LECTURE 6: FIBER BUNDLES

Transcription

1 LECTURE 6: FIBER BUNDLES In this section we will introduce the interesting class o ibrations given by iber bundles. Fiber bundles lay an imortant role in many geometric contexts. For examle, the Grassmaniann varieties and certain iber bundles associated to Stieel varieties are central in the classiication o vector bundles over (nice) saces. The act that iber bundles are examles o Serre ibrations ollows rom Theorem?? which states that being a Serre ibration is a local roerty. 1. Fiber bundles and rincial bundles Deinition 6.1. A iber bundle with iber F is a ma : E X with the ollowing roerty: every oint x X has a neighborhood U X or which there is a homeomorhism φ U : U F = 1 (U) such that the ollowing diagram commutes in which π 1 : U F U is the rojection on the irst actor: U F φ U = 1 (U) π 1 U Remark 6.2. The rojection X F X is an examle o a iber bundle: it is called the trivial bundle over X with iber F. By deinition, a iber bundle is a ma which is locally homeomorhic to a trivial bundle. The homeomorhism φ U in the deinition is a local trivialization o the bundle, or a trivialization over U. Let us begin with an interesting subclass. A iber bundle whose iber F is a discrete sace is (by deinition) a covering rojection (with iber F ). For examle, the exonential ma R S 1 is a covering rojection with iber Z. Suose X is a sace which is ath-connected and locally simly connected (in act, the weaker condition o being semi-locally simly connected would be enough or the ollowing construction). Let X be the sace o homotoy classes (relative endoints) o aths in X which begin at a given base oint x 0. We can equi X with the quotient toology with resect to the ma P (X, x 0 ) X. The evaluation ɛ 1 : P (X, x 0 ) X induces a well-deined ma X X. One can show that X X is a covering rojection. (It is called the universal covering rojection. A later exercise will exlain this terminology.) Let : E X be a iber bundle with iber F. I : X X is any ma, then the rojection (): X X E X is again a iber bundle with iber F (see Exercise 1 o sheet 6). We will need the ollowing deinitions. Deinition 6.3. Let : Y X be an arbitrary ma. (i) A section o over an oen set U X is a ma s: U Y such that s = id U. (ii) The ma : Y X has enough local sections i every oints o X has an oen neighborhood on which some local section o exists. 1

2 2 LECTURE 6: FIBER BUNDLES Thus, by the very deinition, every iber bundle has enough local sections. And i you know a bit o dierential toology, you ll know that any surjective submersion between smooth maniolds has enough local sections. Many interesting examles o iber bundle show u in the context o nice grou actions. For this urose, let us ormalize this notion. Deinition 6.4. Let G be a toological grou and let E be a sace. A (right) action o G on E is a ma µ: E G E : (e, g) µ(e, g) = e g such that the ollowing identities hold: e 1 = e and e (gh) = (e g) h, e E, g, h G. I E also comes with a ma : E X such that (e g) = (e) or all e and g, then the action o G restricts to an action on each iber o, and one also says that the action is iberwise. Given a sace E with a right action by G, then there is an induced equivalence relation on E deined by e e i e g = e or some g G. The quotient sace E/ is called the orbit sace o the action, and is usually denoted E/G. The equivalence classes are called the orbits o the action. They are the ibers o the quotient ma π : E E/G. Deinition 6.5. Let G be a toological grou. A rincial G-bundle is a ma : E B together with a iberwise action o G on E, with the roerty that: (i) The ma φ: E G E B E : (e, g) (e, e g) is a homeomorhism. (ii) The ma : E B has enough local sections. Proosition 6.6. Any rincial G-bundle is a iber bundle with iber G. Proo. Write δ = π 2 φ 1 : E B E E G G or the dierence ma, characterized by the identity e δ(e, e ) = e or any (e, e ) E B E. I b B and b U B is a neighborhood on which a local section s: U E exists, then the ma U G 1 (U): (x, g) s(x) g is a homeomorhism, with inverse given by e ( (e), δ(s((e)), e) ). An imortant source o rincial bundles comes rom the construction o homogeneous saces. Let G be a toological grou, and suose that G is comact and Hausdor. Let H be a closed subgrou o G, and let G/H be the sace o let cosets gh. Then the rojection π : G G/H satisies the irst condition in the deinition o rincial bundles, because the ma φ: G H G (G/H) G: (g, h) (g, gh) is easily seen to be a continuous bijection, and hence it is a homeomorhism by the comact- Hausdor assumtion. So, we conclude that i G G/H has enough local sections, then it is a rincial H-bundle. (For those who know Lie grous: i G is a comact Lie grou and H is a

