let G = Z be aninfinite cyclic group. Then the group G is called the mapping torus andhave received a great deal of attention in recent years.Let G ϕ = F n ⋋ ϕ Z , where ϕ is a homomorphism ϕ : Z → Aut( F n ). Inthe case n = 1 it is easy to prove that G ϕ is residually nilpotent. In thiscase Aut( F ) contains only two automorphisms: trivial and inversion: ϕ ( a ) = a − , where F = h a i . In the first case G ϕ = F × Z = Z isabelian and hence is residually nilpotent. In the second case it easycalculate that G ′ ϕ = γ G ϕ = { a k | k ∈ Z } and if m > γ m G ϕ = { a m − k | k ∈ Z } . Hence ∞ \ i =1 γ i G ϕ = 1 , and G ϕ is residually nilpotent.If G ϕ = F ⋋ ϕ Z then the following example shows that there existsan automorphism ϕ for which our group is not residually nilpotent. Example (see [18], Example 1.32). Let G = F ⋋ h t i , where F = h a, b i and t is the following automorphism of F : t : (cid:26) a −→ a b,b −→ a b. From the relation t b t − = a b we have [ t − , b − ] = a. From the relation t a t − = a b V. G.BARDAKOV, R. MIKHAILOV, V. V. VERSHININ, AND J. WU we have b = a − [ a, t − ] = [ b − , t − ] [ a, t − ] . Hence in this example ∞ \ i =1 γ i G = F . and the group G is not residually nilpotent.We can generalize this example. Let us consider now the followinggroup G ϕ = F n ⋋ h t i , n ≥ , where conjugation by t induces the automorphism ϕ ∈ Aut( F n ) . Let A = [ ϕ ] be the abelianization of ϕ i. e. A ∈ GL n ( Z ) ≃ Aut(F n / F ′ n ) isinduced by ϕ. Let E be the identity matrix from GL n ( Z ). Proposition 2.3.
If the matrix A − E ∈ GL n ( Z ) , then T ∞ i =1 γ i ( G ϕ ) = F n and G ϕ is not residually nilpotent.Proof. Let ϕ : x −→ x α x α . . . x α n n c ,x −→ x α x α . . . x α n n c ,. . .x n −→ x α n x α n . . . x α nn n c n , where α ij ∈ Z , c i ∈ F ′ n . Then [ ϕ ] = ( α ij ) ni,j =1 = A. Since ϕ ∈ Aut( F n )then A ∈ GL n ( Z ) . Consider the commutator[ x i , ϕ ] = x − i ( x i ) ϕ = x α i . . . x α ii − i . . . x α in n c ′ i , c ′ i ∈ F ′ n , i = 1 , , . . . , n. Hence we have the following system x α − x α . . . x α n n = d ,x α x α − . . . x α n n = d ,. . .x α n x α n . . . x α nn − n = d n , where d i = [ x i , ϕ ]( c ′ i ) − ∈ G ′ ϕ . Since A − E ∈ GL n ( Z ) then from thissystem we have that all x i lie in the commutator subgroup G ′ ϕ but G ′ ϕ ≤ F n Hence G ′ ϕ = F n . Similarly we can to prove that γ i G ϕ = F n for all i > . (cid:3) Proof of Theorem 2.2.
Let F n be a free group of rank n ≥ { x , x , ..., x n } and Aut ( F n ) the group ofautomorphisms of F n . Consider the subgroup of Aut ( F n ), generatedby automorphisms of the form ε ij : (cid:26) x i x − j x i x j if i = j,x l x l if l = i. N THE PURE VIRTUAL BRAID GROUP
P V and denote this subgroup by Cb n . This is the group basis conjugatingautomorphisms of a free group. It is torsion-free nilpotent [1] and alsohas topological interpretations. It is the pure group of motions of n unlinked circles in S [8, 11]. On the other hand it is also the purebraid-permutation group [6]. This group is denoted P Σ n in [11] and in[6]. McCool gave the following presentation for it [17]:(2.6) h ε ij , ≤ i = j ≤ n | ε ij ε kl = ε kl ε ij ,ε ij ε kj = ε kj ε ij , ε ij ε kj ε ik = ε ik ε ij ε kj i where distinct letters stand for distinct indexes.There exist a homomorphism ϕ : P V −→ Cb , ϕ ( λ ij ) = ε ij , ≤ i = j ≤ . We will denote the images of elements a i , b i , c i , by α i , β i , γ i , i = 1 , , respectively.In Cb hold the set of defining relations from P V and relations ε ε = ε ε , ε ε = ε ε , ε ε = ε ε . Since ε = γ , ε = γ − γ , ε = γ − β ,ε = β γ − γ , ε = γ − α , ε = α γ − γ , this set of defining relations has the form α = γ − α γ , γ − β α γ − γ = α γ − β , β = γ − γ β γ − γ . Rewrite this set as follows: α γ = α , ( β α γ − ) γ = α γ − β , ( γ β γ − ) γ = β . We have
Lemma 2.4. Cb = h G , γ | γ − A γ = B, ψ i is an HNN-extension with associated subgroups A = h α , β α γ − , γ β γ − i , B = h α , α γ − β , β i , and the isomorphism ψ : A −→ B is defined by the rule ψ : α −→ α ,β α γ − −→ α γ − β ,γ β γ − −→ β . Subgroup G is isomorphic to subgroup from P V . V. G.BARDAKOV, R. MIKHAILOV, V. V. VERSHININ, AND J. WU
Proof.
