Notes on Serre fibrations

Transcription

1 Notes on Serre ibrations Stehen A. Mitchell Auust Introduction Many roblems in tooloy can be ormulated abstractly as extension roblems A i h X or litin roblems X h Here the solid arrows reresent mas that are iven, and the roblem is to ind a ma h commutin in the diaram. Usually the ma i is an inclusion, and is some kind o bundle ma such as a local roduct. Note the secial cases: (i) i i is an inclusion, = A, and = 1 A, then the extension roblem asks whether A is a retract o X; and (ii) i is surjective, X=, and = 1, the litin roblem asks whether admits a section. More enerally, one can combine the two diarams into one: A i h X Here the roblem is to ind a ma h such that both trianles commute. This situation arises, or examle, when a section o a bundle is iven on a subsace and we attemt to extend it to a lobal section. Note that we recover the extension roblem by takin to be a oint, and the litin roblem by takin A to be the emty set. The ma h (i it exists) 1

2 is oten called a iller or solution to the diaram. It is a ma that simultaneously extends and lits. Now rom a homotoy-theoretic ersective, all o these roblems are ill-osed. For examle, suose that in the extension roblem we have a ma : A homotoic to, and extends. Then it does not ollow that extends. Similarly, i the ma in the litin roblem is homotoic to a ma that lits, it does not ollow that itsel lits. The reader can easily suly counterexamles in both cases. One motivation or ibrations and coibrations is simly this: I i is a coibration and is homotoic to a ma that extends, then extends; i is a ibration and is homotoic to a ma that lits, then lits. The dual concets coibration and ibration relect a more eneral duality that is ervasive in homotoy theory. We will not attemt to ormulate this duality recisely. Thus the word dual, as used in these notes, has no technical meanin and should be imlicitly laced in quotation marks. On the other hand, the duality is extremely useul or intuitive uroses, and the reader is ured to become amiliar with it. Note: xamles can be ound in section 6, which should be read simultaneously with the revious sections. 2 Deinitions and basic roerties Let : be a ma o saces. We say that has the homotoy litin roerty with resect to a sace X i or all commutative diarams X i o H X I G the iller H exists. Here I is the unit interval and i 0 is the inclusion x (x, 0). We are iven a homotoy G into and an initial ma into ; the homotoy litin roerty says that we can always lit G to a homotoy H that starts with. Note that relacin i 0 by i 1 would not chane the deinition. The ma has the relative homotoy litin roerty with resect to a air o saces (X, A), i, whenever we are iven a diaram as above and a lit H A already deined on A I, the lited homotoy H can be taken to aree with H A on A I. In other words, there is a iller H in the diaram H A X (A I) i o i H X I G The ma is a Hurewicz ibration i it has the homotoy litin roerty with resect to all saces X, and is a Serre ibration i it has the homotoy litin roerty with resect to 2

3 all CW-comlexes X. Note that any lobal roduct X F X is a Hurewicz ibration and hence also a Serre ibration. Theorem 2.1 Let be a local roduct, and suose is aracomact Hausdor. Then is a Hurewicz ibration. For the roo see [May],. 49. Much easier is to show that any local roduct is a Serre ibration; see below. The ollowin roerties are easily veriied: Proosition 2.2 Any comosition o Hurewicz ibrations [Serre ibrations] is a Hurewicz ibration [Serre ibration ]. Any ullback o a Hurewicz ibration [Serre ibration ] is a Hurewicz ibration [Serre ibration]. Theorem 2.3 Let : be a ma. Then the ollowin are equivalent: a) is a Serre ibration; b) has the homotoy litin roerty with resect to all n-discs D n ; c) has the relative homotoy litin roerty with resect to all airs (D n, S n 1 ); d) has the relative homotoy litin roerty with resect to all CW-airs (X, A). Proo: a) b): This is immediate rom the deinitions. b) c): It is visually obvious, and not hard to rove, that the air (D n I, D n 0 S n 1 I) is homeomorhic to the air (D n I, D n 0). The desired imlication ollows easily rom this. c) d): Suose that a lit H is already iven on A I. We extend H over X n I A I by induction on n. At the inductive ste, we reduce to constructin a iller in a diaram o the orm D n 0 S n 1 I D n I Such a iller exists by assumtion (c). d) a): This is immediate, takin A =. Takin to be a oint in (d), we have incidentally roved: Corollary 2.4 Any CW-air (X, A) has the homotoy extension roerty. Remark: Note that the roo o the theorem actually roves slihtly more: Call : an m-serre ibration i has the homotoy litin roerty with resect to all CW-comlexes o dimension at most m. Deine relative m-serre ibrations in the analoous way. Then the theorem remains valid when all our conditions are relaced by their evident m-analoues. 3

