Homological Representations of the Braid Groups
3.7 Proof of Theorem 3.15
3.7.3 End of the proof
Lemma 3.26.If a self-homeomorphism f of(D, Q)represents an element of the kernelKer (Bn →AutR(H)), then N, f(α)= N, α for any noodle N and any oriented spanning arcαon (D, Q).
Proof. As was already observed above, the homomorphismf∗:H → Htrans- forms anyα-classv∈ Hinto anf(α)-class. Formula (3.31) and the assumption f∗= id imply that
−(q−1)2(qt+ 1)N, f(α)∗=F , f∗(v)
=F , v
=−(q−1)2(qt+ 1)N, α∗.
Therefore,N, f(α)=N, α.
Similarly,f(α1) can be isotopped offN4. By Section 3.6.1, this can be done by a sequence of isotopies eliminating digons for the pair (N4, f(α1)). Since N4
and f(α1) do not meet N3, neither do the digons in question. Hence the isotopies along these digons do not create intersections of f(α1) with N3. Repeating this argument, we can ensure thatf(α1) is disjoint from all the noodles Ni with i = 3,4, . . . , n−1. Drawing these (disjoint) noodles, one easily observes that all spanning arcs in their complement are isotopic toα1. Then, applying one more isotopy, we can arrange thatf(α1) =α1. Note that all self-homeomorphisms of a closed interval keeping the endpoints fixed are isotopic to the identity. Therefore we can further isotopf so that it becomes the identity onα1. Applying a similar procedure to α2, we can ensure that f|α2 = id while keepingf|α1= id. Continuing in this way, we can isotopf so that it preserves the interval [1, n]×{0}pointwise. Applying a further isotopy, we can ensure thatf = id in an open neighborhood of this interval in D. In other words,f = id outside an annular neighborhoodAof∂DinΣ=D−Q.
We identifyAwith∂D×[0,1], so that∂D⊂∂Ais identified with∂D×{0}. The (smooth) homeomorphismf|A:A→A must be isotopic (rel∂A) to the kth power of the Dehn twist about the circle∂D× {1/2} ⊂Afor somek∈Z;
see, for instance, [Iva02, Lemma 4.1.A]. Thus, f is isotopic to gk, where g is the self-homeomorphism of D acting as the Dehn twist on A and as the identity onD−A.
We claim that the homeomorphismg acts onHvia multiplication by the monomial q2ntb for some b ∈ Z. (In fact, b = 2 but we shall not need it.) Thenf∗ :H → His multiplication by q2nktbk. For k= 0, this cannot be the identity map: if it is, then
(q2nktbk−1)H= 0 and the linearity of the function
H →Z[q±1, t±1], v→ FN, v
implies that this function is identically zero for any noodleN. By Lemma 3.25 we must haveN, α= 0 for all N, α. The latter is not true, as was observed before the statement of Lemma 3.20. This contradiction shows thatk= 0, so thatf is isotopic to the identity.
To compute the action ofgonH, consider the homeomorphismg:C → C defined by g({x, y}) = {g(x), g(y)} for distinct x, y ∈ Σ; cf. Section 3.5.3.
Consider the lift g : C → Cof g keeping fixed all points lying over the base point c ={d1, d2} ∈ C. Since g = id outside A, we haveg = id outside the set{(x, y)∈ C |x∈Aory∈A}. LetA⊂Cbe the preimage of this set under the covering projectionC → C . The homeomorphismg has to act on C − A as a covering transformation qatb for some a, b ∈ Z. The set Ais a tubular neighborhood of∂CinCand therefore any 2-cycle inCcan be deformed into C − A. Hence, gacts onHas multiplication byqatb.
We now verify that a = 2n. For i = 1,2, define a path δi : I → A by δi(s) =di×s, wheres∈I= [0,1] andd1, d2∈∂Dare the points used in the construction ofC. Setδ={δ1, δ2}:I → C and let δ:I →Cbe an arbitrary lift ofδ. The pointδ(0) lies over cand thereforeg(δ(0)) =δ(0). The point δ(1) lies in the closure of C − A and therefore g(δ(1)) = qatbδ(1). Therefore the pathg◦δ:I→Cleads fromδ(0) to qatbδ(1). Multiplying by δ−1, we obtain the pathδ−1(g◦δ) leading fromδ(1) to qatbδ(1) in C. By the definition of the coveringC → C , the integeramust be the value of the invariantwon the loop obtained by projecting the latter path toC. This loop is nothing but
δ−1(g◦δ) ={δ−11(g◦δ1), δ−21(g◦δ2)}. Hence,
a=w(δ−1(g◦δ)) =w(δ1−1(g◦δ1)) +w(δ2−1(g◦δ2)).
It remains to observe thatw(δi−1(g◦δi)) =nfori= 1,2. This completes the proof in the casen≥3.
The remaining cases n = 1,2 are easy. For n = 1, there is nothing to prove, since B1 = {1}. The group B2 is infinite cyclic, and the square of a generator is the Dehn twist as in the previous paragraphs, which, as we have just explained, represents an element of infinite order in AutR(H).
Notes
The Burau representationψn was introduced by Burau [Bur36]. A version of Theorem 3.1 was first obtained by Squier [Squ84], who used a different, more complicated, matrix in the role ofΘn. The matrix Θn in Theorem 3.1 was pointed out by Perron [Per06].
The representations ψ2, ψ3 were long known to be faithful; see [Bir74].
Moody [Moo91] first proved thatψn is nonfaithful forn≥9. Long and Pa- ton [LP93] extended Moody’s argument to n ≥ 6. Bigelow [Big99] proved thatψ5is nonfaithful. Our exposition in Section 3.2 follows the ideas and tech- niques of these papers. The examples in Section 3.1.3 are taken from [Big99].
The proof of Lemma 3.5 was suggested to the authors by Nikolai Ivanov; see also [PR00, Prop. 3.7]. Theorem 3.7 is folklore. The reducibility ofψn is well known; see [Bir74].
The Alexander–Conway polynomial is a refinement, due to J. H. Conway, of the Alexander polynomial of links; see [Lic97] for an exposition. Burau computed the Alexander polynomial of the closure of a braid from its Burau matrix; see [Bir74]. The refinement of this result to the Alexander–Conway polynomial (Section 3.3 and the second claim of Theorem 3.13) is due to V. Turaev (unpublished).
The Lawrence–Krammer–Bigelow representation is one of a family of rep- resentations introduced by Lawrence [Law90]. Her work was inspired by a
study of the Jones polynomial of links and was concerned with representa- tions of Hecke algebras arising from the actions of braids on the homology of configuration spaces. Theorem 3.15 was proven independently and from dif- ferent viewpoints by Krammer [Kra02] and Bigelow [Big01] after Krammer proved it forn = 4 in [Kra00]. The theory of noodles (Section 3.6) and the proof of Theorem 3.15 given in Section 3.7 are due to Bigelow [Big01]. (In loc. cit. Bigelow also uses the concept of a “fork” introduced by Krammer in [Kra00]. Here we have avoided the use of forks.) For more on this and related topics, see the surveys [Big02], [Tur02], [BB05].