Lecture notes for Algebraic 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 .

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 | : 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 | : D − S → eα is in fact a .

Thus integer n ≥ 0, known as the 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 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 can√ be taken to be the maps from D to the closed hemispheres of S , mapping tx, where |x| = 1, to (tx, ± 1 − t2). In this CW-structure there are two cells of each dimension ≤ n. ∞ S n ∞ 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 S∞. The only thing that can be questioned here is the ∞ n n n n fact that S is Hausdorff. To see this, assume that x 6= y ∈ 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 ∞ 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 {e1} (e1 being the first vector in the n+1 n standard basis of R ) and S − {e1}. 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 |x| = 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 S∞. 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 ∈ 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 . 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). n −1 n n −1 n Now (c) follow from (b) since (Φ ) (A) is closed in D iff (Φα) (A) is closed in Dα, for all α. 6. A map f : X → Y is continuous iff f| :e ¯ → Y is continuous for each cell e of X. If Φ : Dn → e¯ is a characteristic map for e then f| is continuous iff f| ◦ Φ: Dn → Y is continuous. The first follows immediately from the fact that f −1(F ) ∩ e¯ = (f|)−1(F ) and that X has the weak topology with respect to the closure of its cells. The second is a consequence of Φ being a quotient map. 7. Note that a cell ek of Xk is an open subset of Xk. Indeed, the union of all othere cells of Xk is a subcomplex and hence the complement of ek is closed.

k k k Choose a characteristic map Φα : D → e¯α for each cell of dimension k in X . They amalgamate to a ` k k ` k k−1 k k−1 map Φ : α Dα → X which restricts to a homeomorphism Φ| : α(Dα − Sα ) → X − X . Let Xk → Xk/Xk−1 be the quotient map collapsing Xk−1 to a point. Then the composite

a k Φ k q k k−1 q¯ : Dα → X → X /X α is a closed map. (The map Φ is not closed in general if the number of cells of dimension k is infinite.) ` k Indeed, if F ⊂ α Dα is closed then

 ` k−1 −1 Φ(F ) if F ∩ ( α Sα ) = ∅ q¯ (¯q(F )) = k−1 ` k−1 Φ(F ) ∪ X if F ∩ ( α Sα ) 6= ∅.

k and intersecting this with closures of cells of X and using that each Φα is a quotient map one sees that both alternatives gives closed sets in Xk, soq ¯(F ) is closed since q is a quotient map. In particularq ¯ is a ` k−1 quotient map collapsing Sα to a single point, i.e.

` Dk ∼ Xk α α →= ` k−1 Xk−1 α Sα

On the other hand choose a quoutient map Dk → Sk collapsing Sk−1 to a point p ∈ Sk. To be specific we could take tx 7→ (2pt(1 − t)x, 1 − 2t). Taking disjoint union of spaces we get a qoutient map ` k ` k k−1 α Dα → α Sα and collapsing the subspace consisting of all the points pα ∈ Sα to a point a quotient ` k W k ` k−1 map α Dα → α Sα collapsing α Sα to a singel point. It follows that ` k k _ ∼= D ∼= X Sk ← α α → . α ` k−1 Xk−1 α α Sα

We record this as

8. Xk/Xk−1 is (homeomorphic to) a wedge of spheres of dimension k in one-to-one correspondens with the cells of dimension k. Note that once a choice of characteristic maps for the n-cells is given, we have a specific homeo- morphism, if we agree to choose the homeomorphism Dn/Sn−1 → Sn as above.

3 The following next observation is much used in (algebraic) topology without explicitly mentioning it, especially in case L = I = [0, 1]. The proof is not completely obvious even in this simple case, though. There are easy counter examples when L is not locally compact, e.g. L = Q. What’s alluded to is

9. Suppose q : X → Y is a quotient map (so in particular q is surjective) and L is a (any neighborhood of a point contains a compact neighborhood of the point). Then q × 1 : X × L → Y × L is a quotient map, too. −1 To see this, let (y, p) ∈ O ⊂ Y × L and suppose U = (q × 1) (O) is open in X × L. Pick an x0

with q(x0) = y. Then (x0, p) ∈ U so there are neighborhoods Ux0 and Vp of x0 in X and p in L

such that Ux0 × Vp ⊂ U, since U is open. We can assume that Vp is compact. Now consider the x0 set U = {x ∈ X | {x} × Vp ⊂ U}. This set is saturated with respect to q : X → Y, since U is saturated with respect to q × 1. It’s also open, since if {x} × Vp ⊂ U then Ux × Vp ⊂ U, for some neighborhood Ux of x in X, by compactness of Vp.

x0 Now, q(U ) × Vp ⊂ O is a basic open neighborhood of (y, p) in Y × L. This proves that O ⊂ Y × L is open.