4 Theorem 2.5 Consider a diaram A i h X in which (X, A) is a CW-air and is a Serre ibration. Then a) I is homotoic rel A to a ma such that a iller h exists or, then a iller h exists in the oriinal diaram; b) I is ibrewise homotoic to a ma such that a iller h exists or, then a iller h exists in the oriinal diaram. Proo: a) The assumtion is that (i) there are mas, h such that = ih and h = ; and (ii) there is homotoy G : X I such that G 0 =, G 1 =, and G(i(a), t) = i(a) or all a A and all t. Since G is constant on A, it can be lited on A to the constant homotoy H A (i(a), t) = (a). Now extend H A to a lit H, as in art (d) o Theorem 2.3. Then h = H 1 is the desired iller. b) y a ibrewise homotoy we mean a homotoy F that is only allowed to move around within its ibre 1 ((a)). More recisely, we require that the diaram A I F π A A commute. (This notion is dual to relative homotoy.) Now consider the diaram h X 0 A I F H X I G where F is a ibrewise homotoy that starts with h and ends with, and G(x, t) = (x). Note that the square commutes because F is a ibrewise homotoy. Then the iller H exists, and h = H 1 is the desired iller or the oriinal diaram. Note the ollowin secial cases: (i) Take A = in (a). Then i is homotoic to a ma that lits, itsel lits. (ii) Take =oint in (b). Then i is homotoic to a ma that extends, itsel extends. This is in act true with i relaced by any coibration. In case (ii) the ibrewise homotoies o the theorem are just ordinary homotoies, so this is really a result about coibrations. While we re on the subject, here is another imortant result on CW-airs: 4

5 Theorem 2.6 Let (X, A) be a CW-air, with inclusion ma i : A X. Then a) I A is a homotoy retract o X, then A is a retract o X. b) I i is a weak equivalence, then A is a deormation retract o X. Proo: a) This art holds or any air (X, A) havin the homotoy extension roerty. y homotoy retract we mean that there is a ma r : X A such that ri is homotoic to the identity. Now aly Theorem 2.5, art (b), with =oint, = 1 A, and = ri. b) y Whitehead s theorem, i is a homotoy equivalence. In articular, A is a homotoy retract o X, and so is an actual retract by art (a). Furthermore, i r is the retraction, then ir is homotoic to the identity o X. The remainin roblem is to show that there is a homotoy rel A. For a roo see [Sanier],. 31, Theorem 11 (note that his stron deormation retract is my deormation retract ), or rove it yoursel. (Hint: Start rom a homotoy ir 1 X that may not be a homotoy rel A, and use the homotoy extension roerty to construct a homotoy o homotoies endin with the desired homotoy rel A. Here you only need the homotoy extension roerty or the air (X I, X 0 A I X 1); this is automatic or in the case o a CW-air.) 3 Remarks on ath-comonents I X is an arbitrary tooloical sace, with ath-comonents X α, then the natural ma : X α X is a continuous bijection, but not in eneral a homeomorhism. For examle, i X is totally disconnected (e.., the Cantor set), then X α is just X with the discrete tooloy. To et a homeomorhism we would need the ath-comonents o X to be oen sets; this is true notably when X is locally ath-connected. On the other hand, it is clear that is always a weak equivalence. Now suose that : is a Serre ibration. Let { α } denote the ath-comonents o, and let { αβ } denote the ath-comonents o 1 α or each ixed α. Then there is a ullback diaram αβ α,β α α in which the horizontal mas are weak equivalences (homeomorhisms i and are locally ath-connected) and each individual ma αβ α is a Serre ibration. In this way, most questions about Serre ibrations are easily reduced to the case o a ath-connected base and 5