3 LECTURE 6: FIBER BUNDLES 3 closed subgrou, then G/H is a maniold and G G/H is a submersion, hence has enough local sections.) Remark 6.7. In case you have to rove by hand that π : G G/H has enough local sections, it is useul to observe that it suices to ind a local section on a neighborhood V o π(1) where 1 G is the unit, so π(1) = H G/H. Because i s: V G is such a section, then or another coset gh, the oen set gv is a neighborhood o gh in G/H, and s: gv G deined s(ξ) = gs(g 1 ξ) is a local section on gv. Remark 6.8. A related construction yields or two closed subgrous K H G a ma G/K G/H which gives us a iber bundle with iber H/K under certain assumtions. (See Exercise 5 o sheet 6) 2. Stieel varieties and Grassmann varieties We will now consider some classical and imortant secial cases o these general constructions or grous, namely the cases o Stieel and Grassmann varieties. We begin by the Stieel varieties. Consider the vector sace R n with its standard basis (e 1,..., e n ). A k-rame in R n (or more exlicitly, an orthonormal k-rame) is a k-tule o vectors in R n, (v 1,..., v k ) with v i, v j = δ ij. Thus, v 1,..., v k orm an orthonormal basis or a k-dimensional subsace s(v 1,..., v k ) R n. We can toologize this sace o k-rames as a subsace o R n... R n (k times). It is a closed and bounded subsace, hence it is comact. This sace is usually denoted and called the Stieel variety. (It has a well-deined dimension: what is it?) Note that V n,k V n,1 = S n 1 is a shere. We claim that V n,k is a homogeneous sace, i.e., a sace o the orm G/H as just discussed. To see this, take or G the grou O(n) o orthogonal transormations o R n. We can think o the elements o O(n) as orthogonal n n matrices, or as n-tules o vectors in R n, (v 1,..., v n ) (the column vectors o the matrix) which orm an orthonormal basis in R n. Thus, there is an evident rojection π : O(n) V n,k which just remembers the irst k vectors. The grou O(n k) can be viewed as a closed subgrou o O(n), using the grou homomorhism ( ) I 0 O(n k) O(n): A 0 A where I = I k is the k k unit matrix. One easily checks (Exercise!) that the rojection induces a homeomorhism O(n)/O(n k) = V n,k. Note that it is again enough to show that we have a continuous bijection since the saces under consideration are comact Hausdor. Thus, to see that π : O(n) V n,k is a rincial bundle, it suices to check that there are enough local sections. This can easily be done exlicitly, using the Gram-Schmidt algorithm or transorming a basis into an orthonormal one. Indeed, as we said

