OBSTRUCTION THEORY by TZE-BENG NG B.Sc, University of Warwick, 1971 a THESIS SUBMITTED in PARTIAL FULFILMENT of the REQUIREMENTS

f

OBSTRUCTION THEORY

by

TZE-BENG NG

B.Sc, University of Warwick, 1971

A THESIS SUBMITTED IN PARTIAL FULFILMENT OF

THE REQUIREMENTS FOR THE DEGREE OF

MASTER OF SCIENCE

in the Department

of

MATHEMATICS

We accept this thesis as conforming to the

required standard. ,

THE UNIVERSITY OF BRITISH COLUMBIA March, 1973. In presenting this thesis in partial fulfilment of the iquirements for an advanced degree at the University of British Columb.^,, I agree that the Library shall make it freely available for reference and study.

I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission.

Tze-Beng Ng

Department of Mathematics

The University of British Columbia Vancouver 8, Canada

Date 19th'i March 1973 .ABSTRACT

The aim of this dissertation at the outset is to give a survey of obstruction theories after Steenrod and to describe the various techniques employed by different researchers, the intricate perhaps subtle relation from one technique to another.

Owing to the difficulty in computing higher co- homology operations, one is led naturally to K-theory and the Eilenberg-Moore spectral sequence. However, these and other recent developments especially those in the study of stable Postnikov systems go beyond the intention of this modest survey. iii

TABLE OF .CONTENTS

Chapter'1. Introduction. 1

Chapter 2. Fiber Spaces. 2.1. Definitions and Notations. 10 2.2. Principal Fiber Spaces. 13 2.3. Transgression in Fiber Spaces. 17 2.4. On Decomposition of Fibration and Lifting Problem. 22 2.5. G-spaces. 26

Chapter 3. Classical Obstruction Theory. 3.1. Obstruction to Extension. 30 3.2. Primary Obstruction. 33 3.3. Extension Theorems and Classification of Maps. 35

Chapter 4. Global Obstruction. 4.1. Definitions of Homotopyand Homology Obstructions. 40 4.2. Generalized Cell Complexes and Obstructions. 45 4.3. Extension Theorems. 51 4.4. n-systems. 53 4.5. Secondary Obstruction and Construction of a Postnikov System. 55 4.6. A Postnikov Decomposition. • 57 4.7. Localisation. 61 4.8. Secondary Difference Obstruction. 62

Chapter 5. Principal Fiber Bundles. 5.1. Definitions. 66 5.2. Preliminaries. 67 5.3. Obstructions for Pair of Fiber Spaces 69 5.4. Applications. 72 5.5. Decomposition of Principal Fiber Bundles and Moore-Postnikov Invariants. 74 5.6. The Main Theorems. 79 5.7. Obstruction and Characteristic Classes 81 5.8. Conclusion. 91

Chapter 6. Modified Postnikov Towers and Obstructions to Liftings. 6.1. Modified Postnikov Towers. 94 6.2. Construction of Induced Maps between Postnikov Systems. 98 6.3. The k-invariants for a Lifting. 104 6.4. Obstruction Theory for Orientable Fiber Bundles. 112 6.5. An Illustration. 116 6.6. Some Examples of the Use of Postnikov Towers. 121

Bibliography. 125 ACKNOWLEDGMENT

I must thank Professor Denis Sjerve for his guidance, for the many enlightening conversations and helpful suggestions, for bringing to my attention [50] and [51], very much so for making some of the decisions in cutting down the size of this thesis and for reading the final draft. I must also thank Professor U. Suter for reading this thesis. Finally I am pleased to express my gratitude to the Mathematics Department for generously provided a teaching assistantship and a

Summer research grant while I am a student of Maths, at the University of British Columbia.

Tze-Beng Ng

March, 1973. CHAPTER 1. INTRODUCTION

One of the oldest problems in (algebraic) topology is to find liftings for a map a: K KB to E, the domain of another map £: E —

—> B. An even more interesting problem is the number of non-homotopic liftings of a, and hence to determine algebraic (problem) invariants for a homotopy classification of such liftings.

Suppose we cannot lift a, assuming that we do not know this in advance, can we give an "invariant" to confirm the impossible? In many cases we can do so and such an invariant is branded with the term obstruction (to lifting).

