LECTURE 11: POSTNIKOV AND WHITEHEAD TOWERS

LECTURE 11: POSTNIKOV AND WHITEHEAD TOWERS I the previous sectio we used the techique of adjoiig cells i order to costruct CW approximatios for arbitrary spaces Here we will see that the same techique allows us to modify spaces by killig all homotopy groups above a certai dimesio This will be used to approximate a coected space by a tower of spaces which have oly o-trivial homotopy groups below or above a fixed dimesio where they are isomorphic to the oes of the give space The first case gives rise to the Postikov tower ad the secod oe to the Whitehead tower Moreover, the homotopy groups of two subsequet levels i these towers oly differ i oe dimesio I fact, the maps belogig to the towers are fibratios ad the fibers have precisely oe o-trivial homotopy group 1 The Postikov tower We kow that if α: e +1 represets a elemet [α] π (, x 0 ), the [α] 0 if ad oly if α exteds to a map e +1 Thus if we elarge to a space α e +1 by adjoiig a ( + 1)-cell with α as attachig map, the the iclusio i: iduces a map i : π (, x 0 ) π (, x 0 ) with i [α] 0 We say that [α] has bee killed (Naively, we thik of as a smallest extesio of that does that Some justificatio for this thikig will be provided i the exercises) The followig lemma expresses what happes to the homotopy groups i lower dimesios The proof is similar to the oe that the iclusio of the -skeleto of a CW complex is a -equivalece ad will hece ot be give Lemma 101 Let (, x 0 ) be a poited space, ad let α e +1 be obtaied from by adjoiig a ( + 1)-cell The the iclusio i: iduces a map π k (, x 0 ) π k (, x 0 ) which is a isomorphism for k < ad surjective for k It is difficult to cotrol what happes to the higher homotopy groups For example, sice π 3 (S 2 ) is o-trivial, addig a 2-cell to a elemet i π 1 may well add elemets i π 3 However, we ca kill all of π without chagig π k for k <, by iteratig the procedure of Lemma 101 Lemma 102 Let (, x 0 ) be a poited space The there exists a relative CW complex i: Y, costructed by adjoiig (+1)-cells oly, such that i : π k (, x 0 ) π k (Y, y 0 ) is bijective for k < ad such that π (Y, y 0 ) 0 Proof Let A be a set of represetatives α of geerators [α] of the group π (, x 0 ) Let Y be obtaied from by attachig a ( + 1)-cell e +1 α alog α: e +1 α for each α A: A e +1 A e +1 Y The by a iterated applicatio of Lemma 101, the map i: Y iduces isomorphisms i π k for k <, ad iduces the zero-map o π Sice this map is also surjective, we coclude that π (Y ) has to vaish 1 i

2 LECTURE 11: POSTNIKOV AND WHITEHEAD TOWERS For the proof of the ext theorem, recall that ay map f : U V ca be factored as f p φ, f : U φ p P (f) V, where p is a Serre fibratio ad φ is a homotopy equivalece ( mappig fibratio, see Sectio 5) We say that (up to homotopy), ay map ca be tured ito a fibratio Theorem 103 (Postikov tower) For ay coected space, there is a tower of fibratios P 1 () ψ 1 ψ 2 P 2 () P 3 () ad compatible maps f i : P i () (compatible i the sese that ψ f +1 f : P ()), with the followig properties: (i) π k (P ()) 0 for k > (ii) π k () π k (P ()) is a isomorphism for k (ad hece so is π k P () π k P 1 () for k < ) (iii) The fiber F of ψ 1 has the property that π (F ) π () ad π k (F ) 0 for all k Remark 104 A space like this fiber F with oly oe o-trivial homotopy group is called a Eileberg-MacLae space If Z is such a space with π k (Z) 0 for all k ad π (Z) A, oe says that Z is a K(A, )-space (strictly speakig oe always meas the space Z together with a chose isomorphism π (Z) A) We will discuss these spaces i more detail i a later lecture With this termiology the situatio of the theorem ca be depicted as follows f 3f2 P 3 () ψ 2 P 2 () F 3 K(π 3 (), 3) F 2 K(π 2 (), 2) ψ 1 P 1 () f1 where we used to deote a fibratio Proof of Theorem 103 Let i : Y be a space obtaied from by killig π k () for all k >, ie, such that (i) (i ) : π k () π k (Y ) is a isomorphism for all k (ii) π k (Y ) 0 for all k > Such a space Y ca be obtaied by repeated applicatio of the procedure of Lemma 102, where Y (+1) Y (+1) Y (+2) kills π +1 () by adjoiig ( + 2)-cells, Y (+2), ad so o The resultig space Y m> Y (m) kills π +2 (Y (+1) ) by adjoiig, the uio edowed with ( + 3)-cells to Y (+1) the weak topology, has the desired property, as is immediate from the fact that ay map K Y

