CONFIGURATION SPACES IN ALGEBRAIC TOPOLOGY: LECTURE 2 BEN KNUDSEN We begin our study of configuration spaces by observing a few of their basic properties. First, we note that, if f : X ! Y is an injective continuous map, we have the Σk-equivariant factorization k k f / k XO YO Confk(f) Confk(X) / Confk(Y ) and thus an induced map Bk(f): Bk(X) ! Bk(Y ). This construction respects composition and identities by inspection, so we have functors inj Σk Confk : Top ! Top inj Bk : Top ! Top; where Topinj denotes the category of topological spaces and injective continuous maps, TopΣk the category of Σk-spaces and equivariant maps, and Top the category of topological spaces. Recall that a continuous map is said to be an open embedding if it is both injective and open. Proposition. If f : X ! Y is an open embedding, then Confk(f) : Confk(X) ! Confk(Y ) and Bk(f): Bk(X) ! Bk(Y ) are also open embeddings. Proof. From the definition of the product topology, f k : Xk ! Y k is an open embedding, and the first claim follows. The second claim follows from the first and the fact that π : Confk(X) ! Bk(X) is a quotient map. This functor also respects a certain class of weak equivalences. Definition. An injective continuous map f : X ! Y is an isotopy equivalence if there is an 0 injective continuous map g : Y ! X and homotopies H : g ◦ f =) idX and H : f ◦ g =) idY 0 such that Ht and Ht are both injective for each t 2 [0; 1]. Proposition. If f : X ! Y is an isotopy equivalence, then Confk(f) : Confk(X) ! Confk(Y ) is a homotopy equivalence. Date: 1 September 2017. 1 2 BEN KNUDSEN Proof. In the solid commuting diagram k Xk × [0; 1]k ∼ (X × [0; 1])k H / Xk O 6 O H∆ idXk ×∆k Xk × [0; 1] O Confk(X) × [0; 1] / Confk(X); the diagonal composite is given by the formula ∆ Ht (x1; : : : ; xk) = (Ht(x1);:::;Ht(xk)): By assumption, Ht is injective for each t 2 [0; 1], so the dashed filler exists. By construction, this map restricts to Confk(g ◦ f) = Confk(g) ◦ Confk(f) at t = 0 and to idConfk(X) at t = 1. 0 Applying the same argument to Ht completes the proof. Corollary. If f : X ! Y is an isotopy equivalence then Bk(f): Bk(X) ! Bk(Y ) is a homotopy equivalence. Proof. The homotopies constructed in the proof of the previous corollary are homotopies through Σk-equivariant maps, so they descend to the unordered configuration space. Remark. Another point of view on the previous two results is provided by the fact (which we will not prove here) that Confk and Bk are enriched functors, where the space of injective continuous maps from X to Y is given the subspace topology induced by the compact-open topology on Map(X; Y ). Taking this fact for granted, these results follows immediately, since a homotopy through injective maps is simply a path in this mapping space. Example. If M is a manifold with boundary, then @M admits a collar neighborhood U =∼ t @M × (0; 1]. We define a map r : M ! M by setting rj@M×(0;1](x; t) = (x; 2 ) and extending by the identity. This map is injective, and dilation defines homotopies through injective maps from r ◦ i : M˚ ! M˚ and i ◦ r : M ! M to the respective identity maps. It follows that the induced map Confk(M˚) ! Confk(M) is a homotopy equivalence. These functors also interact well with the operation of disjoint union. Proposition. Let X and Y be topological spaces. The natural map a Confi(X) × Confj(Y ) ×Σi×Σj Σk ! Confk(X q Y ) i+j=k is a Σk-equivariant homeomorphism. In particular, the natural map a Bi(X) × Bj(Y ) ! Bk(X × Y ) i+j=k is a homeomorphism. CONFIGURATION SPACES IN ALGEBRAIC TOPOLOGY: LECTURE 2 3 Proof. From the definitions, the dashed filler exists in the commuting diagram a ' Xi × Y j × Σ / (X q Y )k Σi×Σj k O i+j=k O a / Confi(X) × Confj(Y ) ×Σi×Σj Σk Confk(X q Y ) i+j=k and is easily seen to be a bijection, which implies the first claim, since the vertical arrows are inclusions of subspaces. The second claim follows from the first after taking the quotient by the action of Σk. Thus, we may restrict attention to connected background spaces whenever it is convenient to do so. Our next goal is to come to grips with the local structure of configuration spaces. We assume from now on that X is locally path connected, and we fix a basis B for the topology of the space X