Lecture 14: CW-Complexes

Chapter 2 Computational methods 14 CW-complexes There are various ways to model geometrically interesting spaces. Manifolds provide one important model, well suited to analysis. Another model, one we have not talked about, is given by simplicial complexes. It’s very combinatorial, and constructing a simplicial complex model for a given space involves making a lot of choices that are combinatorial rather than topological in character. A more flexible model, one more closely reflecting topological information, is given by the theory of CW-complexes. In building up a space as a CW-complex, we will successively “glue” cells onto what has been already built. This is a general construction. Suppose we have a pair (B, A), and a map f : A → X. Define a space X ∪f B (or X ∪A B) in the diagram A _  f /X  / X ∪f B B by X ∪f B = X t B/ ∼ where the equivalence relation is generated by requiring that a ∼ f (a) for all a ∈ A. We say that we have “attached B to X along f (or along A).” There are two kinds of equivalence classes in X ∪f B: (1) singletons containing elements of B − A, and (2) {x} t f −1 (x) for x ∈ X. The topology on X ∪f B is the quotient topology, and is characterized by a universal property: any solid-arrow commutative diagram A _  B f /X  j / X ∪f B g can be uniquely filled in. It’s a “push-out.” 35 j )#  Y 36 CHAPTER 2. COMPUTATIONAL METHODS Example 14.1. If X = ∗, then ∗ ∪f B = B/A. Example 14.2. If A = ∅, then X ∪f B is the coproduct X t B. Example 14.3. If both, B/∅ = ∗ ∪∅ B = ∗ t B . For example, ∅/∅ = ∗. This is creation from nothing. We won’t get into the religious ramifications. Example 14.4 (Attaching a cell). A basic collection of pairs of spaces is given by the disks relative to their boundaries: (Dn , S n−1 ). (Recall that S −1 = ∅.) In this context, Dn is called an “n-cell,” and a map f : S n−1 → X allows us to attach an n-cell to X, to form f S n−1 _  /X  / X ∪f D n Dn You might want to generalize this a little bit, and attach a bunch of n-cells all at once: ` f n−1 α∈A  S _ α `  /X / X ∪f n α∈A Dα ` n α∈A Dα What are some examples? When n = 0, (D0 , S −1 ) = (∗, ∅), so you are just adding a discrete set to X: a X ∪f D0 = X t A α∈A More interesting: Let’s attach two 1-cells to a point: S 0 t _ S 0  D1 t D1 f /∗  / ∗ ∪f (D 1 t D 1 ) Again there’s just one choice for f , and ∗ ∪f (D1 t D1 ) is a figure 8, because you start with two 1-disks and identify the four boundary points together. Let me write S 1 ∨ S 1 for this space. We can go on and attach a single 2-cell to manufacture a torus. Think of the figure 8 as the perimeter of a square with opposite sides identified. 14. CW-COMPLEXES 37 The inside of the square is a 2-cell, attached to the perimeter by a map I’ll denote by aba−1 b−1 : aba−1 b−1 S 1 _  / S1 ∨ S1  / (S 1 ∨ S 1 ) ∪f D 2 = T 2 . D2 This example illuminates the following definition. Definition 14.5. A CW-complex is a space X equipped with a sequence of subspaces ∅ = Sk−1 X ⊆ Sk0 X ⊆ Sk1 X ⊆ · · · ⊆ X such that • X is the union of the Skn X’s, and • for all n, there is a pushout diagram like this: ` n−1 α∈An _ Sα `  α∈An Dαn fn gn / Skn−1 X .  / Skn X The subspace Skn X is the n-skeleton of X. Sometimes it’s convenent to use the alternate notation Xn for the n-skeleton. The first condition is intended topologically, so that a subset of X is open if and only if its intersection with each Skn X is open; or, equivalently, a map f : X → Y is continuous if and only if its restriction to each Skn X is continuous. The maps fn are the attaching maps and the maps gn are characteristic maps. Example 14.6. We just constructed the torus as a CW complex with Sk0 T 2 = ∗, Sk1 T 2 = S 1 ∨ S 1 , and Sk2 T 2 = T 2 . Definition 14.7. A CW-complex is finite-dimensional if Skn X = X for some n; of finite type if each An is finite, i.e., finitely many cell in each dimension; and finite if it’s finite-dimensional and of finite type. The dimension of a CW complex is the largest n for which there are n-cells. This is not obviously a topological invariant, but, have no fear, it turns out that it is. In “CW,” the “C” is for cell, and the “W” is for weak, because of the topology on a CW-complex. This definition is due to J. H. C. Whitehead. Here are a couple of important facts about them. Theorem 14.8. Any CW-complex is Hausdorff, and it’s compact if and only if it’s finite. Any compact smooth manifold admits a CW structure. Proof. See [2] Prop. IV.8.1, [6] Prop. A.3. Bibliography [1] M. G. Barratt and J. Milnor, An example of anomalous singular homology, Proc. Amer. Math. Soc. 13 (1962) 293–297. 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