We have to prove that A ≃ B. Find the sets of generators of B and A in old generators of Cb . We have B = h ε ε , ε ε , ε ε i and we see that ε ε is an automorphism of F which is the conjuga-tion by x , ε ε is an automorphism of F which is the conjugationby x , ε ε is an automorphism of F which is the conjugation by x . Hence B = h b x , b x , b x i ≃ F , where b y is an inner automorphism of F which is conjugation by y .Similar, A = h b x , b x , \ x x x − i ≃ F , and A ≃ B. (cid:3) The group G is residually torsion free nilpotent as a subgroup ofresidually torsion free nilpotent group Cb . Recall the following resultof Malcev from [16]: the free product of residually torsion-free nilpotentgroups is residually torsion-free nilpotent. Hence, Proposition 2.1 andthe fact that G is residually torsion free nilpotent imply the statementof Theorem 2.2. (cid:3) Remark on finite type invariants.
Recall the notion of finitetype invariants for virtual braids (see [4]). Let A be an abelian groupand n ≥
2. Consider a set map v : V B n → A , i.e. an A -valued invariantof virtual n -braids. Let J be the two-sided ideal in the integral groupring Z [ V B n ] generated by elements { σ i − ρ i , σ − i − ρ i | i = 1 , . . . , n − } .The filtration of the group ring Z [ V B n ] ⊃ J ⊃ J ⊃ . . . is called Goussarov-Polyak-Viro filtration . We say that an invariant v : V B n → A is of degree d if its linear extension Z [ V B n ] → A vanisheson J d +1 . As usual the augmentation ideal is defined by the formula∆( P V n ) = Ker { Z [ P V n ] → Z } . The Goussarov-Polyak-Viro filtrationfor V B n corresponds to the filtration by powers of augmentation idealof the group ring Z [ P V n ] : Z [ P V n ] ⊃ ∆ ⊃ ∆ ⊃ . . . . Theorem 2.2 implies that the intersection of augmentation powers ofthe group ring Z [ P V ] is zero. Hence, we have the following: Proposition 2.5.
The set of invariants of finite degree is complete forvirtual pure braids on three strands. (cid:3)
N THE PURE VIRTUAL BRAID GROUP
P V Relations among relations
Presentation of
P V .Theorem 3.1. The presentation (2.1) is aspherical.Proof.