6 ath-connected total sace. We cannot assume the ibres are connected, however think o a coverin ma, or examle. Note also that or any sace, the unique ma rom the emty set to is a Hurewicz ibration and hence also a Serre ibration. This shows that ibrations need not be surjective. ut i is a Serre ibration and b is in the imae o, then the entire ath-comonent o b must be in the imae; this ollows immediately rom the deinition, interretin aths as homotoies o a oint. 4 The exact homotoy sequence o a Serre ibration A ointed Serre ibration is a Serre ibration : equied with baseoints e 0, b 0 such that (e 0 ) = b 0. In this case we reer to F = 1 b 0 as the ibre o, and write i : F or the inclusion. Theorem 4.1 Let : be a ointed Serre ibration with ibre F. Then there is a natural lon exact sequence π n F i π n π n π n 1 F Remarks: a) This is a lon exact sequence o rous as ar as π 1. The sequence ends with π 1 π 1 π 0 F π 0 π 0 which is exact as ointed sets. Note that the last ma need not be onto, since could have ath-comonents α such that 1 α is emty. Frequently, however, and are ath-connected, or at least we can easily reduce to that case by considerin one comonent at a time. Hence the sequence tyically ends with π 1 π 0 F 0. For n 1, only the baseoint comonents o and are relevant. b) The sequence is natural in the sense that a commutative diaram leads to a commutative ladder o lon exact sequences. We will derive this sequence rom the lon exact sequence o a air, so we bein by constructin the latter. Let I n denote the n-cube, and or n 2 let bi n I n denote I n U, where U is the interior o the bottom ace x n = 0. In other words, bi n is like a cardboard box with the bottom cut out but with to and sides let intact. For n = 1 we set bi 1 = {0}. Now let (X, A) be a ointed air. This means that X is ointed, A is a subsace o X, and the baseoint lies in A. The baseoint is denoted, althouh it will be omitted rom the notation entirely when no conusion can arise. Now deine the relative homotoy set π n (X, A) or n 1 by 6

7 π n (X, A) = [(I n, I n, bi n ), (X, A, )] In other words, we take homotoy classes o mas o triles (I n, I n, bi n ) (X, A, ), where the homotoies are required to kee I n in A and bi n at the baseoint. For n = 0 we deine π 0 (X, A) = π 0 X/π 0 A. Observe that π n (X, ) is just the usual π n (X, ). Note that π 1 (X, A) consists o homotoy classes o aths that start at and end in A. The homotoies must kee the initial oint o the ath at at each stae, but the endoint is ree to move around inside A. It should be clear that there is no reasonable way to concatenate such aths; thus π 1 (X, A) is only a ointed set, not a rou. I n 2, on the other hand, we can deine a roduct structure by the usual ormula ( )(x 1,..., x n ) = { (2x1,..., x n ) x 1 1/2 (2x 1 1,..., x n ) x 1 1/2 It is easy to check that this ormula ives a ma o triles as required. The reader should also check to see why it doesn t work or n = 1. Proosition 4.2 The roduct is well-deined on homotoy classes in π n (X, A) or n 2, and ives π n (X, A) a rou structure with identity element the constant ma. This rou structure is abelian or n 3. Proo: The roo o the irst statement is identical to the corresondin roo or the undamental rou, since all the action takes lace in the irst coordinate. The roo o the second statement is identical to the roo that π n (X, ) is abelian or n 2. Note that the latter arument requires two derees o reedom; we need n 3 in the relative case because x n is tied down by its secial role in the deinition o π n (X, A). It is clear that π n (, ) deines a unctor rom ointed airs to sets (i n 0), to rous (i n 2), and to abelian rous (i n 3). Note in articular that we have mas o airs i : (A, ) (X, ) and j : (X, ) (X, A). Given a ma o triles : (I n, I n, bi n ) (X, A, ), the restriction o to the bottom ace x n = 0 is a ma o airs (I n 1, I n 1 ) (A, ). We denote this ma by. It is immediate on insection that the assinment is well-deined on homotoy classes, yieldin a ma π n (X, A) π n 1 A or which the same notation will be used. Furthermore, it is immediate that or n 3, ( ) = on the nose that is, no homotoies are required. Thus is a rou homomorhism or n 3. Theorem 4.3 There is a lon exact sequence o rous (ointed sets or n 1) π n A i π n X j π n (X, A) π n 1 A This sequence is natural with resect to mas o ointed airs. Proo: The mas in the sequence have already been deined, and it is obvious that is natural with resect to mas o ointed airs. What remains to be shown is the exactness. This is not hard, but somewhat tedious since there are so many thins to check. We will 7

