BEN KNUDSEN. Conf k (f) Conf k (Y )

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1 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 Σ k -equivariant factorization X k f k Y k Conf k (X) Conf k (f) Conf k (Y ) and thus an induced map B k (f) : B k (X) B k (Y ). This construction respects composition and identities by inspection, so we have functors Conf k : Top inj Top Σ k B k : Top inj Top, where Top inj denotes the category of topological spaces and injective continuous maps, Top Σ k the category of Σ k -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 Conf k (f) : Conf k (X) Conf k (Y ) and B k (f) : B k (X) B k (Y ) are also open embeddings. Proof. From the definition of the product topology, f k : X k Y k is an open embedding, and the first claim follows. The second claim follows from the first and the fact that π : Conf k (X) B k (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 injective continuous map g : Y X and homotopies H : g f = id X and H : f g = id Y such that H t and H t are both injective for each t [0, 1]. Proposition. If f : X Y is an isotopy equivalence, then Conf k (f) : Conf k (X) Conf k (Y ) is a homotopy equivalence. Date: 1 September

2 2 BEN KNUDSEN Proof. In the solid commuting diagram X k [0, 1] k (X [0, 1]) k H k X k id X k k H X k [0, 1] Conf k (X) [0, 1] Conf k (X), the diagonal composite is given by the formula H t (x 1,..., x k ) = (H t (x 1 ),..., H t (x k )). By assumption, H t is injective for each t [0, 1], so the dashed filler exists. By construction, this map restricts to Conf k (g f) = Conf k (g) Conf k (f) at t = 0 and to id Confk (X) at t = 1. Applying the same argument to H t completes the proof. Corollary. If f : X Y is an isotopy equivalence then B k (f) : B k (X) B k (Y ) is a homotopy equivalence. Proof. The homotopies constructed in the proof of the previous corollary are homotopies through Σ k -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 Conf k and B k 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 = M (0, 1]. We define a map r : M M by setting r M (0,1] (x, t) = (x, t 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 Conf k ( M) Conf k (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 Conf i (X) Conf j (Y ) Σi Σ j Σ k Conf k (X Y ) is a Σ k -equivariant homeomorphism. In particular, the natural map B i (X) B j (Y ) B k (X Y ) is a homeomorphism.

3 CONFIGURATION SPACES IN ALGEBRAIC TOPOLOGY: LECTURE 2 3 Proof. From the definitions, the dashed filler exists in the commuting diagram X i Y j Σi Σ j Σ k (X Y ) k Conf i (X) Conf j (Y ) Σi Σ j Σ k Conf k (X Y ) 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 Σ k. 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. (1) We write B k = {U X : U = k U i, U i B}, and we impose the order relation U V U V and π 0 (U V ) is surjective. (2) We write B Σ k = {(U, σ) : U B k, σ : {1,..., k} π 0 (U)}, and we impose the order relation (U, σ) (V, τ) U V and τ = σ π 0 (U V ). Denoting the poset of open subsets of a space Y by O(Y ), there is an inclusion B Σ k O(Conf k(x)) of posets defined by U Conf 0 k(u, σ) := { (x 1..., x k ) Conf k (U) : x i U σ(i) } Confk (X) and similarly an inclusion B k O(B k (X)) defined by U B 0 k(u) := {{x 1,..., x k } B k (U) : {x 1,..., x k } U i, 1 i k} B k (X). Note that these subsets are in fact open, since U X is open and configuration spaces respect open embeddings. Lemma. For any U B k and σ : {1,..., k} π 0 (U), there are canonical homeomorphisms Bk(U) 0 = Conf 0 k(u, σ) k = U σ(i). Proof. It is easy to see from the definitions that the dashed fillers in the commuting diagram X k Conf k (X) B k (X) k U σ(i) Conf 0 k(u, σ) B 0 k (U) 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 4 BEN KNUDSEN (1) The collection {Conf 0 k(u, σ) : (U, σ) B Σ k } O(Conf k(x)) is a topological basis. (2) The collection {B 0 k (U) : U B k} O(B k (X)) is a topological basis. Proof. Since X is Hausdorff, Conf k (X) is open in X k ; therefore, by the definition of the product and subspace topologies, it will suffice for the first claim to show that, given (x 1,..., x k ) V X k such that V = k V i for open subsets x i V i X, and V Conf k (X), there exists (U, σ) B Σ k with (x 1,..., x k ) Conf 0 k(u, σ) V. Now, since B is a topological basis, we may find U i B with x i U i V i. The second condition implies that the V i are pairwise disjoint, so we may set U = k U i and take σ(i) = [U i ]. With these choices (x 1,..., x k ) Conf 0 k(u, σ) k k = U i V i = V, as desired. The second claim follows from first, the fact that π : Conf k (X) B k (X) is a quotient map, and the fact that π(conf 0 k(u, σ)) = Bk 0 (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 : Conf k (X) B k (X) is a covering space. Proof. For U B k, we have Σ k -equivariant identifications π 1 (Bk(U)) 0 = Conf 0 k(u, σ) = Bk(U) 0 Σ k, σ:{1,...,k} =π0(u) where the second is induced by a choice of ordering of π 0 (U). Corollary. If M is an n-manifold, then Conf k (M) and B k (M) are nk-manifolds. Proof. We take B to be the set of Euclidean neighborhoods in M, in which case for any U B k and σ : {1,..., k} π 0 (U). Exercise. When is B k (M) orientable? B 0 k(u) = Conf 0 k = R nk 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 l (M) M l (x 1,..., x l ) Conf k (M) M k (x 1,..., x k ) (we take the projection to be on the last l k coordinates for simplicity, but it is not necessary to make this restriction). Clearly, the fiber over a configuration (x 1,..., x k ) in the base is the configuration space Conf l k (M \ {x 1,..., x k }). Our first theorem asserts that the situation is in fact much better than this.