Consider the standard 2-complex K constructed for the presen-tation (2.1). We have the following exact sequence of abelian groups,called also Hopf exact sequence:(3.1) 0 → H ( P V ) → π ( K ) P V → H ( K ) → H ( P V ) → H ∗ ( P V , − )to the chain complex C ∗ ( ˜ K ) of the universal covering space ˜ K of K .It is easy to prove that H ( P V ) = Z ⊕ . For that we can either usethe result from [5], where all homology of virtual pure braid groupsare described, or consider the semidirect product decomposition (2.2)and observe that H ( P V ) = H ( V , H ( V ∗ )). Since P V is a semidirectproduct of two free groups, its third homology is zero. Now observe that H ( K ) is a free abelian group of a rank less or equal to six, since thereare only six 2-dimensional cells in the 2-complex K . Hence, the map H ( K ) → H ( P V ) is an isomorphism and therefore, π ( K ) P V = 0.Let α ∈ π ( K ). Since π ( K ) P V = 0 , for every n ≥
1, the element α can be presented as a finite sum(3.2) α = X i (1 − g i ) . . . (1 − g in ) α i for some elements g ij ∈ P V . Here (1 − g ) β = β − g ◦ β, g ∈ P V , β ∈ π ( K ) and the action of P V = π ( K ) on π ( K ) is standard. Considerthe monomorphism of Z [ P V ]-modules: f : π ( K ) = H ( ˜ K ) ֒ → C ( ˜ K ) ∼ = Z [ P V ] ⊕ , where C ( ˜ K ) is the second term of the chain complex C ∗ ( ˜ K ). Thecoordinates of the monomorphism f are elements of Z [ P V ] which liein the intersection of powers of augmentation ideal ∆( P V ) due to theexistence of the presentation (3.2) for every n ≥
1. However, for aresidually torsion-free nilpotent group, the intersection of powers ofaugmentation ideals is zero [13], hence α = 0 by Theorem 2.2. Hence,the presentation (2.1) is aspherical. (cid:3) Presentation of
P V n , n ≥ . For n ≥ , as we mentioned inintroduction, the groups P V n admit the following presentation(3.3) h λ ij , ≤ i = j ≤ n | [ λ ij , λ kl ] = 1 , λ ki λ kj λ ij λ − ki λ − kj λ − ij = 1 i Figure 1.
Truncated octahedron and relation amongrelations from (3.3)where distinct letters stand for distinct indices. The classifying spacesof groups
P V n are constructed in [5] as natural quotients of unions ofpermutohedra.Recall that, by a result of Deligne [7] and Salvetti [20], the classifyingspace of a braid group on n strands can be constructed as a quotientof the n -th permutohedron. In [15], the interpretation for this con-struction is given in terms of homotopical syzygies for presentations ofbraid groups. It is shown in [15] that the relations among relations inusual presentation of braid groups can be viewed as labellings of thepermutohedron P , or truncated octahedron. Here we do the same forthe groups P V n , n ≥ . The relations from (3.3) are of length four andsix. The labelling of the truncated octahedron presented in the Figure
N THE PURE VIRTUAL BRAID GROUP
P V ij of an arrowmeans that the generator λ ij corresponds to this arrow.Now one can follow the ideology of the paper [15] and find the in-terpretation of the construction of the classifying space for P V n from[5] in the same way as the result of Deligne [7] and Salvetti [20] is in-terpreted in [15]. It is clear that the higher permutohedra appear inthe construction of higher homotopical syzygies for the presentations(3.3). 4. Cohomology Ring of
P V A Representation of the Ring H ∗ ( G ) . Let X = T { a , b } be the torus T labeled by { a , b } . Let { x , x } be the standard basisfor π ( X ) = π ( T ). According to the defining relation [ a , b ] = 1 in G , there is a map φ : X −→ BG such that in the fundamental groups(4.1) φ ∗ ( x ) = a and φ ∗ ( x ) = b . Similarly, from the defining relation [ a , b ] = 1 in G , there is a torus X = T { a , b } with a map φ : X −→ BG such that in the fundamental groups φ ∗ ( x ) = a and φ ∗ ( x ) = b , where { x , x } is the standard basis for π ( X ). Consider the definingrelation b c = b a ⇐⇒ [ b , c ] = [ b , a ] ⇐⇒ [ b , c ][ a , b ] = 1in G . Let X = S { b , c ; a , b } be the closed oriented surface of genus2 labeled by { b , c ; a , b } with standard generators { y , z , y , z } ofits fundamental group π ( X ) and defining relation [ y , z ][ y , z ] =1. Then there is map φ : X −→ BG such that in the fundamental groups φ ∗ ( y ) = b , φ ∗ ( z ) = c , φ ∗ ( y ) = a and φ ∗ ( z ) = b . Similar constructions apply to the remaining three defining relationsin equation (2.4). So we have the spaces(4.2) X = S { a , c ; b , a } ,X = S { b , c ; a b , b } and X = S { a , c ; b a , a } with the maps φ i : X i → BG , i = 4 , ,
6, such that in fundamentalgroups φ i ∗ sends the standard basis { y ( i − , z ( i − , y ( i − , z ( i − } intothe corresponding words in the labeling bracket for X i , where the stan-dard defining relation in π ( X i ) is [ y ( i − , z ( i − ][ y ( i − , z ( i − ] = 1.For instance, φ ∗ ( y ) = a b .Let X = W i =1 X i and let(4.3) φ : X −→ BG be the map such that φ | X i = φ i for 1 ≤ i ≤