Suppose we can lift a, i.e. we have the following commutative diagram:

where 8: K »• E is a lifting of a i.e. Cog ~ a . Then we know the following diagram:

Hn(E)

H (K) -> H (B) commutes, we may therefore look at the following algebraic problem: H (E) n n

Hn(K) ->Hn(B)

When can we find {<}> : H (K) > H (E)} such that Yn n n £. od> = for all n ? * n *

(1) If so, when is {<)> ) induced by a map 3: K * E ?

< > (2) Do { j n} commute with boundary operator ?

(3) Do {

Consider the following diagram:

H (E)®Z sr n p

H (K) ® Z : y H (B) ® Z n p a ® 1 n p obtained by tensoring the preceding diagram with the integer mod p, Z

(p a positive prime). And we consider liftings of this problem. Again the same question can be asked: When is $ representable as some ®l

The advantage of looking at this problem is its effectiveness.

Just looking at the homology is not enough as some simple examples will show. So we increase the structure by looking at the cohomology ring: (E) y s 5 y * ^ * H (K) •< - H (B) a * *

When can we find homomorphism 6: H (E) >-.H (K) such that

6o£ = a ?

The cohomology ring is more interesting because of its structure. Thus

9 must be a ring homomorphism. We may strengthen the question by

requiring 6 to commute with coefficient operators, with the Poincare

duality operator in case E and K are Poincare complexes, with the *

boundary operator 6 and with the Bockstein operator. But we know we

have cohomology operations and we can regard cohomology as modules

over the Steenrod mod p algebra Gl(p), for example. Our next requirement

on 6 will be that it should commute with all cohomology operations. In

addition to requiring 6 be an Ol(p)-module homomorphism, we may require OUp) * that it should also be an (H (B;Z ))-module homomorphism, where TP * * * the module structures on H (E;Z^) and H (K'"Zp) are induced by 5 and * Gtfp) * a respectively, and (H (B;Z )) is the semi-tensor product algebra XP ®Ol which is equal to H (B;Z ) (p) as a vector space and with P multiplication as follows: * For v«>a, u®0 e H (B,Z )®OUP) /

(V®a)«(u®0) = I (-1) 1 (va u)«aJB where i|>(a) = £ a, © a'. and i

i>: CH(p) -> Ol(p)®0l(p) is the diagonal map.

Once again we have the question: When is the module homomorphism

induced by a map 0: K •> E ?

In another direction one can consider the homotopy lifting

TT problem, that is, when can we find a homomorphism g: a (K) • "%(E) making the following diagram:

* *

(K) , *(B ) # commutative i.e. £„oB = a„ ? When is 8 induced by a map K > E ? In # tt general we cannot tell. But if £: E -*- B is a covering space then we have the following theorem:

LIFTING THEOREM [24, pp.89]. Suppose K is connected and locally pathwise connected. Then a can be lifted to a map 8: K >• E iff the following algebraic problem:

if. (E) pf 1

fl(K) o—"VTri(B) ft is solvable, i.e. iff there exist homomorphism (3 satisfying K^o$ = ct^, Consider the following situation:

K »- B a with a "V a', i.e. a. = a' £ [K, B] , where [K, B] denotes the homotopy

classes of maps from K to B. We want some conditions to ensure that the liftability of a depends only on its homotopy class a e [K, B]. In

case £ is a fiber space we have the following:

COVERING HOMOTOPY THEOREM. Let £=(E, £, B, F) be a fiber space

, X locally compact and paracompact, and A closed in X. Suppose f: X —

s- E is a fixed given map and h: Xxi > B is a homotopy of a = £of:

X • B satisfying h|xx{0} = gof. Suppose further that there is a

" partial lifting " of h on A, i.e. a map h': Axiuxx{0} *• E

satisfying £oh' = h|AxiWxx{0> and h'|x*{0} = f. Then there exist a homotopy of f, h: Xxi >- E satisfying

(1) Koh = h ,

(2) h|xx{0} = f , and

(3) h|Ax!WXx{0} = h".

