Configuration spaces of ordered sets of points Let M be a topological space and let

Braids and Braid Groups

1.4 Configuration spaces

1.4.1 Configuration spaces of ordered sets of points Let M be a topological space and let

Mⁿ =M×M× · · · ×M

be the product ofn≥1 copies ofM with the product topology. Set Fn(M) ={(u1, u2, . . . , un)∈Mⁿ|ui=uj for alli=j}.

This subspace ofMⁿ is called the configuration space of orderedn-tuples of (distinct) points inM.

If M is a topological manifold (possibly with boundary ∂M), then the configuration spaceFn(M) is a topological manifold of dimensionndim(M).

Clearly, any orderedn-tuple of points inM can be deformed into an ordered n-tuple of points in the interiorM^◦=M−∂M ofM. If dim(M)≥2 andM is connected, then any orderedn-tuple of points inM^◦ can be deformed into any other such tuple. Therefore for suchM, the manifoldFn(M) is connected.

Its fundamental group is called thepure braid groupofM onnstrings.

ForM =R², we recover the same pure braid group Pn as above. To see this, assign to a pure geometric braidb⊂R²×Ithe pathI→ Fn(R²) sending t∈I into the tuple (u1(t), u2(t), . . . , un(t)) defined by the condition that the ith string ofbmeetsR²× {t}at the point (u_i(t), t) for alli= 1,2, . . . , n. This path begins and ends at then-tuple

qn = ((1,0),(2,0), . . . ,(n,0))∈ Fn(R²).

Conversely, any path (α1, α2, . . . , αn) :I→ Fn(R²) beginning and ending atqn gives rise to the pure geometric braid

n i=1

t∈I

(αi(t), t).

These constructions are mutually inverse and yield a bijective correspondence between pure geometric braids and loops in (Fn(R²), qn). Under this corre- spondence the isotopy of braids corresponds to the homotopy of loops. Thus, Pn = π1(Fn(R²), qn). The braid group Bn admits a similar interpretation, which will be discussed in Section 1.4.3.

Coming back to an arbitrary connected topological manifoldM of dimen- sion ≥ 2, it is useful to generalize the definition of Fn(M) by prohibiting several points inM^◦=M−∂M. More precisely, fix a finite setQm⊂M^◦ of m≥0 points and set

Fm,n(M) =Fn(M −Qm).

The topological type of this space depends onM, m, andn, but not on the choice ofQm. Clearly,F0,n(M) =Fn(M) andFm,1(M) =M −Qm.

To describe the relationship between various configuration spaces, we need the notion of a locally trivial fibration. For the convenience of the reader, we recall this notion in Appendix B.

Lemma 1.26.Let M be a connected topological manifold of dimension ≥2 with ∂M = ∅. For n > r ≥ 1, the forgetting map p : Fn(M) → Fr(M) defined by p(u1, . . . , un) = (u1, . . . , ur) is a locally trivial fibration with fiberFr,n−r(M).

Proof. Pick a pointu⁰= (u⁰₁, . . . , u⁰_r)∈ Fr(M). The fiberp⁻¹(u⁰) consists of the tuples (u⁰₁, . . . , u⁰_r, v1, . . . , vn−r) ∈ M^r, where all u⁰₁, . . . , u⁰_r, v1, . . . , vn−r

are distinct. SettingQ_r={u⁰₁, . . . , u⁰_r}, we obtain

Fr,n−r(M) ={(v1, . . . , vn−r)∈(M−Qr)^n−r|vi=vj fori=j}. It is obvious that the formula (u⁰₁, . . . , u⁰_r, v1, . . . , vn−r)→(v1, . . . , vn−r) de- fines a homeomorphismp⁻¹(u⁰)≈ Fr,n−r(M).

We shall prove the local triviality of pin a neighborhood ofu⁰. For each i = 1,2, . . . , r, pick an open neighborhood Ui ⊂M of u⁰_i such that its clo- sureUiis a closed ball with interiorUi. Since the pointsu⁰₁, . . . , u⁰_rare distinct, we may assume thatU1, . . . , Ur are mutually disjoint. Then

U =U1×U2× · · · ×Ur

is a neighborhood ofu⁰inFr(M). We shall see that the restriction ofptoU is a trivial bundle, i.e., that there is a homeomorphismp⁻¹(U)→U×Fr,n−r(M) commuting with the projections toU.

We construct below for eachi= 1,2, . . . , ra continuous map θ_i:U_i×U_i→U_i

such that for everyu∈U_i, the mapθ_i^u:U_i→U_i sendingv ∈U_i toθ_i(u, v) is a homeomorphism sending u⁰_i to u and fixing the boundary sphere ∂U_i pointwise. Foru= (u1, . . . , ur)∈U, define a mapθ^u:M →M by

θ^u(v) =

θi(ui, v) ifv∈Uifor some i= 1,2, . . . , r,

v ifv∈M −

i Ui.

It is clear thatθ^u:M →M is a homeomorphism continuously depending onu and sending the pointsu⁰₁, . . . , u⁰_rtou1, . . . , ur, respectively. The formula

(u, v1, . . . , vn−r)→(u, θ^u(v1), . . . , θ^u(vn−r))

defines a homeomorphism U × Fr,n−r(M) → p⁻¹(U) commuting with the projections toU. The inverse homeomorphism is defined by

(u, v1, . . . , vn−r)→(u,(θ^u)⁻¹(v1), . . . ,(θ^u)⁻¹(vn−r)). Thus,p|U :p⁻¹(U)→U is a trivial fibration.