6. The abelianization G → G ab3 = Z ⊕ induces a map(4.4) ψ : BG −→ B Z ⊕ = T . Lemma 4.1.
The following properties hold: The algebra map φ ∗ : H ∗ ( BG ) −→ H ∗ ( X ) is a monomorphism. Moreover φ ∗ : H ( BG ) −→ H ( X ) is an isomorphism of abelian groups. The algebra map ψ ∗ : H ∗ ( T ) −→ H ∗ ( BG ) is an epimorphism.Proof. Let θ = ψ ◦ φ : X → T . We first prove that θ ∗ : H ( T ) −→ H ( X )is onto. Observe that H ( X ) is the free abelian group with a basis { x i , y i , z i , y i , z i | ≤ i ≤ } and H ( T ) = H ( BG ) is the free abelian group with a basis { a , b , a , b , c } , N THE PURE VIRTUAL BRAID GROUP
P V where we use the same notation for the induced generators in the firsthomology group from the fundamental group. From the constructionof the map φ , the linear transformation θ ∗ : H ( X ) → H ( T )is given by the following formula:(4.5) θ ∗ ( x ) = a , θ ∗ ( x ) = b , θ ∗ ( x ) = a , θ ∗ ( x ) = b ,θ ∗ ( y ) = b , θ ∗ ( z ) = c , θ ∗ ( y ) = a , θ ∗ ( z ) = b ,θ ∗ ( y ) = a , θ ∗ ( z ) = c , θ ∗ ( y ) = b , θ ∗ ( z ) = a ,θ ∗ ( y ) = b , θ ∗ ( z ) = c , θ ∗ ( y ) = a + b , θ ∗ ( z ) = b ,θ ∗ ( y ) = a , θ ∗ ( z ) = c , θ ∗ ( y ) = b + a , θ ∗ ( z ) = a . Consider the dual basis { x ∗ i , y ∗ i , z ∗ i , y ∗ i , z ∗ i | ≤ i ≤ } for H ( X ) and { a ∗ , b ∗ , a ∗ , b ∗ , c ∗ } for H ( T ). The linear transformation θ ∗ : H ( T ) → H ( X ) is givenby the formula(4.6) θ ∗ ( a ∗ ) = x ∗ + y ∗ + z ∗ + y ∗ ,θ ∗ ( b ∗ ) = x ∗ + y ∗ + z ∗ + y ∗ ,θ ∗ ( a ∗ ) = x ∗ + y ∗ + y ∗ + y ∗ + z ∗ ,θ ∗ ( b ∗ ) = x ∗ + y ∗ + y ∗ + y ∗ + z ∗ ,θ ∗ ( c ) = z ∗ + z ∗ + z ∗ + z ∗ . From the algebra structure on H ∗ ( X ), we have(4.7) θ ∗ ( a ∗ b ∗ ) = x ∗ x ∗ , θ ∗ ( a ∗ b ∗ ) = x ∗ x ∗ , θ ∗ ( a ∗ c ∗ ) = y ∗ z ∗ ,θ ∗ ( b ∗ c ∗ ) = y ∗ z ∗ , θ ∗ ( a ∗ c ∗ ) = y ∗ z ∗ , θ ∗ ( b ∗ c ∗ ) = y ∗ z ∗ . Since { x ∗ x ∗ , x ∗ x ∗ , y ∗ z ∗ , y ∗ z ∗ , y ∗ z ∗ , y ∗ z ∗ } forms a basis for H ( X ), the linear map θ ∗ : H ( T ) −→ H ( X )is onto.Now since θ ∗ = φ ∗ ◦ ψ ∗ , the linear map φ ∗ : H ( BG ) −→ H ( X )is onto and so it is an isomorphism because H ( BG ) ∼ = H ( P V ) ∼ = H ( P V ) ∼ = Z ⊕ and H ( X ) ∼ = Z ⊕ . This proves assertion 1 because clearly φ ∗ : H ( BG ) → H ( X )is a monomorphism. Assertion 2 also follows because H i ( BG ) = 0 for i > (cid:3) The Cohomology Ring H ∗ ( G ) . From formulas (4.6) and (4.7),we have θ ∗ ( b ∗ a ∗ ) = ( x ∗ + y ∗ + y ∗ + y ∗ + z ∗ )( x ∗ + y ∗ + z ∗ + y ∗ )= y ∗ z ∗ + z ∗ y ∗ = y ∗ z ∗ − y ∗ z ∗ = θ ∗ ( a ∗ c ∗ ) − θ ∗ ( b ∗ c ∗ )and so we have the relation(4.8) a ∗ c ∗ + a ∗ b ∗ + c ∗ b ∗ ≡ H ∗ ( BG ). Similarly, by computing θ ∗ ( a ∗ b ∗ ), we have(4.9) b ∗ c ∗ + b ∗ a ∗ + c ∗ a ∗ ≡ H ∗ ( BG ). From formula (4.6), we have θ ∗ ( b ∗ b ∗ ) = ( x ∗ + y ∗ + y ∗ + y ∗ + z ∗ )( x ∗ + y ∗ + z ∗ + y ∗ )= 0 . Similarly, θ ∗ ( a ∗ a ∗ ) = 0 and so we have the relations(4.10) a ∗ a ∗ = b ∗ b ∗ ≡ H ∗ ( BG ). Theorem 4.2.
