Lecture notes for Algebraic Topology 11 J A S, S-11 1 CW-complexes There are two slightly different (but of course equivalent) definitions of a CW-complex. Basically, one version is suitable when you have a given space and want to provide it with a CW-structure, the other one is better when you want to construct a space (with structure). It's fair to say that the concept of a CW-complex is what you need to do nice inductive arguments in algebraic topology. Definition 1.1 A CW-complex is a Hausdorff space X with a partition into disjoint subset, called (open) cells, typically denoted eα where α runs trough some indexing set (usually not denoted by anything). The requirements on this partition are as follows: n 1. For each cell eα there is a map (not part of the structure) Φα : D ! X; such that n n−1 Φα j : D − S ! eα is a bijection. n−1 Here Φα is said to be a characteristic map for the cell eα. The restriction φα of Φα to S is an attaching map for the cell. n n n Since D is compact and X Hausdorff Φα(D ) is closed, soe ¯α ⊂ Φα(D ). On the other hand, by n n n−1 n n−1 n continuity, Φα(D ) = Φα(D − S ) ⊂ Φα(D − S ) =e ¯α; so Φα(D ) =e ¯α: n −1 n The map Φα : D ! e¯α is a quotient map (F ⊂ e¯α is closed iff Φα (F ) is closed) since D is compact n n−1 ande ¯α is Hausdorff. As a consequence of this, not that the bijection Φα j : D − S ! eα is in fact a homeomorphism. Thus integer n ≥ 0; known as the dimension of the cell, dim eα, is uniquely determined since Dn − Sn−1 is homeomorphic to Dm − Sm−1 only if m = n; being open subsets of Euclidean spaces. n If eα is known to have dimension n it's somtimes indicated by eα. n n n 2. For each cell eα the set e¯α − eα is contained in a finite union of cells of dimension < n. This property is called closure finiteness and accounts for the C in CW. 3. A subset F of X is closed iff F \ e¯α is closed for each cell eα of X. This is expressed by saying that X has the weak topology with respect to the closures of the cells and explains the W in CW. Note that if X is a finite CW-complex (i.e there are only finitely many cells in the partition) property 2. n n reduces to:e ¯α − eα is contained in a union of cells of dimension < n and property 3. is automatically satisfied. 1. Having made a choice of characteristic map Φα for each cell, note thate ¯α \ F is closed in e¯α −1 −1 n exactly when Φα (¯eα \ F ) = Φα (F ) is closed in Dα; since Φα is a quotient map. Thus, if Φ is the ` n amalgamation of all Φα to a (surjective) map Φ : α Dα ! X; then the condition in 3 is equivalent to Φ−1(F ) being closed. Hence, property 3. is equivalent to Φ being a quotient map. Example 1.1 We give Sn a CW-structure by partitioning it inductively. Assume Sn−1 is partitioned, then partition the rest of Sn into the upper and lower (open) hemispheres. Characteristic maps for these 1 n n two additional cells canp be taken to be the maps from D to the closed hemispheres of S ; mapping tx; where jxj = 1; to (tx; ± 1 − t2). In this CW-structure there are two cells of each dimension ≤ n. 1 S n 1 n Defining S = n S and giving it a topology by defining F ⊂ S to be closed iff F \ S is closed (in Sn) for each n ≥ 0 we have a CW-structure on S1: The only thing that can be questioned here is the 1 n n n n fact that S is Hausdorff. To see this, assume that x 6= y 2 S are separated by Ox and Oy in S then n+1 n+1 n+1 n n (exercises!) there are open disjoint set Ox and Oy in S whose intersection with S are Ox and n S i S i 1 Oy ; respectively. Then Ox = i≥n Ox and Oy = i≥n Oy are open in S and separates x and y: n Another CW-structure on S is given by the partition consisting of fe1g (e1 being the first vector in the n+1 n standard basis of R ) and S − fe1g. As a characteristic