8 sketch what is needed and leave urther details to the reader. We recommend that the reader try to rove the theorem hersel beore even lookin at the sketch. Consider irst the exactness o π n (X, A) π n 1 A i π n 1 X A ma o triles (I n, I n, bi n ) (X, A, ) is the same thin as a ma o airs : (I n 1, I n 1 ) (A, ) toether with a nullhomotoy o i. Hence the exactness at π n 1 A is immediate rom the deinitions. Next, consider the sement π n X j π n (X, A) π n 1 A Then or any we have ( j) = on the nose; no homotoies are required. Now suose : (I n, I n, bi n )(X, A, ) and is nullhomotoic. Let F be a nullhomotoy o. y the homotoy extension roerty, we can extend F to a homotoy G : I n I such that G on bi n I. Hence [] = [G 1 ] in π n (X, A). Since G 1 ( I n ) =, this shows that Ker Im j. Finally, consider the sement π n A i π n X j π n (X, A) In this case we will need a simle lemma. Consider the (n + 1)-cube I n I and sinle out the three aces A 0, A 1, deined by x n+1 = 0, x n+1 = 1, and x n = 0 resectively. I one ictures the case n = 3 in xyz-sace, with the usual orientation o the axes and viewed rom the ositive x-axis, these are resectively the bottom, to and back aces o the cube. Thus a homotoy between two mas, : (I n, I n ) (X, ) is a ma o the (n + 1)-cube that has on the bottom ace, on the to ace, and on all remainin aces. Lemma 4.4 and are homotoic i and only i there is a ma F : I n I X that has on the back ace, on the bottom ace, and on all remainin aces. This is visually obvious, and not hard to rove riorously. Now suose iven a ma : (I n, I n ) (A, ). Since is homotoic to itsel we can ind F as in the lemma, with =. Then F is a nullhomotoy o ji(), rovin that ji = 0. Conversely, suose : (I n, I n ) (A, ) and we are iven a nullhomotoy F o j. On the back ace F mas (I n, I n ) (A, ). Hence by the lemma, is homotoic to a ma into A. In other words, Ker j Im i. This comletes the roo o the theorem. We now turn to the roo o Theorem 4.1. Modulo noise in low derees, this theorem ollows rom Theorem 4.3 and the ollowin lemma: Lemma 4.5 For n 1, the natural ma : π n (, F ) π n is an isomorhism. Proo: Suose α π n, and reresent α by a ma o airs : (I n, I n ) (, b 0 ). Then there is a iller in the diaram 8

