Groups of homeomorphisms vs. configuration spaces

Braids and Braid Groups

1.7 Groups of homeomorphisms vs. configuration spaces

(b) Consider r≥2 spanning arcs on (M, Q) with one common endpoint a∈Qand disjoint otherwise. Moving around ain the direction given by the orientation ofM, denote these arcs byα₁, α₂, . . . , α_r. Prove that

τ_α⁻¹

1τ_α2τ_α1 =τ_α2τ_α1τ_α⁻¹

2

commutes withτ_αi for 3≤i≤r. Deduce that τα1β τα2τα1 =τα2τα1β τα2

for any elementβ of the group generated byτα3, τα4, . . . , ταr.

Exercise 1.6.2.Prove that M(S¹) = {1}. (Hint: Composing an arbitrary self-homeomorphismf ofS¹ with a rotation of S¹ into itself, we can assume that f has a fixed point. Cutting out S¹ at a fixed point of f, we obtain a self-homeomorphism of a closed interval, which, as we know, is isotopic to the identity.)

It is known that a map f from a topological space X to Top(M, Q) is continuous if and only if the map X×M →M sending (x, y) ∈X×M to f(x)(y) is continuous. Applying this to X = I, we conclude that two self- homeomorphisms of (M, Q) are isotopic if and only if they can be connected by a path in Top(M, Q), i.e., if and only if they lie in the same connected component of Top(M, Q). Therefore,

M(M, Q) =π0(Top(M, Q)). (1.15) Set Top(M) = Top(M,∅). The obvious embedding Top(M, Q)→Top(M) makes Top(M, Q) into a closed subgroup of the topological group Top(M).

The group Top(M) is intimately related to the configuration spaces of nonordered points ofM^◦ introduced in Section 1.4.3. Forn≥1, set

Cn=Cn(M^◦) =Fn(M^◦)/Sn.

To describe the relation between Top(M) and Cn, pick a set Q ⊂M^◦ con- sisting ofn points. We define an evaluation map e=eQ : Top(M)→ Cn by e(f) = f(Q), where f ∈ Top(M). It is easy to deduce from the definitions thateis a surjective continuous map.

Lemma 1.35.The map e : Top(M) → Cn is a locally trivial fibration with fiberTop(M, Q).

Proof. Let Fn = Fn(M^◦) be the configuration space of n ordered points in M^◦. We can factor e as the composition of a map c : Top(M) → Fn

with the coveringFn → Cn. To construct c, fix an order in the set Q and define c by c(f) = f(Q), where f ∈ Top(M) and the order in f(Q) is in- duced by the one in Q. To prove the lemma, it suffices to prove that c is a locally trivial fibration. The proof of the latter is very similar to the proof of Lemma 1.26. Let us prove the local triviality ofc in a neighborhood of a point u⁰ = (u⁰₁, . . . , u⁰_n) ∈ Fn. For each i = 1,2, . . . , n, pick an open neigh- borhoodU_i ⊂M^◦ ofu⁰_i such that its closureU_i is a closed ball with interior U_i. Since the points u⁰₁, . . . , u⁰_n are distinct, we can assume that U₁, . . . , U_n are mutually disjoint. ThenU =U₁×U₂× · · · ×U_n is a neighborhood ofu⁰ in Fn. We shall prove that the restriction ofc to U is a trivial bundle. For everyi= 1,2, . . . , n, there is a continuous map θ_i :U_i×U_i →U_i such that settingθ^u_i(v) =θi(u, v), we obtain a homeomorphismθ^u_i :Ui→Uithat sends u⁰_i to u and fixes the sphere ∂Ui pointwise (see the proof of Lemma 1.26).

For u = (u1, . . . , un) ∈ U, we define a homeomorphism θ^u : M → M by θ^u(v) =θ^u_iⁱ(v) ifv∈Uiwithi= 1,2, . . . , nandθ^u(v) =vifv∈M−

i Ui. It is clear thatθ^u:M →M sendsu⁰₁, . . . , u⁰_ntou1, . . . , un, respectively. Observe thatc⁻¹(u⁰) is the closed subgroup of Top(M) consisting of all f ∈Top(M) such that f(u⁰_i) = u⁰_i for i = 1,2, . . . , n. The formula (u, f) → θ^uf de- fines a homeomorphism U ×c⁻¹(u⁰) → c⁻¹(U) commuting with the pro- jections toU. The inverse homeomorphism sends any g∈c⁻¹(U) to the pair

(c(g),(θ^c(g))⁻¹g)∈U×c⁻¹(u⁰).

Remark 1.36.Two elements of Top(M) have the same image under the eval- uation map e if and only if they lie in the same left coset of Top(M, Q) in Top(M). Although we shall not need it, note thateinduces a homeomorphism from the quotient homogeneous space Top(M)/Top(M, Q) ontoCn.