The cohomology ring H ∗ ( G ) is the quotient algebra ofthe exterior algebra E ( a ∗ , b ∗ , a ∗ , b ∗ , c ∗ ) subject to the four defining relations given in formulas (4.8), (4.9)and (4.10).Proof. Let A be the quotient algebra of H ∗ ( T ) = E ( a ∗ , b ∗ , a ∗ , b ∗ , c ∗ )subject to the four defining relations given in formulas (4.8), (4.9)and (4.10). Then A i ∼ = H i ( BG )for i ≤
2. It suffices to check that A = 0. From formula (4.10), wehave a ∗ a ∗ b ∗ ≡ a ∗ a ∗ b ∗ ≡ a ∗ a ∗ c ∗ ≡ b ∗ b ∗ a ∗ ≡ b ∗ b ∗ a ∗ ≡ b ∗ b ∗ c ∗ ≡ . From formulas (4.8) and (4.10), we have a ∗ c ∗ b ∗ ≡ a ∗ c ∗ b ∗ ≡ . N THE PURE VIRTUAL BRAID GROUP
P V From formulas (4.9) and (4.10), we have a ∗ c ∗ b ∗ ≡ a ∗ c ∗ b ∗ ≡ ( b ∗ c ∗ − b ∗ a ∗ ) b ∗ ≡ . Thus A = 0 and hence the result. (cid:3) The Cohomology Ring H ∗ ( P V ) . From the decomposition of
P V in Proposition 2.1, there is a group isomorphism δ : G ∗ h c i = G ∗ Z −→ P V with δ ( a ) = λ λ , δ ( b ) = λ λ ,δ ( a ) = λ λ , δ ( b ) = λ λ ,δ ( c ) = λ λ , δ ( c ) = λ . Thus δ ∗ : H ( G ∗ Z ) −→ H ( P V )is the linear transformation given by(4.11) δ ∗ ( a ) = λ + λ , δ ∗ ( b ) = λ + λ ,δ ∗ ( a ) = λ + λ , δ ∗ ( b ) = λ + λ ,δ ∗ ( c ) = λ + λ , δ ∗ ( c ) = λ and so its dual δ ∗ : H ( P V ) → H ( G ∗ Z ) is given by the formula(4.12) δ ∗ ( λ ∗ ) = a ∗ + b ∗ + c ∗ + c ∗ ,δ ∗ ( λ ∗ ) = a ∗ + b ∗ + c ∗ ,δ ∗ ( λ ∗ ) = b ∗ ,δ ∗ ( λ ∗ ) = b ∗ ,δ ∗ ( λ ∗ ) = a ∗ ,δ ∗ ( λ ∗ ) = a ∗ . It follows that ( δ ∗ ) − : H ( G ∗ Z ) → H ( P V ) is given by(4.13) ( δ ∗ ) − ( a ∗ ) = λ ∗ , ( δ ∗ ) − ( b ∗ ) = λ ∗ , ( δ ∗ ) − ( a ∗ ) = λ ∗ , ( δ ∗ ) − ( b ∗ ) = λ ∗ , ( δ ∗ ) − ( c ∗ ) = λ ∗ − λ ∗ − λ ∗ , ( δ ∗ ) − ( c ∗ ) = ( λ ∗ − λ ∗ ) + ( λ ∗ − λ ∗ ) + ( λ ∗ − λ ∗ ) . In the cohomology ring H ∗ ( G ∗ Z ) = H ∗ ( BG ∨ S ), we have c ∗ α = 0for any α ∈ H ( G ∗ Z ). Thus we have relations(4.14) ( λ ∗ − λ ∗ ) λ ∗ ij ≡ ( λ ∗ − λ ∗ ) λ ∗ ij + ( λ ∗ − λ ∗ ) λ ∗ ij in H ∗ ( P V ) for 1 ≤ i = j ≤
3. From formula (4.10), we have(4.15) λ ∗ λ ∗ ≡ λ ∗ λ ∗ ≡ in H ∗ ( P V ). Together with formulas (4.8) and (4.9), we have(4.16) λ ∗ λ ∗ ≡ λ ∗ λ ∗ + λ ∗ λ ∗ ,λ ∗ λ ∗ ≡ λ ∗ λ ∗ + λ ∗ λ ∗ in H ∗ ( P V ). Theorem 4.3.