5 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 E f Y [0,1] p X f Y The inclusion X E f given by the constant paths is a homotopy equivalence, and evaluation at 1 defines a map π f : E f Y, which is a fibration [May99, 7.3]. The homotopy fiber of f at the basepoint y Y is the fiber hofib y (f) := π 1 f (y) of this fibration. The construction of E f, and hence the homotopy fiber, is functorial, and we say that a diagram X X f Y Y is homotopy Cartesian, or a homotopy pullback square, if the induced map on homotopy fibers is a weak equivalence for all choices of basepoint. The primary benefit of knowing that a square is homotopy Cartesian is the induced Mayer-Vietoris long exact sequence in homotopy groups. f p(0). Theorem (Fadell-Neuwirth). Let M be a manifold and 0 k l <. The diagram Conf l k (M \ {x 1,..., x k }) Conf l (M) is homotopy Cartesian. (x 1,..., x k ) Conf k (M) Remark. In fact, Fadell-Neuwirth [FN62] prove that this map is a locally trivial fiber bundle, and they give an identification of its structure group. We will not need this full statement, and our alternate proof of this weaker form will allow us to illustrate the efficacy of hypercover methods at a later point in the course. The proof is a debt that we will return to pay after having developed a few more advanced homotopy theoretic techniques. For the time being, we concentrate on exploiting this result. Corollary. If M is a simply connected n-manifold with n 3, then Conf k (M) is simply connected for every k 0. In particular, π 1 (B k (M)) = Σ k. Proof. The case k = 0 is trivial and the case k = 1 is our assumption. The Van Kampen theorem and our assumption on n imply that M \ {pt} is simply connected, so the first claim follows by induction using the exact sequence π 1 (M \ {pt}) π 1 (Conf k (M)) π 1 (Conf k 1 (M)). The second claim follows from the observation that Conf k (M) B k (M) is a Σ k -cover with simply connected total space and hence the universal cover. Corollary. If M is a connected surface of finite type different from S 2 or RP 2, then Conf k (M) is aspherical for every k 0. In particular, B k (M) is aspherical.

6 6 BEN KNUDSEN Proof. The case k = 0 is obvious and the case k = 1 follows from our assumption on M. This assumption further guarantees that M \{x 1,..., x k 1 } is also aspherical, so the first claim follows by induction using the exact sequence π i (M \ {x 1,..., x k 1 }) π i (Conf k (M)) π i (Conf k 1 (M)) with i 2. The second claim follows from the first and the fact that π : Conf k (M) B k (M) is a covering space. In order to proceed further, it will be useful to have a criterion for splitting these exact sequences. Proposition. If M is the interior of a manifold with non-empty boundary, then the map π k,l : Conf l (M) Conf k (M) admits a section up to homotopy. Proof. Write M = N, and fix a collar neighborhood N U and an ordered set {x k+1,, x l } of distinct points in U. By retracting along the collar, we obtain an embedding ϕ : M M that is isotopic to the identity and misses our chosen points. Then the assignment (x 1,..., x k ) (ϕ(x 1 ),..., ϕ(x k ), x k+1,, x l ) defines a continuous map s : Conf k (M) Conf l (M) such that π k,l s = Conf k (ϕ) id Confk (M), since ϕ is isotopic to the identity. Corollary. For n 3, k 0, and i 0, there is an isomorphism π i (Conf k (R n )) = k 1 j=1 π i j S n 1 Proof. For k {0, 1} the claim is obvious, as is the claim for π 0, and the claim for π 1 has already been established. In the generic case, we proceed by induction using the exact sequence π i+1 (Conf k 1 (R n )) π i (R n \ {x 1,..., x k 1 }) π i (Conf k (R n )) π i (Conf k 1 (R n )). The section up to homotopy constructed above induces a section at the level of homotopy groups, so the lefthand map is trivial and the righthand map is surjective. The result now follows from the homotopy equivalence R n \ {x 1,..., x k 1 } k 1 Sn 1 and the fact that all groups in sight are Abelian. The higher homotopy groups of bouquets of spheres being very complicated objects [Hil55], this result is a striking contrast to the situation in dimension 2. Remark. It should be noted that the product decomposition of the previous corollary is additive only. Viewed as shifted Lie algebra via the Whitehead bracket, π (Conf k (R n )) has a very rich structure see [FH12, II]. The following result is proved by essentially the same argument. Corollary. The fundamental group of Conf k (R 2 ) is an iterated extension of free groups. References [FH12] E. Fadell and S. Husseini, Geometry and topology of configuration spaces, Springer Monographs in Mathematics, Springer Science and Business Media, [FN62] E. Fadell and L. Neuwirth, Configuration spaces, Math. Scand. 10 (1962), [Hil55] P. J. Hilton, On the homotopy groups of the union of spheres, J. London Math. Soc. s1-30 (1955), [May99] J. P. May, A concise course in algebraic topology, Chicago Lectures in Mathematics, The University of Chicago Press,