Homological Representations of the Braid Groups
3.5 The Lawrence–Krammer–Bigelow representation
Subtracting In−1, then adding the first row to the second one, and finally multiplying the first row by−t, we obtainD− =−t−1detA−, where
A−=
⎛
⎝a+t b x c−1 d−1 y
p q M−In−3
⎞
⎠.
The matricesA0,A+,A− differ only in the first columns, which we denote by A10,A1+,A1−, respectively. Clearly,
−tA1++A1−= (1−t)A10. We conclude thatD+−tD−= (1−t)D0.
The functionL →f(L) satisfies all conditions of the Alexander–Conway polynomial except that a priori it takes values in the field of rational functions in s rather than in its subring of Laurent polynomials Z[s, s−1]. However, applying the skein relation and an induction on the number of crossings of a link diagram as at the beginning of the proof, one observes that all the values offare integral polynomials ins−s−1. In particular, all the values offare
Laurent polynomials ins.
projection F → C is a 2-fold covering. In the notation of Section 1.4.3, we haveC=C2(Σ) =Cn,2(D).
In the sequel, a nonordered pair of distinct points x, y ∈ Σ is denoted by{x, y}. Note that{x, y}={y, x} ∈ C. A (continuous) pathξ:I→ C, where I= [0,1], can be written in the formξ={ξ1, ξ2} for two (continuous) paths ξ1, ξ2:I→Σ. The equalityξ={ξ1, ξ2}means that ξ(s) ={ξ1(s), ξ2(s)} for alls∈I. The pathξis a loop if {ξ1(0), ξ2(0)}={ξ1(1), ξ2(1)}, so that either
ξ1(0) =ξ1(1)=ξ2(0) =ξ2(1) or
ξ1(0) =ξ2(1)=ξ1(1) =ξ2(0).
In the first case, the pathsξ1, ξ2are loops onΣ. In the second case, the paths ξ1, ξ2 are not loops but their productξ1ξ2is well defined and is a loop onΣ.
We introduce two numerical invariantswanduof loops in C. Consider a loopξ={ξ1, ξ2}inC as above. Ifξ1,ξ2are loops, thenw(ξ) =w(ξ1) +w(ξ2), wherew(ξi) is the total winding number ofξi around{(1,0), . . . ,(n,0)}; see Section 3.2.2. Ifξ1(1) =ξ2(0), then the product pathξ1ξ2is a loop onΣ and we setw(ξ) =w(ξ1ξ2).
To define the second invariantu(ξ), consider the map s→ ξ1(s)−ξ2(s)
|ξ1(s)−ξ2(s)| :I→S1⊂C. (3.14) This map sendss= 0,1 either to the same numbers or to opposite numbers.
Therefore, the map s→
ξ1(s)−ξ2(s)
|ξ1(s)−ξ2(s)| 2
:I→S1 (3.15)
is a loop onS1. The counterclockwise orientation ofS1 determines a gener- ator ofH1(S1;Z) ∼=Z. The loop (3.15) onS1 is homologous tok times the generator withk∈Z, and we setu(ξ) =k. Note thatu(ξ) is even ifξ1, ξ2 are loops and odd otherwise. The invariantsw(ξ) and u(ξ) are preserved under homotopy ofξ and are additive with respect to the multiplication of loops.
For example, consider the loopξ={ξ1, ξ2}, whereξ1is the constant loop in a pointz∈Σ andξ2 is an arbitrary loop inΣ− {z}. Thenw(ξ) =w(ξ2) andu(ξ) = 2v, wherev is the winding number ofξ2 aroundz. In particular, ifξ2is a small loop encircling counterclockwise a point ofQandz∈∂Σ=∂D, then w(ξ) = 1 andu(ξ) = 0. To give another example, pick a small closed diskB ⊂Σ and two distinct pointsa, b∈∂B. Letξ1 (resp.ξ2) parametrize the arc on∂B leading fromato b (resp. from b to a) counterclockwise. For the loopξ={ξ1, ξ2}, we havew(ξ) =w(ξ1ξ2) = 0 andu(ξ) = 1.
3.5.2 The covering spaceCand the module H
We fix once for all two distinct pointsd1, d2∈∂Σ =∂Dand takec={d1, d2} as the base point ofC. The formula
ξ → qw(ξ)tu(ξ)
defines a group homomorphismϕfrom the fundamental groupπ1(C, c) to the multiplicative free abelian group with generatorsq, t. The examples in the previous subsection show that this homomorphism is surjective.