1.7.2 Parametrizing isotopies

We show here that geometric braids naturally give rise to one-parameter fam- ilies of homeomorphisms of the 2-disk. This construction will be instrumental in the sequel.

Letn≥1 andD⊂R² be a closed Euclidean disk containing the set Q=Q_n ={(1,0),(2,0), . . . ,(n,0)} (1.16) in its interior. An isotopy{ft:D→D}t∈Iin the class of self-homeomorphisms ofD isnormal iff0(Q) =Qandf1= idD. In other words, a normal isotopy is a path in Top(D) leading from a point of Top(D, Q) to the identity homeo- morphism idD∈Top(D). For any normal isotopy{ft:D→D}t∈I, the set

t∈I

(ft(Q), t)⊂R²×I

is a geometric braid onnstrings. We say that the isotopy{ft}t∈I parametrizes this geometric braid.

Lemma 1.37.For any geometric braid b⊂ D^◦×I on n strings, there is a normal isotopy parametrizingb.

Proof. Consider the evaluation mape=eQ: Top(D)→ Cn=Cn(D^◦) sending f ∈Top(D) tof(Q). As already observed in Section 1.4.3, the braidb gives rise to a loopf^b:I→ Cn sending anyt∈Iinto the unique n-point subsetbt

ofR² such that b∩(R²× {t}) =bt× {t}. This loop begins and ends at the pointq=e(idD)∈ Cn represented byQ. By Lemma 1.35 and the homotopy lifting property of locally trivial fibrations (see Appendix B), the loopf^blifts to a pathf^b :I→Top(D) beginning at a point ofe⁻¹(q) = Top(D, Q) and ending at idD. The path f^b is a normal isotopy and the equality ef^b = f^b

means that this isotopy parametrizesb.

1.7.3 Proof of Theorem 1.33

LetD,Q=Qn, Cn =Cn(D^◦),e=eQ : Top(D)→ Cn, andq =e(idD)∈ Cn

be the same objects as in Section 1.7.2. By Lemma 1.35,eis a locally trivial fibration withe⁻¹(q) = Top(D, Q). This fibration induces a mapping

∂:π1(Cn, q)→π0(Top(D, Q)) =M(D, Q).

Recall the definition of∂ following Appendix B. Let β ∈π1(Cn, q) be repre- sented by a loopf :I→ Cnbeginning and ending atq. By the homotopy lifting property ofe, this loop lifts to a pathf:I→Top(D) beginning at a point of e⁻¹(q) = Top(D, Q) and ending at id_D. Then∂(β) = [f(0)]∈π₀(Top(D, Q)) is the homotopy class off(0). That ∂(β) depends only onβ can be seen di- rectly: iffis another loop representingβ, then a homotopy betweenf, flifts to a homotopy between arbitrary liftsf ,f in Top(D). This homotopy yields a path in Top(D, Q) connectingf(0) tof(0). Hence [f(0)] = [f(0)].

The mapping∂:π1(Cn, q)→M(D, Q) is a group homomorphism. Indeed, consider two loops f, g in Cn beginning and ending at q and representing β, γ ∈ π1(Cn, q), respectively. Let f ,g : I → Top(D) be lifts of f, g ending at idD. Observe that for anyt∈I,

e(f(t) g(0)) =f(t)g(0)(Q) =f(t)(Q) =f(t).

Therefore the path t → f(t)g(0) : I → Top(D) is a lift of f ending at f(1)g(0) = g(0). The product of this path with g is a lift of f g : I → Cn

ending at idD and beginning atf(0)g(0). Thus,

∂(βγ) = [f(0)g(0)] = [f(0)] [g(0)] =∂(β)∂(γ).

Recall that Bn = π1(Cn, q); see Exercise 1.4.4. The homomorphism

∂ : B_n = π₁(Cn, q) → M(D, Q) can be described in terms of parametriz- ing isotopies as follows. If b is a geometric braid representingβ ∈ B_n, then for any normal isotopy{f_t:D→D}t∈I parametrizingb as in Section 1.7.2,

∂(β)∈M(D, Q) is the isotopy class off₀: (D, Q)→(D, Q).

We claim that ∂ = η, where η : B_n → M(D, Q) is the homomorphism introduced in Section 1.6.3. It suffices to verify that∂ andη coincide on the generatorsσi, wherei= 1,2, . . . , n−1. Sinceη(σi) =τα_i, we need only check that ∂(σi) = τα_i. Let {gt : D → D}t∈I be the isotopy of the identity map g0 = id : D → D into g1 = τα_i obtained by rotating αi in D about its midpoint counterclockwise. Then

{ft=g1−t:D→D}t∈I

is an isotopy off0=ταintof1= id. It is easy to see that the geometric braid

t∈I

(ft(Q), t)⊂R²×I representsσi ∈Bn. Thus,∂(σi) = [f0] =ταi.