Now, since we know that given any map £: E >• B we can convert it into a map E" >-B , which is a Hurewicz fibration and where E is a deformation retract of E' (see for example [44, pp.84]), we may assume that £ is a fibration right from the beginning. Dual to the lifting problem we have the extension problem:

When can we find a map 8: E > K such that the following diagram,

E _

> K is commutative, i.e. p|x = a ? In general the answer is not always

"yes". There are some obvious counter examples. In case 8 exists we want some conditions to ensure that all o' ^ a be extendable over E.

In this direction we have the well known homotopy extension theorem of

Borsuk: -

HOMOTOPY EXTENSION THEOREM.' Let g: E > K be a map, X closed in E and 3|X = a. Suppose one of the following conditions is satisfied

(1) E and X are triangulable.

(2) K is triangulable.

(3) K is an ANR.

(4) (E, X) is an ANR pair.

Then any homotopy h of a can be extended to a homotopy of 8 .

( see [20] or [24].)

We now have the following algebraic problem: When can we find

4>: H#(E) • H^(K) satisfying 4>oi# = ? HA(E)

HA (X) - HA(K)

When is induced by a map E • K ? Again some easy examples will show that looking at the homology is not enough so we consider the corresponding problem in cohomology: * H (E) * v. i N-

- * H (X) «- H. (K)

& it it When can we find $: H (K) -*• H (E) such that i o$ = 0. ?

When is it induced by a map E > K ? Since we know H (E) can be considered as a module over the algebra of cohomology operations, we require $ to' commute with cohomology operations, the boundary operator and the Bockstein operator, in addition to $ being a ring homomorphism.

We have a broad and sophisticated spectrum of cohomology theories, which all together make this problem interesting and at the same time revealing the difficulties involved in trying to obtain even a partial solution.

We may consider the corresponding problem in K-theory: K*(E)

,1 v-

Kff(X) *• K (K)

We require that $: K (K) >• K (E) , if it exists, should corranute with

the Adams operations t/j . Suppose we consider in the category of

G-spaces,.where G is a topological group, a diagram

. E s

or

X -> B B

and seek either an equivariant lifting or an equivariant extension.

Then take the corresponding KG-theory problem:

K*(E) K! (E)

0 or

KMX) <- K#(B) KG(B) «-

seeking in each case 6 or $ such that they should commute with the ij>

If we consider the KU-theory, we can then exploit our existing knowledge of the theory of characteristic classes in the direction of giving a negative answer to the problem. (This is not the only way in which the theory of characteristic classes can be exploited). In some cases this has been done successfully without using K-theory, (see

[28] and [40]) . In summary, let F: "\7 *•

Gl(p)-modules, etc.), then F takes the geometric problem into one which is algebraic. This gives an obstruction. In particular, F can be any cohomology theory.

This dissertation can be said to have grown out of studying the foundation laid down by researchers, in trying to understand this very complicated and old problem in algebraic topology. CHAPTER 2. FIBER SPACES

This chapter is a discussion of the preliminaries needed for the later chapters. In what follows we shall assume fiber spaces in the semi-simplicial category {43] , unless otherwise stated. In this sense, we shall discuss properties of fiber spaces as if we were discussing them in the semi-simplicial category without specifically mentioning it.

§2.1. Definitions and Notations

2.1.1. By a fiber space in the sense of Serre we mean a quadruple (E, p, B, F) such that p: E •> B satisfies the covering homotopy property for any finite complex. E is called the total space, -1 B its base space, and p (x) = E - F is called the fiber over x e B.

2.1.2. Given a map f: X >• Y in the category of based spaces and based maps, we assume the definitions of the mapping cylinder of f and of the mapping cone C^ of f. We note the following property:

The diagram

X - > Y

Mf homotopy commutes and Y is a strong deformation retract of M^. 2.1.3. Given a fiber space (E, p, B, F), we can replace it by a homotopy equivalent fiber space, still denoted by (E, p, B, F) by abuse of notation, such that there exists a subset MC B and a homotopy

equivalence X: M >• E with poX = iG , where i0 is the inclusion of

M in B.

Proof. By 2.1.2, we get the following homotopy commutative diagram:

E

r B ^ • M < p r Let r be the retraction and form the induced fiber space (E , p', M , r p F).

E < E

P

B M r p

In particular, p is homotopically equivalent to p" and if we let X': E

>• E^ be the homotopy equivalence such that p"oX'= i0= inclusion of

E in M^ , then by the covering homotopy theorem we have the result simply replace (E, p, B, F) by (E , p", M , F).