4 4 LECTURE 6: FIBER BUNDLES above, it is enough to ind a local section on a neighborhood o π(1) = π(e 1,..., e n ) = (e 1,..., e k ). Let U = {(v 1,..., v k ) v 1,..., v k, e k+1,..., e n are linearly indeendent} and let s(v 1,..., v k ) be the result o alying the Gram-Schmidt to the basis v 1,..., v k, e k+1,..., e n. (This leaves v 1,..., v k unchanged, changes e k+1 into e k+1 v i, e k+1 v i divided by its length, and so on.) Let us observe that this construction o the Stieel varieties also shows that they it into a tower O(n) = V n,n V n,n 1... V n,k V n,k 1... V n,1 = S n 1 in which each ma V n,k V n,k 1 is a rincial bundle with iber: O(n k + 1)/O(n k) = V n k+1,1 = S n k From these Stieel varieties we can now construct the Grassmann varieties. In act, the grou O(k) obviously acts on the Stieel variety V n,k o k-rames in R n. The orbit sace o this action is called the Grassmann variety, and denoted G n,k = V n,k /O(k). The orbit o a k-rame (v 1,..., v k ) only remembers the subsace W sanned by (v 1,..., v k ), because any two orthogonal bases or W can be related by acting by an element o O(k). Thus, G n,k is the sace o k-dimensional subsaces o R n. Since V n,k = O(n)/O(n k), the Grassmann variety is itsel a homogeneous sace G n,k = O(n)/ ( O(k) O(n k) ) where O(k) and O(k) O(n k) are viewed as the subgrous o matrices o the orms ( ) ( ) B 0 B 0 and 0 I 0 A resectively. The quotient ma q : O(n) G n,k is again a rincial bundle (with iber O(k) O(n k)), because q again has enough local sections. Indeed, it suices to construct a local section on a neighborhood o q(i). As a k-dimensional subsace o R n this is R k {0}. Let U = {W R n W R n k = R n } be the subsace o comlements o the subsace R n k R n sanned by e k+1,..., e n, and deine a section s on U as ollows: write w i or the rojection o e i on W, 1 i k, i.e., e i = w i + j>k λ j e j. Then (w 1,..., w k, e +1,..., e n ) still san all o R n, and we can transorm this into an orthonormal basis by Gram-Schmidt, the result o which deines s(w ). It ollows that V n,k G n,k also has enough local sections (why?), so this is a rincial bundle too (or the grou O(k)). Summarizing, we have a diagram o three rincial bundles O(n) V n,k G n,k

5 LECTURE 6: FIBER BUNDLES 5 constructed as O(n) O(n)/O(n k) O(n)/ ( O(k) O(n k) ). 3. Fiber bundles are ibrations The relation o the revious considerations to the revious lecture is given by the ollowing result. Theorem 6.9. A iber bundle is a Serre ibration. Beore roving this theorem, we draw some immediate consequences by alying the long exact sequence o homotoy grous associated to a Serre ibration to our examles o iber bundles. More alications o this kind can be ound in the exercises. Alication (i) Let : E B be a covering rojection, let e 0 E and let b 0 = (e 0 ). I we denote the iber by F (a discrete sace), then we have ointed mas (F, e 0 ) (E, e 0 ) (B, b 0 ). Then : π i (E, e 0 ) π i (B, b 0 ) is an isomorhism or all i > 1. Moreover, i E is connected then there is short exact sequence 0 π 1 (E) π 1 (B) F 0 where we have omitted base oints rom notation, and where we view F as a ointed set (F, e 0 ). Thus, or the covering R S 1 this gives us π i (S 1 ) = 0 or i > 1, since R is contractible. More generally, or the universal covering rojection X X with iber π 1 (X, x 0 ) we have π i ( X) = π i (X) or i > 1 and π 1 ( X) = 0. These statements all ollow by alying the long exact sequence o a Serre ibration. (ii) In the second lecture we stated that π i (S n ) = 0 or i < n (a act that can easily be roved using a bit o dierential toology, but which we haven t given an indeendent roo yet). Using this, we can analyze the long exact sequence associated to the iber bundle O(n) V n,1 = S n 1 with iber O(n 1), to conclude that the ma π i (O(n 1)) π i (O(n)) induced by the inclusion (always with the unit o the grou as the base oint) is an isomorhism or i + 1 < n 1 and a surjection or i < n 1. Writing O(n k) O(n) as a comosition we ind that O(n k) O(n k + 1)... O(n 1) O(n), π i (O(n k)) π i (O(n)) is an isomorhism i i + 1 < n k and is surjective i i < n k. Feeding this back in the long exact sequence or the iber bundle we conclude that O(n) O(n)/O(n k) = V n,k, π i (V n,k ) = 0, i < n k.