9 e 0 I n bi n where the to ma is the constant ma. Since mas the bottom ace into F by construction, deines an element o π n (, F ) that mas to α. Hence is onto. Now suose iven β 0, β 1 π n (, F ) with β 0 = β 1. Reresent β i by a ma o triles h i : (I n, I n, bi n ) (, F, e 0 ). Then by assumtion there is a homotoy G : I n I rom β 0 to β 1, keein I n I at the baseoint. Choose a iller H in the diaram W H I n I where W = I n 0 I n 1 bi n I, = h i on I n i, and is constant on bi n I. Then H is a homotoy showin β 0 = β 1. Hence is one-to-one. This comletes the roo o the lemma. In view o Theorem 4.3, this yields the exact sequence o Theorem 4.1 excet or the sement π 0 F π 0 π 0. This case is clear because i x and (x) can be joined by a ath to b 0, then a lit o this ath with initial oint x ives a ath rom x to some oint o the ibre. (Note, however, that π 0 (, F ) need not biject to π 0 ; thus the last ma may not be onto.) We next take a closer look at the boundary ma π 1 π 0 F. Note that the imae o π 1 in π 1 need not be a normal subrou, and that π 0 F is only a set. Proosition 4.6 Suose and are ath-connected. Then the boundary ma : π 1 π 0 F induces a bijection π 1 / π 1 = π 0 F. (Here π 1 / π 1 denotes the set o cosets.) Proo: Unravellin the deinitions, we ind that is deined as ollows: Given α π 1, choose a loo λ : I reresentin it. y the homotoy litin roerty there is a lit λ : I startin at the baseoint. Then α is the ath-comonent o λ(1). Notice that this is exactly how one deines the action o π 1 on a ibre in coverin sace theory. The only dierence is that in coverin sace theory the lit λ is unique; here it is not. In any event, the rest o the roo also resembles coverin sace theory, and is let to the reader. As an alication we rove: 9

10 Proosition 4.7 Suose iven a commutative diaram o ointed saces with all our saces ath-connected and, Serre ibrations. Let h : F F denote the induced ma on ibres. Then i any two o,, h are weak equivalences, so is the third. Proo: (I the ibres are also ath-connected, this ollows immediately rom the lon exact homotoy sequence and the 5-lemma. In the eneral case, more care is needed.) Suose that and are weak equivalences. Consider the commutative diaram π 1 π 1 π 0 F = = π 0 h π 1 π 1 π 0 F Note that the mas and π 0 h are only mas o sets, so we must be careul about alyin the 5-lemma. It is clear that π 0 h is onto, but without urther structure there is no reason that π 0 h should be one-to-one. (The roblem can be traced to the ollowin simle act: I a rou homomorhism has trivial kernel, then it is one-to-one, but this is alse or mas o ointed sets.) Fortunately, however, we do have the urther structure rovided by Proosition 4.6. It ollows that π 0 h is bijective. The 5-lemma then shows that with any choice o baseoints, π n h is an isomorhism or all n 1. Thus h is a weak equivalence. The other two cases o the roosition are let to the reader (use the 5-lemma, but with caution). Theorem 4.8 Let : be a ma, and suose ath-connected. Then a) I is a local roduct and is locally ath-connected, then any two ibres o are homeomorhic; b) I is a Hurewicz ibration, any two ibres o are homotoy equivalent; c) I is a Serre ibration, any two ibres o are weakly equivalent. The roo o (a) is an easy exercise. For (b), see [Sanier],. 101, Corollary 13. For (c), note that by ullin back over a ath I, we reduce at once to the case contractible. Then the lon exact homotoy sequence shows that or any ibre 1 b, the inclusion 1 b is a weak equivalence. Hence any two ibres are weakly equivalent. 10

11 5 The main litin theorem We now come to a articularly eleant litin/extension theorem. y a subcomlex inclusion we mean the inclusion ma o a subcomlex o a CW-comlex. Theorem 5.1 In the diaram A i h X suose that is Serre ibration and i is a subcomlex inclusion. Then i either i or is a weak equivalence, the iller h exists. Proo: Suose irst that is a weak equivalence. We will construct h inductively over X n A. The case n = 0 is easy, since X 0 is discrete. At the inductive ste, we reduce to the secial case S n 1 α i h α D n α Here α = φ α and α = h n 1 ψ α, where φ α, ψ α are resectively the characteristic ma and attachin ma or a tyical n-cell e n α. Now any ma D n is homotoic rel S n 1 to a ma that is constant on D n (1/2), the disc o radius 1/2. In view o Proosition 2.5, we may thereore assume that α (D n (1/2)) b 0 or some b 0. Let W denote the annulus consistin o {x D n : 1/2 x 1}. Then W is homeomorhism to S n 1 I. Since is a Serre ibration, there is a lit h α deined on W. Now observe that h α mas the shere o radius 1/2 into the ibre 1 b 0. Since is a weak equivalence, the lon exact homotoy sequence shows that this ibre is weakly contractible. Hence h α extends to a ma h α : D n, and by construction it lits α. This comletes the roo in the case is a weak equivalence. Now suose i is a weak equivalence. Then by Theorem 2.6, A is a deormation retract o X. Let r denote the retraction. Since ir 1 X rel A, ir rel A. ut ir clearly admits a lit in the diaram namely, r and hence lits by Proosition 2.5. Corollary 5.2 Let : be a nonemty Serre ibration, with base sace a contractible CW-comlex. Then admits a section. Proo: Take X =, = 1, A a oint o, and any ma. Then h is the desired section. Remark: As one miht exect, a much stroner statement holds: I the base is contractible then the ibration itsel is ibre-homotoy equivalent to the trivial ibration F. See [Sanier],. 102, Corollary