2.1.4. Given any based map f: X *- Y, there is a standard construction by which we can replace f by a fibration f": X'" >• Y and

X may be taken to be a deformation retract of X^. (see [44, pp.84]). 2.1.5. We shall assume the knowledge of the standard construction of the Barratt-Puppe sequences with respect to a based map f: X y Y. That is, there are two sequences:

f i a Sf Si X > Y —=-> y SX y SY —y SC^ >- • • • f f and fij ttf a j f -y y QX y Q.Y y y x y SIE Ef Y such that given any pointed space, W, we have the following long exact sequences:

[X, W] < [Y, W] •< [C , W] -< [SX, W] •< [SY, W] •< ••• and

••• y [w, fix] >• [w, fiY] y [w, E ] y [w, x] y [w, Y]

They are exact sequences of groups as far as SX (fiY) and are exact sequences of abelian groups as far as S2X (Q2Y) .

2.1.6. We shall assume the following consequence from the definition of fiber spaces. For a fibration F 1 •> E ——y B, the sequence: ijj Pn 3n TT (F) ~5—y TT (E) ~5—>• TT (B) y TT , (F) —> •' n n n n-l is exact.

2.1.7. From the Serre spectral sequence we obtain the following result:. THEOREM (SERRE). Let (E, p, B, F) be a fiber space with B simply connected. Suppose that H^(B) = 0 for 0 < i < p and that

H_. (F) = 0 for 0 < j < q. Then there is an exact sequence:

±* P* T H , (F) >• H , (E) >• H , .(B) > H „ (F) • ••• p+q-lV np+q-lv P+q-1 p+q-2 y y h ... H (E) 0 where T is the transgression in the Serre spectral sequence.

§2.2. Principal Fiber Spaces

2.2.1. Consider the standard Hurewicz fibration: —— Q.Y y PY > Y .

We can form from it the induced fibration if we are given a map f: B >• Y . This induced fibration is called the principal fibration induced by f. It is denoted by (E , p, B, fiY). So we have the fiber square:

fiY fiY

E. -> PY

Pullback f B >• Y

In particular, E = {(b, 6) e BxpY ; f(b) = a(6)}. This construction has the following obvious properties:

(1) f h implies E - E . (2) If a: B' ->• B and b: Y Y^ are homotopy equivalences, then E, . - E_ . bofoa f (3) The sequence: [W, E ] »• [W, B] y [W, Y]

is exact for any space W.

2.2.2. From 2.2.1(3) we have the following immediate

consequence:

LEMMA. If g: W >• B is a map, then g lifts to E iff f^g

2.2.3. There is an action of ClY on the total space E^ given by

u: fiY>

"adding" a and 8. Therefore there exists a natural action of the group

[W, fiY] on the set [W, E ].

The foregoings, 2.1.5, 2.2.1 and 2.2.3, have corresponding

statements for maps of pairs and fibrations of pairs. Below we shall discuss n-connectedness and n-connected spaces. This is important as very often we would like to have n-connected spaces as fibers in some fibrations. This is indeed always the case and n-connected spaces occur quite naturally.

2.2.4. Using usual fiber space arguments we can obtain the following useful theorem: THEOREM. Suppose X and B are simply connected and that f: X y B is a map. Then f^: H_^ (X) >• H^ (B) is an isomorphism for i < n and an epicmorphism for i < n iff the same is true for f „: TT. (X) • TT. (B) . #11

As a corollary we note that the cofibre (mapping cone) of f is n-connected iff the fibre is (n-1)-connected. If this is the case we call f n-connected.

Using the relative Serre homology (exact) sequence (see 2.3.1)

, we can prove the following easy corollary to 2.2.4.

2.2.5. COROLLARY, Suppose B is (n-1)-connected,- n > 2, and F is (m-1)-connected, ni > 2. Then for p: (E, F) • (B, *), H^(p) is an isomorphism for i < m+n-1, and an epi morphism for i < m+n. In other words, p: (E/F) y B is (m+n)-connected. A dual statement holds for cohomology.