consisting of connected subsets. We define two partially ordered sets as follows. ∼ k (1) We write Bk = fU ⊆ X : U = qi=1Ui;Ui 2 Bg, and we impose the order relation U ≤ V () U ⊆ V and π0(U ⊆ V ) is surjective: Σ ' (2) We write Bk = f(U; σ): U 2 Bk; σ : f1; : : : ; kg −! π0(U)g, and we impose the order relation (U; σ) ≤ (V; τ) () U ≤ V and τ = σ ◦ π0(U ⊆ V ): Σ Denoting the poset of open subsets of a space Y by O(Y ), there is an inclusion Bk ! O(Confk(X)) of posets defined by 0 U 7! Confk(U; σ) := (x1 : : : ; xk) 2 Confk(U): xi 2 Uσ(i) ⊆ Confk(X) and similarly an inclusion Bk ! O(Bk(X)) defined by 0 U 7! Bk(U) := ffx1; : : : ; xkg 2 Bk(U): fx1; : : : ; xkg \ Ui 6= ?; 1 ≤ i ≤ kg ⊆ Bk(X): Note that these subsets are in fact open, since U ⊆ X is open and configuration spaces respect open embeddings. ' Lemma. For any U 2 Bk and σ : f1; : : : ; kg −! π0(U), there are canonical homeomorphisms k 0 ∼ 0 ∼ Y Bk(U) = Confk(U; σ) = Uσ(i): i=1 Proof. It is easy to see from the definitions that the dashed fillers in the commuting diagram Xk o Conf (X) / B (X) O kO k O k Y 0 0 Uσ(i) o Confk(U; σ) / Bk(U) i=1 exist and are bijections. Since the lefthand map is the inclusion of a subspace and the righthand map is a quotient map, the claim follows. Proposition. Let X be a locally path connected Hausdorff space and B a topological basis for X consisting of connected subsets. 4 BEN KNUDSEN 0 Σ (1) The collection fConfk(U; σ):(U; σ) 2 Bk g ⊆ O(Confk(X)) is a topological basis. 0 (2) The collection fBk(U): U 2 Bkg ⊆ O(Bk(X)) is a topological basis. k Proof. Since X is Hausdorff, Confk(X) is open in X ; therefore, by the definition of the product and subspace topologies, it will suffice for the first claim to show that, given (x1; : : : ; xk) 2 V ⊆ Xk such that ∼ Qk • V = i=1 Vi for open subsets xi 2 Vi ⊆ X, and • V ⊆ Confk(X), Σ 0 there exists (U; σ) 2 Bk with (x1; : : : ; xk) 2 Confk(U; σ) ⊆ V . Now, since B is a topological basis, we may find Ui 2 B with xi 2 Ui ⊆ Vi. The second condition implies that the Vi are `k pairwise disjoint, so we may set U = i=1 Ui and take σ(i) = [Ui]. With these choices k k 0 ∼ Y Y (x1; : : : ; xk) 2 Confk(U; σ) = Ui ⊆ Vi = V; i=1 i=1 as desired. The second claim follows from first, the fact that π : Confk(X) ! Bk(X) is a quotient map, 0 0 and the fact that π(Confk(U; σ)) = Bk(U) for every σ. These and related bases will be important for our later study, when we come to hypercover methods. For now, we draw the following consequences. Corollary. Let X be a locally path connected Hausdorff space. The projection π : Confk(X) ! Bk(X) is a covering space. Proof. For U 2 Bk, we have Σk-equivariant identifications −1 0 [ 0 ∼ 0 π (Bk(U)) = Confk(U; σ) = Bk(U) × Σk; σ:f1;:::;kg∼=π0(U) where the second is induced by a choice of ordering of π0(U). Corollary. If M is an n-manifold, then Confk(M) and Bk(M) are nk-manifolds. Proof. We take B to be the set of Euclidean neighborhoods in M, in which case 0 ∼ 0 ∼ nk Bk(U) = Confk = R ' for any U 2 Bk and σ : f1; : : : ; kg −! π0(U). Exercise. When is Bk(M) orientable? From now on, unless specified otherwise, we take our background space to be a manifold M. In this case, we have access to a poweful tool relating configuration spaces of different cardinalities. The starting point is the observation that the natural projections from the product factor through the configuration spaces as in the following commuting diagram: ` Conf`(M) / M (x1; : : : ; x`) _ k Confk(M) / M (x1; : : : ; xk) (we take the projection to be on the last ` − k coordinates for simplicity, but it is not necessary to make this restriction). Clearly, the fiber over a configuration (x1; : : : ; xk) in the base is the configuration space Conf`−k(M n fx1; : : : ; xkg). Our first theorem asserts that the situation is in fact much better than this. CONFIGURATION SPACES IN ALGEBRAIC TOPOLOGY: LECTURE 2 5 Recollection. Recall that, if f : X ! Y is a continuous map, the mapping path space of f is the space of paths in Y out of the image of f. In other words, it is the pullback in the diagram [0;1] Ef / Y p _ f X / Y p(0): The inclusion X ! Ef given by the constant paths is a homotopy equivalence, and evaluation at 1 defines a map πf : Ef ! Y , which is a fibration [May99, 7.3].