12 Corollary 5.3 Let : be a Serre ibration with base sace a CW-comlex, and suose the ibre F is weakly contractible. Then admits a section s. Furthermore, i A is a subcomlex o and a section s A is already iven on A, s can be chosen to extend s A. Proo: Since F is weakly contractible, the lon exact homotoy sequence shows that is a weak equivalence. Now take X = and = 1. Note that Theorem 5.1 also contains the very deinition o a Serre ibration, by takin the let-hand vertical ma to be the inclusion i 0 : X X I. Further exloration o secial cases is let to the reader. 6 xamles Theorem 6.1 A local roduct is a Serre ibration. Proo: We sketch the roo and let the reader suly the details. Suose : is a local roduct. y Theorem 2.3, it is enouh to show that has the homotoy litin roerty with resect to all n-discs. We roceed by induction on n. At the inductive ste we can assume (see the Remark ollowin Theorem 2.3) that has the homotoy litin roerty with resect to all CW-comlexes o dimension less than n. Usin this toether with the Lebesue coverin lemma, one can reduce to showin that a lobal roduct U F U has the relative homotoy litin roerty with resect to the air (D n, S n 1 ). This in turn amounts to showin that the extension h always exists in the diaram D n 0 S n 1 I h D n I F ut clearly D n 0 S n 1 I is a retract o D n I (han a lihtbulb above the center o the cylinder and ollow its rays). Hence the extension h exists, comletin the roo. One can also roduce ibrations by startin rom a coibration and takin unction saces: Proosition 6.2 Suose X is a locally comact Hausdor sace and A X is a coibration. Then or any sace Y, the restriction ma F (X, Y ) F (A, Y ) on unction saces is a ibration. Proo: xercise. The comact-oen tooloy is comatible with recomosition in the irst variable (and ostcomosition in the second; the roo is easy); in articular the restriction ma is continuous. Now note that A is necessarily closed (c. [Hatcher],. 14) and hence locally comact Hausdor. The homotoy litin roerty can then be deduced directly rom the homotoy extension roerty, usin Proosition A.14b rom Hatcher. 12

13 The next eneral examle does not arise as a local roduct. Let Y I denote the athsace o the sace Y; that is, the set o all continuous mas I Y, equied with the comact-oen tooloy. Given a ma : X Y, deine N by the ullback diaram N Y I q X e 0 Y where e 0 denotes evaluation at 0. Thus N is the sace o airs (x, λ) with x X and λ a ath in Y that starts at (x); this construction is dual to the main cylinder construction. Now deine a ma : N Y by (x, λ) = λ(1). Proosition 6.3 : N Y is a Hurewicz ibration. Proo: There is a ullback diaram N Y I (q,π Y ) X Y Id Y Y (e 0,e 1 ) The ma (e 0, e 1 ) is a ibration by Proosition 6.2, since it is just restriction to {0, 1} I. Hence (q, π Y ) is a ibration, and thereore so is since it is the comosition N X Y Y. As an imortant secial casse, note that the ibre over a chosen baseoint y 0 is the sace o airs (x, λ) that start at (x) and end at y 0. In articular, takin X to be a oint and the inclusion o the baseoint, we obtain the ath-loo ibration P Y Y. In this case, the ibre over the baseoint is the loo-sace o Y, denote ΩY. The dual construction starts rom a ma : X Y and orms the coibration X M, where M is the reduced main cylinder. The coibre is then the reduced main cone. Takin Y to be a oint, the dual o the ath-sace is the reduced cone on X, and the dual o the loo sace is the reduced susension o X. Proosition 6.4 The rojection ma q : N X is a ointed homotoy equivalence. A homotoy inverse s : X N is iven by s(x) = (x, α(x)), where α(x) is the constant ath at (x). The roo is an easy exercise: Clearly qs is actually equal to the identity, while a homotoy rom sq to the identity is obtained by ollowin each λ out to time t. Corollary 6.5 Any ma : X Y can be actored in the orm X j X Y with j a homotoy equivalence and a Hurewicz ibration. 13