map for the large cell we can take a quotient n n p n−1 map D ! S ; tx 7! (2t − 1; 2 t(1 − t)x) (where jxj = 1), identifying S to e1. In this CW-structure there is one cell of dimension 0 and one in dimension n (and no others). A CW-complex is of dimension n if all cells have dimension ≤ n and there is at least one cell of dimension n. It is of finite type if there are only finitely many cells of each dimension. A subcomplex A of a CW-complex X is a union of cells e in X such that e ⊂ A ) e¯ ⊂ A. In the first CW-structure on Sn above Sn−1 is a subcomplex of Sn and Sn is a subcomplex of S1. For any CW-complex X the n-skeleton Xn consisting of all cells of dimension ≤ n is a subcomplex (by property 2.) Any union and intersection of subcomplexes is a subcomplex. 2. Any cell e of X is contained in a finite subcomplex K(e). This is proved by induction on the dimension of the cell. A 0-dimensional cell is just a point and a subcomplex. Assuming the result for cells of dimension < n. Let en be a cell of dimension n. Thene ¯n −en is contained in a finite union of cells of dimension < n. Each of these is contained in a finite subcomplex. The union of these is a finite subcomplex containinge ¯n − en. Adjoining the cell en results in a finite subcomplex containing en. A finite subcomplex B of X is obviously closed in X being a finite union of the setse ¯ for e ⊂ B. In fact: 3. Any subcomplex A of X is closed in X. To see this, let e be a cell of X and K(e) a finite subcomplex of X containing e. Thene ¯ \ A = e¯ \ K(e) \ A. Heree ¯ is closed and so is K(e) \ A; being a finite subcomplex. 4. A subcomplex A of X is a CW-complex. In the definition of a CW-complex 1. and 2. are immediate. Suppose F ⊂ A hase ¯α \ F closed for each cell eα in A and let e be a cell of X contained in the finite subcomplex K(e). Then F \ e¯ = F \ e¯ \ K(e) \ A. Here K(e) \ A ⊂ A is a finite subcomplex of X and hence closed in A. By assumption on F the set F \ K(e) \ A is closed, hence so is F \ e¯ = F \ e¯ \ K(e) \ A. Any finite subcomplex of X is a compact subspace, since it is a finite union of compact setse ¯. In fact: 4. Any compact subset K of X is contained in a finite subcomplex. To see this, choose a point xα 2 K \ eα for each cell such that this intersection is non-empty. Let S be the subspace of all points so chosen and T ⊂ S. Then for any cell e we have T \e¯ = T \K(e)\e;¯ where K(e) is a finite subcomplex containing e. Then T \K(e) is finite and hence closed. Thus any subset of S is closed and S a discrete closed subspace of K and consequently a discrete compact space. It follows that S is finite and K is contained in a finite union of cells and hence in a finite subcomplex. Thus any CW-complex is a union of an ascending sequence of closed subspaces (the skeletons): X0 ⊂ X1 ⊂ · · · ⊂ Xn ⊂ · · · ⊂ X; where X0 is a discret space. 2 5. (a) A subset A ⊂ X is closed in X iff A \ Xn is closed in Xn for all n ≥ 0: n n n−1 n−1 n −1 (b) A subset A ⊂ X is closed in X iff A \ X is closed in X and (Φα) (A) for a choice n n of characteristic maps for all n-cells eα in X . n n ` n n n n (c) If Φ : D = α Dα ! X ; denotes the amalgamation of the Φα :s, then a subset A ⊂ X is closed in Xn iff A \ Xn−1 is closed in Xn−1 and (Φn)−1(A) is closed in Dn For (a) only () needs attention. For this just note that for any cell e of X; there's an n such that e¯ ⊂ Xn ande ¯ \ A =e ¯ \ (A \ Xn) is closed in X; since Xn is closed in X. For (b) again only () needs attention. If e is a cell of Xn of dimension < n; thene ¯ ⊂ Xn−1 and n−1 n n−1 n n n e¯ \ A =e ¯ \ (A \ X ) is closed in X ; since X is closed in X . If eα is an n-cell of X ; then n n −1 n n −1 e¯α \ A is closed iff (Φa ) (eα \ A) = (Φα) (A) is closed (by item 1).