Let X is a CW-complex and L a locally compact (Hasusdorff) space (each point in L has an open neighborhood with compact closure, e.g. L compact). Then:

10. A subset F ⊂ X × L is closed iff F ∩ (¯e × L) is closed for each cell e of X. ` n Choose a characteristic map for each cell and recall the quotient map Φ : α Dα → X. Crossing n with L we know that Φ × 1 as well as each Φα × 1 : Dα × L → e¯× L are quotient maps. Thus F is −1 −1 −1 closed ⇔ (Φ × 1) (F ) is closed ⇔ (Φα × 1) (F ) = (Φα × 1) (F ∩ (¯eα × L)) is closed for all α ⇔ F ∩ (¯eα × L) is closed for each cell eα.

As a corollary of this we note:

11. A map F : X × L → Y is continuous iff F | :e ¯ × L → Y is continuous for all cells e of X.

Suppose now that X and Y are CW-complexes with cells eα and eβ respectively. Then X×Y is partitioned n m into sets eα × eβ. Given characteristic maps Φα : D → X and Φβ : D → Y for these cells and choosing n+m n m a homeomorhism h : D → D × D we get a characteristic map (Φa × Φb) ◦ h for eα × eβ. The conditions 1. and 2. in the definition of a CW-complex is satisfied by this partition of X × Y, but generally not the third one. The problem is that X × Y doesn’t necessarily have the weak topology with respect to the closures of the cells. One way to fix this is to simply givit the weak topology, but then this might not agree with the . The new space, which then is a CW-complex, is denoted X ×c Y and generally has more than X × Y has.

12. If X and Y are CW-complexes then so is X ×c Y if it’s partitioned by the cells e × f for all cells e and f of X and Y, respectively.

If Y is locally compact it follows from item 10 that X ×c Y = X × Y . In particular this applies when Y is a finite CW-complex.

Example 1.2 If we give S5 and S7 the CW-structure with one cell in dimension 0 and one in 5 and 7 respectively, we have a CW-structure on S5 × S7 with in total four cells of dimensions 0, 5, 7 and 5 + 7 = 12.

Let A ⊂ X (X not necessarily a CW-complex!) be a closed subspace and f : A → Y a map. Let i : A,→ X denote the inclusion as a map. In this situation there is a partition of Y ` X into the disjoint ` −1 ` sets {y} f (y), for y ∈ Y and {x}, for x ∈ X. Let q : Y X → Y ∪f X be the quotient map quotient space with respect to the equivalence relation corresponding to this partition. Then maps h : Y ∪f X → Z corresponds bijectively to pairs of maps hY : Y → Z ← X : hX , such that hY ◦ f = hX ◦ i. Restricting ` q : Y X → Y ∪f X to Y and X we get maps q|Y : Y → Y ∪f X ← X : q|X .

4 Thus there’s a commutative diagram

i A / X

f q|X

 q|Y  Y / Y ∪f X and for any commutative diagram as to the left bellow, there’s a unique h as to the right

i i A / X A / X

q|X

f hX { f Y ∪f X hX ; q|Y h  hY  Y / Z  hY #  Y / Z such that hY = h◦q|Y and hX = h◦q|X . This is codified by saying that the commutative square involving Y ∪f X is a push-out (diagram).

We have that Y ∪f X is the union of the closed subset q(Y ) and the disjoint open subset q(X − A). The ∼ ∼ map q restricts to Y = q(Y ) and X − A = q(X − A). The space Y ∪f X is said to be the adjuction of X to Y along the map f.