LECTURE 11: POSTNIKOV AND WHITEHEAD TOWERS 3 with K compact (eg, K S k or K S k [0, 1]) must factor through some Y (m) If you see what this costructio does, the it is clear that there is a caoical iclusio φ : Y +1 Y makig the followig diagram commute (we eed to adjoi more cells for Y tha for Y +1 ): i +1 Y +1 φ Y i Thus, is approximated by smaller ad smaller relative CW complexes Y +1 Y Y 2 Y 1 Now let P 1 () Y 1, ad let f 1 : P 1 () be i 1 : P 1 () Let P 2 () be the space fittig ito a factorizatio of Y 2 φ 1 Y 1 id P 1 () ito a homotopy equivalece j 2 followed by a fibratio ψ 1 Next factor j 2 φ 2 i a similar way as ψ 2 j 3, ad so o, all fittig ito a diagram i 3 j 3 Y 3 P 3 () φ 2 ψ 2 i 2 j 2 Y 2 P 2 () φ 1 ψ 1 i1 Y 1 P 1 () Write f : P () for the compositio j i, ad deote the fiber of ψ 1 : P () P 1 () by F P () Now let us look at the homotopy groups By costructio we have (i) ad (ii) above, ad hece the same is true for P () istead of Y : (i) (f ) : π k () π k (P ()) is a isomorphism for all k (ii) π k (P ()) 0 for all k > We ca feed this iformatio i the log exact sequece of the fibratio F P () ψ 1 P 1 (), a part of which looks like π k+1 (P ) π k+1 (P 1 ) π k (F ) π k (P ) π k (P 1 ) where for simplicity we write P for P (), ad omit all base poits from the otatio So, we clearly have: (i) For k >, the group π k (F ) lies betwee two zero groups, hece is itself the zero-group

4 LECTURE 11: POSTNIKOV AND WHITEHEAD TOWERS (ii) For k <, the group π k (F ) lies betwee a surjectio ad a isomorphism, π k (F ), hece is zero agai (iii) For k, the relevat part of the sequece looks like 0 0 π (F ) π (P ) 0 whece π (F ) is isomorphic to π (P ) π () This tells us that F is a K(π (), )-space ad hece proves the theorem Remark 105 Much more ca be said about these Postikov towers: uder some coditios, the fibratio P P 1 is eve a fiber budle 2 The Whitehead tower The Postikov tower builds up the homotopy groups of (together with all relatios betwee them, such as the actio of π 1 o π ) from below, by costructig for each a space with homotopy groups π 1,, π oly There is also a costructio from above, called the Whitehead tower of, as described i the followig theorem Theorem 106 (Whitehead tower) Let be a coected space There exists a tower W 1 () W 2 () W 3 () with the followig properties: (i) π k (W ()) 0 for k (ii) The map π k (W ()) π k () is a isomorphism for all k > (iii) The map W () W 1 () is a fibratio whose fiber is a K(π (), 1)-space Proof As i the proof of the Postikov tower, ca be approximated by extesios Y +1 Y Y 2 Y 1, where π k (Y ) 0 for k > ad π k () π k (Y ) is a isomorphism for k For Y, let W () be the space of paths i Y from the base poit to, as i the pullback W () Y [0,1] 1 x0 i Y Y So W () is a fibratio (Remember we used this fibratio to describe relative homotopy groups of the pair (Y, ) i the exercises to Sectio 4) These spaces fit aturally ito a sequece W1 () W2 () W3 ()