6 6 LECTURE 6: FIBER BUNDLES Now back to the roo o Theorem??. Instead o roving this theorem, we will rove a slightly more general result (Theorem??), which can inormally be hrased by saying that being a Serre ibration is a local roerty. Theorem?? immediately ollows rom this result and the act that trivial ibrations are Serre ibrations. Theorem Let : E B be a ma with the roerty that every oint b B has a neighborhood U B such that the restriction : 1 (U) U is a Serre ibration. Then : E B is itsel a Serre ibration. The roo o this theorem is relatively straightorward i we assume the ollowing lemma. Recall the ollowing notation rom a revious lecture. Let F = {F a a A} be a amily o aces o the cube I n, and let be the inclusion. J n (F ) = (In {0}) ( a F a I) I n I Lemma A ma : E B is a Serre ibration i and only i it has the RLP with resect to all mas o the orm J n (F ) In I. Note that the i -art is clear because the case F = gives the deinition o a Serre ibration. Earlier on, we have also used the case where F is the amily o all the aces, when J n (F ) In+1 is homeomorhic to I n {0} I n+1. The same is actually true or an arbitrary amily F, but one can also use an inductive argument to reduce the general case to the two cases where F = or F consists o all aces. We will do this ater the roo o Theorem??. Proo o Theorem??. Let : E B be as in the statement o the theorem, and consider a diagram o solid arrows o the orm I n 1 {0} I n in which we wish to ind a diagonal h as indicated. By assumtion on and comactness o I n, we can ind a natural number k large enough so that or any sequence (i 1,..., i n ) o numbers 0 i 1,..., i n k 1, the small cube [i 1 /k, (i 1 + 1)/k]... [i n /k, (i n + 1)/k] is maed by into an oen set U B over which is a Serre ibration (use the Lebesgue lemma!). Now order all these tules (i 1,..., i n ) lexicograhically, and list them as C 1,..., C k n. We will deine a lit h by consecutively inding lits h r on C 1... C r I n making the diagram I n 1 {0} h g g E B E C 1... C r h r B

7 LECTURE 6: FIBER BUNDLES 7 commute. We can ind h 1 because (I n 1 {0}) C 1 is a small coy o I n 1 {0} I n. And given h r, we can extend it to h r+1 by deining h r+1 Cr+1 as a lit in C r+1 ( (I n 1 {0}) (C 1... C r ) ) (g,hr) E C r+1 h r+1 B. Such a lit exists, because C r+1 ( (I n 1 {0}) (C 1... C r ) ) C r+1 is (essentially) a small coy o an inclusion J(F n ) In {0}. (You should draw some ictures or yoursel in the cases n = 2, 3 to see what is going on.) Proo o Lemma??. As we already said it only remains to establish the only i -direction which we already know in the cases o F = or the collection o all aces. We will reduce the intermediate cases to the case o all the aces by induction on n. For n = 0, I 0 has no aces so only the case A = alies and there is nothing to rove. For n = 1, there are our cases, A =, A = {0}, A = {1}, and A = {0, 1}, o which the irst and the last have already been dealt with. For the intermediate case A = {0}, or examle, consider a diagram o the orm I {0} {0} I E where : E B is a Serre ibration. Now one can irst ind a lit or I 2 {1} {0} E {1} I B by the case n = 0. Then next ill the ollowing diagram g B ( ) I {0} {0} I {1} I (,g) E I 2 by the case where F consists o all the aces (the ourth case). The induction rom n to n + 1 roceeds in exactly the same way: suose F = {F a a A} is a amily o aces o I n+1 or which we wish to ind a lit in a diagram o the orm B J n+1 (F ) I n+1 I E B

8 8 LECTURE 6: FIBER BUNDLES I F does not consist o all the aces, we add a ace G and extend to J n+1 F (G I) = J n+1 F {G} by liting in G {0} a (G F a) I E G I B which is ossible by the earlier case o the induction, because a (G F a ) is a amily o aces o a cube o lower dimension. Ater having done this or all the aces not in F, we arrive at the case where A is the set o all aces which was already settled.