2.2.6. THEOREM (SERRE EXACT SEQUENCE FOR HOMOTOPY). Given

X f > Y —-—y c„ , with X (n-1)-connected and C. (m-1)-connected f f

(n, m >_ 2) . Then the induced map p: X >• E^ is (n+m-2)-connected.

Hence there is an exact sequence:

TT „(x) y ir „(Y) y TT , _(c_) y n+m-3 n+m-3 n+m-3 f Proof. Consider the homotopy exact sequence for

E. >• Y -^—y C. : l f -* Vi(Y) ~*" V^V ~* W ~~" VY> W and the homology exact sequence for the cofibration

X y Y > C„ :

H (Y) H .. (C.) -»• H (X) + H (Y) -Hp(Cf> p+1 p+1 f P P X (n-1) -connected implies i_: H (Y) >• H (C,_) is an isomorphism for * P P f p < n-1 and an epi morphism for p < n . Apply 2.2.4 to give the same conclusion for i„: TT (Y) > TT (C.) . Looking at the homotopy exact # P P f sequence above implies that E^ is at least (n-1)-connected. We then have the homology exact sequence for the fibration, E. -> Y C, and hence the following diagram:

Hp(Y) H.Hp(Cf) -> H (E, ) -v H (Y) p-1 1 P-1,

Hp(Y) ->Hp(Cf) — Hp_x(X) -Hp_l(Y) with the rows exact for p <_ n+m-1 . Apply the five lemma to get an isomorphism for p <_ n+m-2 . Apply 2.2.4 to get p an isomorphism for Tf k < n+m-3 . Hence there is an exact sequence:

TT „ (X) -> TT „ (Y) ~y it _(C_) n+m-3 n+m-3 n+m-3 f This completes the proof.

2.2.7. THEOREM. Suppose XCY, X is (n-1)-connected and that

Y/X (m-1)-connected (n, m >_ 2) . Then : Tr (Y, X) >- (Y/X) is an

isomorphism for i <_ m+n-2 and an epi morphism for i <^ m+n-1 .

Proof. Apply the following diagram:

„E.

(lifting)

-> Y -*• Y/X with the homotopy exact sequences:

TT (X) > TT (Y) • TT (Y, X) y TT , (X) > rr . (Y) P P P P-l P-l

y IT (E. ) IT (Y) • TT (Y/X) IT , (E.) — TT . (Y) p 1 p p p-l 1 p-l and the fact that p is (n+m-2)-connected by.2.2.6. An application of the five lemma gives that is an isomorphism for p <_ n+m-2 and an epi morphism for p <_ n+m-1 .

2.2.8. FREUDENTHAL SUSPENSION THEOREM. If B is (n-1)-connected

(n > 2), then s: TT . (B) • IT. , (SB) is an isomorphism for i < 2n-l — l l+l and an epi morphism for i <_ 2n-l . f i

Proof. Consider the obvious cofibration B > * y SB and the principal fibration induced by i which is ClSB . Then there is a lifting : B •> fiSB given by (x) (t) = t*x the adjoint to Identity:

SX >- SX . <{>,,: TT. (B) y TT. (QSB) is the suspension. Clearly is ft l i (2n-l)-connected by 2.2.6; and the result follows.

§2.3. Transgression in Fiber Spaces.

2.3.1. Leray-Serre Spectral Sequence.

Let (E, p, B, F) be a fiber space with B and F, (locally finite) complexes. Let B CB be a subcomplex and E = p 1(B ). ( B may o ooo be empty ). Then there is a filtration of H.(E, E ; T), where T is a * o local coefficient system with fiber, G, an A-module and A, a PID, and a canonical spectral sequence for cohomology { (Er, dr), r >_ 2 } such that (1) E _ > Gr H (E, E ; T), where Gr.H.(E, E ; V) is the p,q p q 0 * * 0 differential graded group coming from the filtration, and

(2) E2 = H (B, B ; H (F; G)), where H (F; G) is a system of P/q P 0 q q local coefficient over B [7] .

* There is also dually a filtration of H (E, E ; T) and a o canonical spectral sequence for cohomology, { (E^, d^_) ; r >_ 2}, with

P,Q P Q (.1) E > Gr H (E, EQ; T) and

(2) EP,Q HP(B, B ; Hq(F; G) ) . 2 0

2.3.2. Remark. If H (F; G) is a simple system as is the case

when (B, Bq) is simply connected or the stuctural group is connected,

then we may canonically identify H (F^; G) with H (F; G) and write

Hq(F; G) in place of Hg(F; G).