LetC → C be the covering corresponding to the subgroup Kerϕofπ1(C, c).
The generatorsqandtact onCas commuting covering transformations, and C=C/(q, t). A loopξ inC lifts to a loop inCif and only ifw(ξ) =u(ξ) = 0.
The 2-fold coveringF → C is a quotient of the coveringC → C , as we now explain. Observe that a loopξ={ξ1, ξ2}onC lifts to a loop onF if and only ifξ1, ξ2 are loops on Σ. The latter holds if and only if u(ξ) is even. Hence, the coveringF → C is determined by the subgroup ofπ1(C, c) formed by the homotopy classes of loopsξ with u(ξ) ∈2Z. Therefore F =C/(q, t2) is the quotient ofCby the group of homeomorphisms generated byqandt2.
The action ofq, tonCinduces an action ofq, ton the abelian group H=H2(C;Z).
This turnsHinto a module over the commutative ring R=Z[q±1, t±1].
The moduleHcan be explicitly computed using a deformation retraction ofC onto a 2-dimensional CW-space; see [Big03], [PP02]. The computation shows thatHis a freeR-module of rank n(n−1)/2, that is,
H ∼=Rn(n−1)/2. (3.16)
For more on the structure ofH, see Section 3.5.6.
3.5.3 An action of Bn onH
As we know from Section 1.6, the braid groupBn is canonically isomorphic to the mapping class groupM(D, Q). In the remaining part of this chapter, we make no distinction between these two groups. We now construct an ac- tion of Bn on H. Any self-homeomorphism f of the pair (D, Q) induces a homeomorphismf:C → C by
f({x, y}) ={f(x), f(y)},
wherex, y are distinct points ofΣ=D−Q. Clearly,f(c) =c, so that we can consider the automorphismf#ofπ1(C, c) induced byf.
Lemma 3.14.We have ϕ◦f#=ϕ.
Proof. We need to prove thatw◦f#=wandu◦f#=u. The first equality is proven by the same argument as in Section 3.2.2. To prove the second equality, consider the inclusion of configuration spacesC=C2(Σ)→ C2(D) induced by the inclusionΣ →D. The definition of the numerical invariantufor loops inC extends to loops in C2(D) word for word and gives a homotopy invariant of loops inC2(D). The Alexander–Tietze theorem stated in Section 1.6.1 implies that the self-homeomorphism of C2(D) induced by f is homotopic to the identity. Hence,u◦f#=uand thereforeϕ◦f#=ϕ.
The equalityϕ◦f#=ϕimplies thatflifts uniquely to a mapf:C → C keeping fixed all points ofClying overc. The same equality ensures thatfcom- mutes with the covering transformations ofC. The mapfis a homeomorphism with inversef−1. Therefore the induced endomorphismf∗ of H=H2(C;Z) is anR-linear automorphism. Consider the mapping
Bn =M(D, Q)→AutR(H)
sending the isotopy class off tof∗:H → H. This mapping is a group homo- morphism. It is called theLawrence–Krammer–Bigelow representationofBn. A fundamental property of this representation is contained in the following theorem.
Theorem 3.15.The Lawrence–Krammer–Bigelow representation of the braid groupBn is faithful for all n≥1.
This theorem is proven in Sections 3.6 and 3.7. One can give explicit matrices describing the action of the generators σ1, . . . , σn−1 ∈ Bn on H; see [Kra02], [Big01], [Bud05]. The proof of Theorem 3.15 given below uses neither these matrices nor the isomorphism (3.16).
3.5.4 The linearity ofBn
We say that a groupGislinear if there is an injective group homomorphism G→ GLN(R) for some integer N ≥1. We state an important corollary of Theorem 3.15.
Theorem 3.16.For all n≥1, the braid group Bn is linear.
This theorem follows from Theorem 3.15 and the isomorphism (3.16). In- deed, choosing a basis of theR-moduleH, we can identify AutR(H) with the matrix group GLn(n−1)/2(R). The ringR =Z[q±1, t±1] can be embedded in the field of real numbers by assigning to q, t algebraically independent real values. This induces an embedding
GLn(n−1)/2(R)→GLn(n−1)/2(R).