The cohomology ring H ∗ ( P V ) is the quotient algebraof the exterior algebra E ( λ ∗ , λ ∗ , λ ∗ , λ ∗ , λ ∗ , λ ∗ ) subject to the following relations λ ∗ ij λ ∗ ji ≡ for ≤ i < j ≤ ,
2) ( λ ∗ − λ ∗ ) λ ∗ ij ≡ ( λ ∗ − λ ∗ ) λ ∗ ij + ( λ ∗ − λ ∗ ) λ ∗ ij for ≤ i = j ≤ and λ ∗ λ ∗ ≡ λ ∗ λ ∗ + λ ∗ λ ∗ .Proof. According to the above computations, the defining relations for H ∗ ( P V ) are given by the formulas (4.14), (4.15) and (4.16). The as-sertion follows from the fact by adding two equations in formula (4.16)together with using formula (4.14), formula (4.16) can be placed by (cid:26) λ ∗ λ ∗ ≡ λ ∗ λ ∗ + λ ∗ λ ∗ ,λ ∗ λ ∗ ≡ . (cid:3) Note.
In the cohomology ring H ∗ ( G ∗ Z ), the element c ∗ creates fiverelations a ∗ c ∗ ≡ a ∗ c ∗ ≡ b ∗ c ∗ ≡ b ∗ c ∗ ≡ c ∗ c ∗ ≡ H ∗ ( G ∗ Z ). Thus the six equations incondition 2 are linearly dependent.5. The Associated Graded Lie Algebra for
P V Let G be a group and let L ( G ) = ∞ M i =1 γ i ( G ) /γ i +1 ( G )be the associated graded Lie algebra of G . Consider the short exactsequence h a , b , a , b i G ։ Z = h c i . Since c acts trivially on h a , b , a , b i ab , there is a short exact sequenceof Lie algebras L ( h a , b , a , b i ) L ( G ) ։ L ( h c i ) N THE PURE VIRTUAL BRAID GROUP
P V by [10]. Let A , A , B , B , C be the elements in the Lie algebra in-duced by a , b , a , b , c , respectively. Observe that L ( h a , b , a , b i ) = L ( Z ∗ Z ) is the quotient of the free Lie algebra L ( A , B , A , B )by the relations(5.1) [ A , B ] = 0[ A , B ] = 0 . From the defining relations of G , we have the following identities:(5.2) [ C , B ] = [ A , B ][ C , A ] = [ B , A ][ C , B ] = [ A , B ][ C , A ] = [ B , A ] Theorem 5.1.
The Lie algebra L ( P V ) is the quotient of the freeLie algebra L ( A , B , A , B , C , C ) subject to the relations defined by( 5.1) and ( 5.2).Proof. From the decomposition
P V ∼ = G ∗h c i , it suffices to show that L ( G ) is the quotient of L ( A , B , A , B , C ) subject to the relationsdefined by ( 5.1) and ( 5.2).Let L be the quotient of the free Lie algebra L ( A , B , A , B , C )subject to the relations defined by ( 5.1) and ( 5.2). Then the morphismof Lie algebras L ( A , B , A , B , C ) ։ L ( G )factors through L . Thus is an epimorphism of Lie algebras φ : L ։ L ( G ) . Let L ′ be the sub Lie algebra of L generated by A , B , A , B . Then L ′ is a Lie ideal of L by the defining relations of L with a short exactsequence of Lie algebras L ′ L ։ L ( C ) . The assertion follows by applying 5-Lemma to the following commuta-tive diagram L ′ L ։ L ( C ) ↓ ↓ k L ( h a , b , a , b i ) L ( G ) ։ L ( h c i ) . (cid:3) References [1] S. Andreadakis, On the automorphisms of free groups and free nilpotentgroups,
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P V Sobolev Institute of Mathematics, Novosibirsk 630090, Russia
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