LECTURE 11: POSTNIKOV AND WHITEHEAD TOWERS 5 Now tur these iclusios ito fibratios (by factorig ito a homotopy equivalece followed by a fibratio as before) to obtai a diagram W 1 () W2 () W3 () W 1 () W 2 () W 3 () where the lower horizotal maps are all fibratios ad the vertical oes are homotopy equivaleces Now let us look at the homotopy groups: We kow π k ( W ) π k (W ), ad there are two fibratios to play with, viz W () ad W () W 1 () The fiber of the first oe is the loop space ΩY of Y, ad the fiber of the secod oe will be deoted G The the log exact sequece of W () looks like or equivaletly π k (ΩY ) π k ( W ) π k () π k 1 (ΩY ) π k+1 (Y ) π k ( W ) π k () π k (Y ) But π k (Y ) 0 for k > ad π k () π k (Y ) is a isomorphism for k, so π k ( W ()) π k (), k >, ad π k ( W ) 0, k, ad hece the same is true for W istead of W Next, the log exact sequece associated to W () W 1 () looks like π k+1 (W ) π k+1 (W 1 ) π k (G ) π k (W ) π k (W 1 ) (where we write W for W (), etc), ad we otice: (i) if k > the π k (G ) is squeezed i betwee two isomorphisms, so π k (G ) 0 (ii) if k 2 the π k (G ) sits betwee two zero groups hece is zero itself (iii) for k we obtai π +1 (W ) π +1 (W 1 ) π (G ) 0 ad the first map is a isomorphism so that π (G ) 0 (iv) i the remaiig case k 1 the sequece looks like 0 π (W 1 ) π 1 (G ) 0, so that we have a isomorphism π () π (W 1 ) π 1 (G ) Thus, this tells us that G is a K(π (), 1)-space Note that the spaces W () used i the proof of the Whitehead tower are precisely the homotopy fibers of the maps i : Y costructed i the proof of the Postikov tower The remaiig work i the proof of Theorem 106 the cosists of turig a certai sequece of maps betwee the homotopy fibers i a sequece of fibratios ad aalyzig what happes at the level of homotopy groups This observatio is sometimes referred to by sayig that the Whitehead tower is obtaied from the Postikov tower by passig to homotopy fibers 3 Examples of Eileberg Mac Lae spaces I the costructio of the Postikov ad Whitehead towers approximatig a give space, K(π, )- spaces aturally came up We will coclude this lecture by givig a few of the most elemetary examples of K(π, )-spaces