Composing it with the Lawrence–Krammer–Bigelow representation, we obtain a faithful homomorphismBn →GLn(n−1)/2(R).
We give another proof of Theorem 3.16 entirely avoiding the use of the isomorphism (3.16). This proof gives an embedding of Bn into GLN(R) forN =n(n+ 1). We begin with a simple algebraic lemma.
Lemma 3.17.Let L = Z[x±11, x±21] be the ring of Laurent polynomials in the variables x1, x2. Let C be a free L-module of finite rank N ≥ 1. For an arbitraryL-submoduleH ofC, the groupAutL(H)ofL-automorphisms ofH embeds intoGLN(R).
Proof. Let Q = Q(x1, x2) be the field of rational functions in the variables x1, x2 with rational coefficients. Clearly,Qis the field of fractions ofL. Con- sider the Q-vector space H = Q⊗LH. Since H is a submodule of a free L-module, it has noL-torsion, and hence the natural homomorphismH →H sending h ∈ H to 1⊗h is injective. Any L-automorphism of H extends uniquely to aQ-automorphism ofH. In this way, the group AutL(H) embeds into GLm(Q), wherem= dimQH. The fieldQcan be embedded inRby as- signing tox1, x2algebraically independent real values. This gives embeddings AutL(H)⊂GLm(Q)⊂GLm(R). Note that the inclusioni:H →C induces a homomorphism ofQ-vector spacesH →C, where C =Q⊗LC. This ho- momorphism is injective: any element of its kernel can be multiplied by an element ofLto give an element of Ker(i) = 0. Thereforem≤dimQC =N,
so that AutL(H)⊂GLm(R)⊂GLN(R).
Note that for any topological manifold M with boundary ∂M, the in- clusion M◦ = M −∂M → M is a homotopy equivalence. The homotopy inverseM → M◦ can be obtained by pushing M into M◦ using a cylinder neighborhood of∂M in M.
We can now prove Theorem 3.16. It is clear that F◦ = F −∂F is the complement of the diagonal {(x, x)}x∈Σ◦ in Σ◦×Σ◦. By Lemma 1.26, as- signing to any ordered pair of points the first point, we obtain a locally trivial fiber bundleF◦ →Σ◦ whose fiber is the complement of a point in Σ◦. The baseΣ◦ of this bundle deformation retracts onto a wedge of ncircles, while the fiber deformation retracts onto a wedge of n+ 1 circles. This implies that F◦ deformation retracts onto a 2-dimensional CW-complex, X ⊂ F◦, with one zero-cell, 2n+ 1 one-cells, andn(n+ 1) two-cells. Since the inclusion F◦→ F is a homotopy equivalence, the inclusionX → F also is a homotopy equivalence.
Recall from Section 3.5.2 thatCcan be viewed as the covering ofFwith the group of covering transformationsZ×Zgenerated byqandt2. The covering C → F restricts to a covering X → X with the same group of covering transformations. Here X is the preimage of X ⊂ F in C, and the inclusion X ⊂Cis a homotopy equivalence. The cellular chain complex ofX has the formC2→C1→C0, where eachCi is a free module over the ring
R0=Z[q±1, t±2]⊂R .
The rank of theR0-moduleCiis equal to the number ofi-cells inX. Therefore H=H2(C;Z) =H2(X;Z) = Ker(∂:C2→C1)
is anR0-submodule of C2. We now apply Lemma 3.17, where we substitute x1=q, x2=t2, C=C2, H=H, andN =n(n+ 1).
By this lemma, AutR0(H) embeds into GLN(R). Composing with the embed- dings
Bn →AutR(H)⊂AutR0(H),
we obtain the claim of the theorem.
3.5.5 A sesquilinear form onH
The moduleHcarries a naturalR-valued sesquilinear form defined as follows.
The orientation of C lifts to Cand turns the latter into an oriented (four- dimensional) manifold. Consider the associated intersection formH ×H →Z.
Its valueg1·g2 on homology classes g1, g2 ∈ H is obtained by representing these classes by transversal 2-cyclesG1, G2inCand counting the intersections ofG1, G2 with signs ±determined by the orientation of C. The intersection formH × H →Zis symmetric and invariant under the action of orientation- preserving homeomorphismsC → C. In particular, this form is invariant under the action of the covering transformationsq, t.