To constructθi, we may assume thatUi =U is the open unit ball in Eu- clidean spaceR^dimM with center at the originui= 0. Fix a smooth function of two variables λ : [0,1[×[0,1] → R such that λ(x, y) = 1 if x ≥ y and λ(x, y) = 0 if (x+ 1)/2≤y, wherex∈[0,1[ andy∈[0,1]. Foru∈U, define a vector fieldf^u on the closed unit ballU ={v∈R^dimM| v ≤1}by

f^u(v) =λ(u,v)u .

The choice of λensures that f^u = u on the ball of radiusu with center at the origin and f^u = 0 outside the ball of radius (u+ 1)/2 with center at the origin. Let {θ^u,t : U → U}t∈R be the flow determined by f^u, that is, the (unique) family of self-diffeomorphisms ofU such thatθ^u,0 = id and dθ^u,t(v)/dt=f^u(v) for allv ∈U, t ∈R. The diffeomorphismθ^u,t smoothly depends on u, t, fixes the sphere ∂U pointwise, and sends the origin to tu.

Therefore the mapθ_i :U ×U →U defined by θ_i(u, v) =θ^u,1(v) for u∈ U,

v∈U satisfies all the required conditions.

Lemma 1.27.Let M be a connected topological manifold of dimension ≥2 with∂M =∅. For anym≥0,n > r≥1, the forgetting map

p:Fm,n(M)→ Fm,r(M)

defined by p(u₁, . . . , u_n) = (u₁, . . . , u_r) is a locally trivial fibration with fiberFm+r,n−r(M).

Proof. This lemma is obtained by applying Lemma 1.26 toM −Qm.

Recall that a connected manifoldM isasphericalif its universal covering is contractible or, equivalently, if its homotopy groupsπi(M) vanish for alli≥2.

Lemma 1.28.For any m≥0, n≥1, the manifoldFm,n(R²)is aspherical.

Proof. Consider the fibration Fm,n(R²)→ Fm,1(R²) =R²−Q_m with fiber Fm+1,n−1(R²) defined above. The homotopy sequence of this fibration gives an exact sequence

· · · −→πi+1(R²−Qm)−→πi(Fm+1,n−1(R²))

−→π_i(Fm,n(R²))−→π_i(R²−Q_m)−→ · · ·. Observe thatR²−Qmcontains a wedge ofmcircles as a deformation retract.

A wedge of circles is aspherical since its universal covering is a tree and hence is contractible. ThereforeR²−Qmis aspherical, so thatπi(R²−Qm) = 0 for i≥2. We conclude that for alli≥2,

πi(Fm,n(R²))∼=πi(Fm+1,n−1(R²)). An inductive argument shows for alli≥2,

πi(Fm,n(R²))∼=πi(Fm+n−1,1(R²))∼=πi(R²−Qm+n−1) = 0. 1.4.2 Proof of Theorem 1.16

Applying Lemma 1.26 to M = R², we obtain a locally trivial fibration p : Fn(R²) → Fn−1(R²) with fiber Fn−1,1(R²). This gives a short exact sequence

1−→π1(Fn−1,1(R²))−→π1(Fn(R²))−→^p# π1(Fn−1(R²))−→1, (1.9) where we use the triviality ofπ₂(Fn−1(R²)) (by Lemma 1.28) and the triviality ofπ₀(Fn−1,1(R²)) (sinceFn−1,1(R²) is connected).

Under the isomorphismsπ1(Fn(R²))∼=Pnandπ1(Fn−1(R²))∼=Pn−1, the homomorphismp# in (1.9) is identified with the forgetting homomorphism fn:Pn→Pn−1of Section 1.3.2. We can rewrite (1.9) as

1−→π1(Fn−1,1(R²))−→Pn fn

−→Pn−1−→1. (1.10) To computeπ1(Fn−1,1(R²)) = π1(R²−Qn−1), we take as Qn−1 ⊂R² the set {(1,0),(2,0), . . . ,(n−1,0)} and take a0 = (n,0) as the base point of R²−Qn−1. Clearly, the groupπ1(R²−Qn−1, a0) is a free group of rankn−1 with the free generatorsx1, . . . , xn−1, shown in Figure 1.13.

The homomorphismπ1(Fn−1,1(R²))→Pn =π1(Fn(R²)) in (1.10) is in- duced by the inclusionR²−Qn−1 =Fn−1,1(R²)→ Fn(R²) assigning to a pointa∈R²−Qn−1the tuple ofnpoints ((1,0),(2,0), . . . ,(n−1,0), a). Com- paring Figures 1.10 and 1.13, we observe that this homomorphism sendsxi

toAi,n for all i. Now the exact sequence (1.10) directly implies the claim of

Theorem 1.16.

y

x · · · · · ·

(1,0) (i,0) (n−1,0)

a₀= (n,0)

x₁

x_i

xn−1

Fig. 1.13. The generatorsx1, . . . , xn−1 ofπ1(R²−Qn−1, a0)