14 Proo: Take X = N, j = s, as above. We recall here that there is a result dual to the last corollary: Proosition 6.6 Any ma : X Y can be actored in the orm X i Y r Y with i a coibration and r a homotoy equivalence. Here one takes Y the main cylinder o, i the obvious inclusion at one end o the cylinder, and r the obvious deormation retraction onto Y. Remark: In the late 60 s Quillen introduced an axiomatic aroach to homotoy theory. The idea is to start with a cateory C equied with three distinuished classes o morhisms, called weak equivalences, ibrations and coibrations. These classes are subject to certain axioms, the most siniicant o these bein modeled on Theorem 5.1 (with i relaced by any distinuished coibration), Corollary 6.5, and Proosition 6.6. The cateory C, toether with the three classes as morhisms,constitutes a model cateory. In the cateory o saces there are several dierent interestin model cateory structures. For examle, one can take the weak equivalences to be the homotoy equivalences, the ibrations to be the Hurewicz ibrations, and the coibrations to be the closed mas that are coibrations in the usual sense. Another model cateory structure on saces, more relevant or our uroses, takes the weak equivalences to be the weak equivalences in the usual sense, the ibrations to be the Serre ibrations, and the coibrations to be all mas that are retracts o cellular coibrations. Without deinin the latter class recisely, we remark that it includes all subcomlex inclusions, and is contained in the class o all ordinary coibrations. Theorem 5.1 remains valid i i is relaced by one o these more eneral coibrations. This axiomatization has been alied alied to other cateories havin nothin to do with tooloical saces er se. For examle, there is a model cateory structure on the cateory o chain comlexes in which the weak equivalences are the so-called quasi-isomorhisms; that is, the mas inducin an isomorhism on homoloy. For a nice introduction to model cateories, see [Dwyer-Salinski]. 7 Homotoy-ibres All saces, mas and homotoies in this section are ointed. Let : X Y be an arbitrary ointed ma. Then the ibre 1 y 0 is somewhat irrelevant rom a homotoy-theoretic standoint, because it is not homotoy-invariant. More recisely, suose we are iven a commutative diaram X Y h X Y 14

15 with and h homotoy-equivalences. Then it does not ollow that the induced ma on ibres 1 y 0 1 y 0 is a homotoy equivalence, or even a weak equivalence. A dramatic illustration is rovided by X P X = X Here the ibre o the to ma is just a oint, while the ibre o the bottom ma is the loo sace ΩX. To make matters worse, i the iven diaram is only homotoy-commutative then there is no induced ma on ibres at all. What we need here is a homotoy-invariant relacement or the eometric ibre 1 y 0. In act such a relacement is already at hand. Given : X Y, deine the homotoy-ibre L as the eometric ibre o the Hurewicz ibration N Y. This deinition is motivated by two key oints: (i) N is homotoyequivalent to X, and (ii) as already shown in Proosition 4.7, eometric ibres o ibrations are homotoically well-behaved. For examle, the homotoy-ibre o Y is ΩY. In eneral, L is the sace o airs (x, λ) X P Y such that λ starts at (x) and ends at the baseoint y 0. It ollows that or any sace W, a ma W L is the same thin as a ma φ : W X toether with a nullhomotoy o φ. Theorem 7.1 Suose iven a homotoy-commutative diaram X Y h X Y Then there exists a ma α : L L such that the diaram L q X α L X q is homotoy commutative. Furthermore (a) I, h are homotoy equivalences, then α is a homotoy equivalence; (b) I, h are weak equivalences, then α is a weak equivalence. Proo: Form the diaram 15