Define a pairing
,:H × H →R by
g1, g2=
k,∈Z
(qktg1·g2)qkt. (3.17) The sum on the right-hand side is finite, since the 2-cyclesG1,G2as above lie in compact subsets ofCand therefore the cycles qktG1 and G2 are disjoint except for a finite set of pairs (k, ).
The pairing (3.17) is invariant under the action of orientation-preserving homeomorphisms C → Ccommuting with the covering transformations q, t.
In particular, it is preserved under the action of the braid groupBn onH. Lemma 3.18.For any g1, g2∈ H andr∈R,
g2, g1=g1, g2, g1, rg2=rg1, g2, rg1, g2=rg1, g2, (3.18) where r→r is the involutive automorphism of the ring R sending q to q−1 andttot−1.
Proof. We have
g2, g1=
k,∈Z
(qktg2·g1)qkt
=
k,∈Z
(g1·qktg2)qkt
=
k,∈Z
(q−kt−g1·g2)qkt
=
k,∈Z
(qktg1·g2)q−kt−
=g1, g2.
To verify the equalities g1, rg2= rg1, g2 and rg1, g2 =rg1, g2, it suffices to consider the caser=qitj withi, j∈Z. We have
g1, qitjg2=
k,∈Z
(qktg1·qitjg2)qkt
=qitj
k,∈Z
(qk−it−jg1·g2)qk−it−j
=qitjg1, g2 and
qitjg1, g2=
k,∈Z
(qk+it+jg1·g2)qkt
=q−it−j
k,∈Z
(qk+it+jg1·g2)qk+it+j
=q−it−jg1, g2.
According to Budney [Bud05], the form,:H × H →R is nonsingular in the sense that the determinant of its matrix with respect to a basis ofH is nonzero. Moreover, replacingq, t with appropriate complex numbers, one obtains a negative definite Hermitian form; see [Bud05]. This gives an injective group homomorphism fromBn into the unitary groupUn(n−1)/2.
3.5.6 Remarks
We make a few remarks aimed at familiarizing the reader with the moduleH. These remarks will not be used in the sequel.
It is quite easy to see that the module His nontrivial and in fact rather big. Let X, X be the same spaces as in the proof of Theorem 3.16. Note that the ringR0=Z[q±1, t±2] embeds into the fieldQ=Q(q, t2) of rational
functions in the variablesq, t2. For anR0-moduleH, denote the dimension of theQ-vector spaceQ⊗R0H by rkH. We verify that rkH ≥n(n−1). Indeed,
rkH0(X;Z)−rkH1(X;Z) + rkH=χ(X) =n(n−1),
whereχ(X) is the Euler characteristic ofX. For every 0-cellxofX there is a path inX leading fromxtoqx, so that (1−q)xis the boundary of a 1-chain.
HenceQ⊗RH0(X; Z) = 0 and rkH0(X;Z) = 0. Therefore, rkH ≥n(n−1).
The isomorphism (3.16) implies thatH ∼=Rn(n0 −1).
Specific elements of H may be derived from arbitrary disjoint spanning arcs α, β on (D, Q). Consider the associated loops α, β : S1 → Σ as in Figure 3.1. Choosing these loops closely enough toα, β, we may assume that they do not meet. The formula
(s1, s2)→ {α(s1), β(s2)} ∈ C
fors1, s2∈S1 defines an embedding of the torusS1×S1 intoC. The induced homomorphism of the fundamental groups sends π1(S1×S1) to the kernel ofϕ. Therefore this embedding lifts to an embedding of the torus into C. It can be shown that the fundamental class of the torus represents a nontrivial homology class inH. Such classes, corresponding to variousα, β, are permuted by the action ofBnonH. A similar but subtler construction applies to pairs of spanning arcs on (D, Q) meeting at one common endpoint; it gives a mapping of an orientable closed surface of genus 2 to C; see [Big03]. Moreover, each spanning arc on (D, Q) gives rise to a mapping of an orientable closed surface of genus 3 toC; see [Big03] and Section 3.7.1. Applying these constructions to the arcs
[1,2]× {0}, [2,3]× {0}, . . . ,[n−1, n]× {0}
on (D, Q) and to pairs of such arcs, one obtainsn(n−1)/2 homology classes inHforming anR-basis ofH.