By the Alexander–Tietze theorem (Section 1.6.1), any point of the set Top(D, Q)⊂Top(D) can be connected to idD∈Top(D) by a path in Top(D).

This implies that the homomorphism

η =∂:π1(Cn, q)→π0(Top(D, Q)) =M(D, Q)

is surjective. The commutativity of the diagram (1.14) and Theorem 1.31 imply thatη is injective. Thereforeη is an isomorphism.

Remark 1.38.The proof of the Alexander–Tietze theorem in Section 1.6.1 actually shows that the point {idD} is a deformation retract of Top(D).

Therefore, π_i(Top(D)) = 0 for all i ≥ 0 and the homotopy sequence of the fibratione: Top(D)→ Cn(D^◦) directly implies that the homomorphism

∂:π1(Cn, q)→π0(Top(D, Q)) is an isomorphism.

1.7.4 Applications

We state two further applications of the techniques introduced above.

Theorem 1.39.For any geometric braid b on nstrings, the topological type of the pair(R²×I, b)depends only on n.

Proof. Pick a disk D ⊂ R² such that b ⊂ D^◦×I. Then the set Q = Qn

defined by (1.16) lies in D^◦. By Lemma 1.37, there is a normal isotopy {ft : D → D}t∈I parametrizing b. The formula (x, t) → (ft(x), t) defines a homeomorphismF : D×I → D×I mapping Q×I onto b and keeping

∂D×I pointwise. Extending F by the identity on (R²−D)×I, we obtain a homeomorphismR²×I →R²×I mapping Q×I ontob. Note that this homeomorphism is level-preserving in the sense that it commutes with the

projection toI.

Theorem 1.40.Every isotopy of a geometric braid in R²×I extends to an isotopy ofR²×I in itself constant on the boundary.

Proof. SetT =R²×I. Let b⊂T be a geometric braid onnstrings and let F :b×I→T be an isotopy ofb. Thus, for eachs∈I, the mapF_s :b→T sendingx ∈b to F(x, s) is an embedding whose image is a geometric braid and F₀ = id_b. We shall construct a (continuous) map G : T ×I → T such that for each s ∈ I, the map Gs : T → T sending x ∈ T to G(x, s) is a homeomorphism fixing∂T pointwise and extendingFs, andG0= idT.

Let Q ⊂ R² be the set {(1,0),(2,0), . . . ,(n,0)} and let D be a closed Euclidean disk in R² such that Q ⊂ D^◦ and F(b×I) ⊂ D^◦×I. For any s, t∈I, denote byf(s, t) the uniquen-point subset ofD^◦ such that

Fs(b)∩(D× {t}) =f(s, t)× {t}.

The formula (s, t) → f(s, t) defines a continuous map f : I² → Cn(D^◦).

Clearly,f(s,0) =f(s,1) =Qfor alls∈I andb=

t∈If(0, t)× {t}.

Consider the evaluation fibration e = e_Q : Top(D) → Cn(D^◦). By the homotopy lifting property ofe, the loopt→f(0, t) lifts to a patht→f(0, t) in Top(D) ending at idD and beginning at a point of Top(D, Q). By the homotopy lifting property of e with respect to the pair (I, ∂I), the latter path extends to a lift f: I² → Top(D) of f such that f(s,1) = idD and f(s,0) =f(0,0) for alls∈I.

We define a homeomorphism g(s, t) : R² → R² to be the identity on R²−Dand to bef(s, t)◦(f(0, t))⁻¹onD. It is clear thatg(s, t) is a continuous function ofs, t∈I and

g(0, t) =g(s,0) =g(s,1) = id for alls, t∈I. We have

g(s, t)(f(0, t)) =g(s, t)(f(0, t)(Q)) =f(s, t)(Q) =f(s, t).

It is now straightforward to check that the mapG:T×I→Tsending (a, t, s) to (g(s, t)(a), t) fora∈R²,s, t∈Ihas all the required properties.

Exercise 1.7.1.Let f be a self-homeomorphism of the 2-sphereS² fixing a pointa∈S²and isotopic to the identity id :S²→S². Prove thatf is isotopic to the identity in the class of self-homeomorphisms ofS²fixinga.

Solution. Applying Lemma 1.35 to M = S², Q = {a}, n = 1, we ob- tain a locally trivial fibration Top(S²) → S² with fiber Top(S²,{a}). Since π0(Top(S²,{a})) = M(S²,{a}) and π0(Top(S²)) = M(S²), this fibration yields an exact sequence

π1(S²)→M(S²,{a})→M(S²).

Sinceπ1(S²) = 0, the kernel of the homomorphismM(S²,{a})→M(S²) is trivial. This implies the required property of self-homeomorphisms ofS².