It’s not hard to construct examples where Y and X are both Hausdorff, yet Y ∪f X is not. We note the following (which suffices in this course):

13. Suppose X is a and f : A → Y a map where A is a closed subspace of X. Given two 0 0 disjoint open set U and V of Y, there are two disjoint open subsets U and V of Y ∪f X such that q(U) = U 0 ∩ q(Y ) and q(V ) = V 0 ∩ q(Y ). The two open sets f −1(U) and f −1(V ) of A are disjoint in the metric space A. In a metric space any closed subset is the zero-set of some map g : A → [0, ∞). Choose gU and gV with zero- −1 −1 sets the complements of f (U) and f (V ) respectively. Put g = gv − gu, then g(x) = 0 if x 6∈ f −1(U) ∪ f −1(v), g(x) < 0 if x ∈ f −1(U) and g(x) > 0 if x ∈ f −1(V ). By Tietzes extension theorem G has an extension to a map G˜ : X → R. Letting U˜ and V˜ be the open subsets where G < 0 and G > 0, respectively, we have the open saturated subsets U ` U˜ and V ` V˜ of Y ` X. Putting U 0 = q(U ` U˜) and V 0 = q(V ` V˜ ) shows the result.

14. Under hypothesis as in item 13, Y ∪f X is a Hausdorff space if Y (and X) is. Any two different points x, x0 in X − A can be simultaneously separated by open set in X − A and hence the same holds for q(x), q(x0). If x ∈ X − A and y ∈ Y the complement in Y ` X of an open around x not intersecting A gives an open saturated set in Y ` X disjoint from the open ball of half the radius around x, which is also an open saturated set in Y ` X . Two different points y, y0 ∈ Y can be separated by open sets and by item 13 there are open disjoint saturated sets in Y ` X extending them.

n−1 n n n Let X be a CW-complex. Consider X ⊂ X and choose characteristic maps Φα : Dα → X for the n- n n−1 n n−1 n ` n n−1 ` n−1 cells. Then Φα is a map of pairs (Dα,Sα ) → (X ,X ). Let D = α Dα and S = α Sα . The maps assembles to a map Φn :(D, Sn−1) → (Xn,Xn−1). Let φn : Sn−1 → Xn−1 denote the restriction to Sn−1

n ∼ n−1 n 15. X = X ∪φn D In other words, the n-skeleton of a CW-complex is the adjunction of n-disks to the (n − 1)-skeleton. n−1 n n n n−1 ` n n The maps i : X ,→ X ← D :Φ gives a map h˜ : X D → X , which is a quotient map by item 5 (c), and injective when restricted to Xn−1 and Dn −Sn−1. It follows that the induced map n−1 n n h : X ∪φn D → X is a bijective map which is a quotient map. Hence h is a homeomorphism.

5 0 1 n S n 16. If X is a CW-complex the skeletons gives a filtration X ⊂ X ⊂ · · · ⊂ X ⊂ · · · of X = n X by closed subspaces, such that (a) X0 is a discrete set (hence Hausdorff). (b) Xn is the adjunction of n-disks to Xn−1, n ≥ 1. (c) F ⊂ X is closed exactly when F ∩ Xn is closed in each Xn, n ≥ 0. This just summarizes the information gained so far.

On the other hand suppose given a space X with a filtration by closed subspaces satisfying (a)–(c). n n−1 ∼ n n−1 n Then X − X = D − S , which is a disjoint union of open n-disks. The (path-)components eα n n−1 n n n of X − X , gives a partion of X and the adjunctions gives maps Φα : Dα → X ⊂ X, restricting to n n−1 n bijections on Dα − Sα → eα. This verifies property 1. in the definition of a CW-complex. n n We can give D a metric by using the standard one on each individual Dα and putting d(x, y) = 2 if x and y are in different copies of the n-disk. By item 14 and induction every Xn, n ≥ 0, is Hausdorff, n n n which, as in the definition of a CW-complex, gives thate ¯ = Φα(D ), (since X is closed in X) so n n n n−1 n−1 k n n e¯α − eα = Φα(S ) is compact (and contained in X ). For each k < n the compact set X ∩ (¯eα − eα) can only intersect a finite number of the cells in Xk − Xk−1, since they are open and disjoint in Xk. Hence property 2. in the definition of a CW-complex is satisfied. Supposee ¯∩F is closed for all cells. By induction we can assume F ∩Xn−1 is closed. Then (Φn)−1(F ∩Xn) n n n ∼ n−1 n n is closed in D sincee ¯α ∩ F is closed for all n-cells. As X = X ∪φn D , this shows that F ∩ X is closed in Xn. By (c) this implies that F is closed in X and verifies property 3. the definition of a CW-complex. The only thing left to check is that X it self is Hausdorff. Pick any two points x, y ∈ X, and choose n n n n so that x, y ∈ X which is Hausdorff. Choose open separating sets Ox and Oy containing x and n n+1 n+2 n+i y, respectively. By item 13 there is then a sequence Ox ⊂ Ox ⊂ Ox ⊂ · · · , where Ox is open n+i n+i n+i−1 n+i−1 in X and Ox ∩ X = Ox and a similar one with x replaced by y, such that in addition n+i n+i S n+i Ox ∩ Oy = ∅, for all i ≥ 0. Defining Ox = i Ox and Oy similarly we have two disjoint open sets in X separating x and y. This gives the following