16 N Y ḡ h N Y where ḡ is the comosite N q X X s N. It ollows rom the hyothesis and the deinitions that this diaram is also homotoycommutative. Since is a Hurewicz ibration, we can relace ḡ by a homotoic ma such that the new diaram N Y h N Y is strictly commutative. We deine α : L L to be the induced ma on eometric ibres. Now consider the diaram L N X L N X The irst square strictly commutes by deinition. The second square is homotoy-commutative because it is strictly commutative when is relaced by ḡ. Hence the outer rectanle is homotoy-commutative, as desired. We omit the roo o (a), since we are in any case willin to work u to weak equivalence. Part (b) ollows rom Proosition 4.7. xamle 1. In a ew cases L can be identiied (u to weak equivalence at least) in more eometric terms. A strikin examle is the ma o classiyin saces H G associated to a subrou H G. I H is suiciently nice subrou (e.., a closed subrou o a Lie rou G), one can show that L is weak equivalent to the homoeneous sace G/H. This is very useul or various uroses; e.., (i) we et lon exact sequence on homotoy rous with π G/H in the ibre slot; and (ii) we can use sectral sequences to relate the cohomoloy o G/H, H, and G. For details and seciic examles, see Notes on rincial bundles. Suose iven saces X, Y, Z and mas X Y Z. Two such adets are said to be weakly equivalent i they are equivalent under the equivalence relation enerated by commutative diarams 16

17 X Y Z = = = X Y Z Call X Y Z a ibre sequence i it is weakly equivalent to some F with a ointed Serre ibration and F the ibre o. More enerally, any sequence o mas... X 1 X 2... X n... (with or without initial/end terms) is a ibre sequence i any three consecutive terms ive a ibre sequence in this sense. Note that i X Y Z is a ibre sequence, then we et a lon exact sequence To be continued π n X π n Y π n Z π n 1 X... 8 Aendix: Whitehead s theorem In this section we use Theorem 5.1 to ive a very slick roo o Whitehead s theorem (I believe this roo is due to Quillen). We need to be careul to avoid a circular arument here, because we used Whitehead s theorem to rove hal o Theorem 5.1 the case when i is a weak equivalence. ut in the roo below we only use the other case, when is a weak equivalence. The roo o this case did not involve Whitehead s theorem. Theorem 8.1 Suose Y and Z are CW-comlexes, and : Y Z is a weak equivalence. Then is a homotoy equivalence. This theorem ollows rom (and in act is equivalent to): Theorem 8.2 Suose Y and Z are arbitrary saces, and : Y Z is a weak equivalence. = Then i X is any CW-comlex, induces a bijection [X, Y ] [X, Z] The deduction o Theorem 8.1 rom Theorem 8.2 is ure (and trivial) cateory theory. For in any cateory C, a morhism : Y Z is an isomorhism i and only i or all = objects X, induces a bijection Hom C (X, Y ) Hom C (X, Z). Here we take C to be the homotoy cateory o CW-comlexes. Proo o Theorem 8.2: y Proosition 6.5 we can actor as a homotoy equivalence ollowed by a Hurewicz ibration: Y Y Z. We thereore reduce at once to the case that is both a weak equivalence and a Hurewicz ibration (hence also a Serre ibration). Alyin Theorem 5.1 to the diaram X Y Z 17

18 then shows that [X, Y ] [X, Z] is surjective. Now suose, h : X Y and there is a homotoy F rom to h. Alyin Theorem 5.1 to the diaram X {0} X {1} h X I F Y Z shows that is homotoic to h. Hence [X, Y ] [X, Z] is also injective. 9 Reerences [Dwyer-Salinski], Homotoy theory and model cateories. Handbook o Alebraic Tooloy, edited by Ioan James. [Maunder], Alebraic Tooloy. [May], A Concise Introduction to Alebraic Tooloy. [Sanier], Alebraic Tooloy. I think this aeared in the 18