6 LECTURE 11: POSTNIKOV AND WHITEHEAD TOWERS Remark 107 Recall that a K(π, )-space, or a Eileberg Mac Lae space of type (π, ), is a space (, x 0 ) such that π i (, x 0 ) for all i together with a isomorphism π (, x 0 ) π Here π ca be a poited set if 0, a group is 1, or a abelia group if 2 It ca be show that for such a π, a K(π, )-space always exists, ad is uique up to homotopy, although we will ot give the geeral costructio i this lecture Example 108 (Examples of K(π, )-spaces) (i) The circle S 1 is a K(Z, 1)-space Ideed, it is a coected space with fudametal group Z, ad oe way to see that the higher homotopy groups vaish is to cosider the uiversal coverig space R S 1 This is a fiber budle with discrete fiber F ad cotractible total space, so the log exact sequece gives us isomorphisms 0 π i (F ) π i+1 (S 1 ) for i > 0 (ii) The same argumet applies to wedges of spheres Cosider for example the figure eight S 1 S 1 Its fudametal group is the free group o two geerators Z Z The uiversal cover of S 1 S 1 ca be explicitly described i terms of the grid i the plae, G (Z R) (R Z) R 2 The map w : G S 1 S 1 ca be described by wrappig each edge of legth 1 i the grid aroud oe of the circles (i a way respectig orietatios): say the vertical edges to the left had circle ad the horizotal edges to the right had oe The uiversal cover E of S 1 S 1 is the space of homotopy classes of paths i G which start i the origi, ad E S 1 S 1 is the compositio E ɛ1 G w S 1 S 1 (where ɛ 1 is evaluatio at the edpoit) The fiber of ɛ 1 : E G over a give grid poit (, m) with, m Z is the set of combiatorial paths from (0, 0) to (, m): a sequece of alteratig decisios: go left or go right, go up or go dow, where successios of up-dow ad left-right cacel each other Sice each homotopy class of paths i E has a uique such combiatorial descriptio, the space E is clearly cotractible (iii) Recall that RP, the real projective space of dimesio, is the space of lies i R +1 It ca be costructed as S / Z 2 where the group Z 2 {0, 1} acts by the atipodal map o the uit sphere S {(x 0,, x ) R +1 x 2 0 + + x 2 1} The embeddig S S +1 sedig (x 0,, x ) to (x 0,, x, 0) seds S to the equator iside S +1, ad is compatible with this atipodal actio so that we get a commutative diagram S 0 S 1 S 2 RP 0 RP 1 RP 2 There is a atural CW decompositio of S +1, give iductively by a CW decompositio of S with two ( + 1)-cells attached to it: the orther ad the souther hemispheres This makes S ito a CW complex with exactly two k-cells i each dimesio k Oe

LECTURE 11: POSTNIKOV AND WHITEHEAD TOWERS 7 ca also take the uio alog the upper row of the diagram (with the weak topology) to obtai the ifiite-dimesioal sphere S S, a CW complex with exactly two -cells i each dimesio Note that sice π i (S ) 0 for i < k we also obtai π i (S ) 0 for all i 0 I other words, S is a weakly cotractible CW complex, ad hece by Whitehead s theorem is cotractible I a similar way, we ca take the uio alog the lower row i the above diagram to obtai RP RP, a CW complex, the ifiite-dimesioal real projective space, with exactly oe -cell i each dimesio The log exact sequece of the coverig projectio S RP with discrete fiber Z 2 shows that π i (RP ) 0, 1 < i <, or i 0, π 1 (RP ) Z 2, ad by passig to the limit, oe cocludes that RP is a K(Z 2, 1)-space (Alteratively, oe ca show that S RP is still a coverig projectio with fiber Z 2 to coclude that RP is a K(Z 2, 1)) (iv) Recall that CP, the complex projective space of (complex) dimesio, is the space of (complex) lies i C +1 It ca be costructed as (C +1 {0})/ C where C C {0} acts by multiplicatio; or, by choosig poits o the lie of orm 1, as the quotiet of the uit sphere i C +1, CP S 2+1 /S 1, where S 1 C agai acts by multiplicatio The quotiet S 2+1 CP has eough local sectios (check this!), hece is a fiber budle with fiber S 1 The embeddig iduces maps C +1 C +2 : (z 0,, z ) (z 0,, z, 0) S 2+1 S 2+3 CP CP +1 ad oe ca agai take the uio, to obtai a map S CP with CP the ifiitedimesioal complex projective space The space CP is a quotiet of S by S 1, ad the map is agai a fiber budle The spaces CP have compatible CW complex structures, give by exactly oe k-cell i each dimesio k Oe way to see this is to represet a lie i C +1 by a poit z (z 0,, z ), z R, z 0, ad z z 2 0 + + z 2 1 There is a uique way of doig this The the last coordiate t z is uiquely determied by z (z 0,, z 1 ) (sice t 1 z ), ad these (z 0,, z 1 ) form a disk of dimesio 2 The boudary of this disk is give by z 1, i other words t 0, ad this is exactly the part already i CP 1 I ay case, either of the two argumets at the ed of the previous example shows that CP is a K(Z, 2)-space