17. Suppose X is a space X with a filtration by closed subsets X0 ⊂ X1 ⊂ · · · ⊂ Xn ⊂ · · · ⊂ X = S n n X , such that (a) X0 is discrete, (b) Xn is the adjunction of n-disks to Xn−1, (c) F ⊂ X is closed iff F ∩ Xn is closed in Xn, for all n ≥ 0. Then X is Hausdorff and there is a unique CW-structure on X, such that Xn is the n-skeleton of X, for all n ≥ 0.

When using the excision theorem and the Mayer-Vietoris sequence it’s important to know that sub- complexes of a CW-complex are neighborhood deformation retracts. So, let A be a subcomplex of the n CW-complex X with a choice of characteristic maps Φα for the cells. For each n ≥ 0 X ∪ A is a subcomplex of X which is the adjunction of n-cells which are not already in A to Xn−1 ∪ A Let P the set consisting of all points Φα(0) for each cell eα not in A. Then P ⊂ X is closed and VA = X − P is n n n an open subset containing A. Let VA = (X − P ) ∪ A, which is an open subset of X ∪ A. Note that 0 VA = A. Now, quite generally, if q : Y → Z is a quotient map and V ⊂ Z is open, then q| : q−1(V ) → V is a quotient map as well. This is easily checked. n−1 ` n n n n−1 Applying this to the quotient map (X ∪ A) D → X ∪ A, defining X ∪ A as X ∪ A with n−1 ` n n n n-disks adjoined, we get that VA (D − O) → VA defines VA as the adjunction of punctured n-cells n−1 n to VA . Here D − O is a disjoint union of disks with the origins removed.

6 Crossing with the interval I and appealing to item 9, we see that the diagram to the left bellow

i×1 i×1 Sn × I / (Dn − O) × I Sn × I / (Dn − O) × I

φ×1 Φ|×1 φ×1 Φ|◦G   n−1 i×I n  n−1 p1◦i  n VA × I / VA × I VA × I / VA is a push-out. The standard deformation of a disk with the origin removed, to the bounding sphere defines a map G :(Dn − O) × I → Dn − O, giving the commutative square to the right above. n n n n n−1 The push-out property now defines a deformation retraction F : VA × VA of VA to VA . Denote the n n n−1 retraction F1 by rn : VA 7→ VA . n ∞ If x ∈ VA , then rk(x) = x, when k > n, so it makes sense to define the infinite composite rn = ∞ i n ∞ n rn ◦ · · · ri ◦ ri+1 ◦ · · ·. We have rn |VA = rn ◦ · · · ◦ ri and that F ◦ (rn+1 × I): VA × I → VA is a ∞ ∞ n−1 rn+1 ' rn rel VA . 0 0 0 If F,F : Y × I → Z are with F1 = F0, they can be composed to F ∗ F by putting one on top of the other doing F the first half of the time and F 0 the second. This is then a homotopy between 0 F0 and F1. Let G : V × I → V be the infinite composit

1 ∞ 2 ∞ n ∞ G = (F ◦ (r2 × 1)) ∗ (F ◦ (r3 × 1)) ∗ · · · ∗ (F ◦ (rn+1 × 1)) ∗ · · · .

n 1 n+1 ∞ This make sense since G(x, t) = x if x ∈ VA and t ≤ 2 . Then G is a homotopy 1 ' r1 and which ∞ is stationary on A and r1 is a retraction VA → A. We record this as

18. If A is a subcomplex of X and V is the open set obtained by removing a point in each cell not in A, then A is a deformation retract of V .

7