2.3.3. Some Exact Sequence.

(A) Let (E, p, B, F) be a fiber space with F, (m-1)-connected

and base B, arcwise connected. Let B C B and E = p ^(B ) C E. Then ,0 o ^ o there is an exact sequence:

m+1 * ••• *• Hn(B, B ; D -P—*- Hn(E, E ; D > Hn~m(B, B ; Hm(F; G) • 0 0 0 *

M m m ... y o > H (B, BQ; D H (E, Eq; D —-»• H°(B, Bq; H (F; G) ) jm+l ,-. * . -i i „ ,m+l ^—y H^B, B ; D P—y (E, E ; T) y H^B, B ; ^(F; G) )

0 0 0

m+2 H (B, Bo; D

for n <_ m , where T is a coefficient system over B, G is its fiber and p*: H3 (B, B ; T) y HD (E, E ; T) (j < m+1) is actually the map o o — induced by p: (E, E ) > (B, BQ) .

(B) Suppose further that the pair (B, BQ) satisfies

rr. (B, B ) = 0 , j < k-1 , k > 2 .

Then, using spectral homology argument, we have a similar spectral sequence to (A). *

y y y y y y N N 0 H (B, Bq; D -2— H (E, EQ; D 0 0

M+K M+K y H (B, Bq; D -JU H (E, EQ ; D — H°(B, Bq ; ^(F; G))

M+K+1 M+K+1 H (B, Bq; D H (E, EQ, T) »• H^B, BQ ; ^(F; G) ) —

m+k+1 m.. ,0

M+K+2 ^—> H (B, BO; r>. In particular, if V is a simple system of coefficient, we have that p : HN(B, B ) >• HN(E, E ) is an isomorphism for n < m+k-1 and o o " — * an monomorphism for n <_m+k . We note that p in the sequence is

actually the map induced by p: (E, EQ) >• (B, BQ) .

We remark that dual statements and sequences for homology hold for (A) and (B). 2.3.4. Transgression.

Let T be a local coefficient system with fiber, G, an A-module and A, a PID . Let {(E^, d^); r >_ 2} be the cohomology Serre spectral sequence coming from 2.3.1 for a fiber space (E, p, B, F). The 0 r ir+1 0 differential d , : E ' >• E ' is called the transgression. Let r+1 r+1 r+1 2 r r T (F, G) denote the submodule of H (F; G) which corresponds under the Or 0 ^r r f isomorphism, E ' = H (B; H (F; G)) ~ H (F; G) (= the fixed

r Or submodule under the action of TT^ (B) in H (F; G) ) , to the term E ^ . So, we have the following commutative diagram:

«0/r r+1 r+1, 0 E E r+1 r+i

Tr(F, G) ^Hr+1(B; G)/Mr+1

„r+l,0 v „r+k+l,l-k . _ , . „ , Since we know d, : E, • E, is zero for k > 2 so that k k k —

r+1,0 „r+l,0 . . , . -r+1,0^ „r+l ,n ^ r+1,0

Ek >• E ' is onto, implying E2 — H (B; G) > r+1 r+1 r+1 r+1 E 0 H B M r is onto, i.e., r+1' = (<" G)/ where M is some submodule of H (B; G). In.fact we can even calculate what M is. Call T also the transgression. Elements of T (P, G) are said to be transgressive r [48]. In fact it can be shown that if we define T (F, G) = -1—* r+1 <5 p H (B, *; G), where 6 is the coboundary homomorphism for the

r —*-l r pair (E, F) ,and S (B, G) = p <$H (F; G) , .then T is the same as the r r+1 —* map also denoted by x, T (F, G) »- S (B, G)/Ker p , given by

—* — T(U) = [u'] where 6 (u) = p (u") and p: (E, F) >• (B, *) .

T has the following non-trivial properties: * (1) Ker T = Im 0 , where 0 is the inclusion FCE. * it (2) S (B, G) = Ker p . * * * (3) If we define a: S (B, G) • T (F, G)/Im 0 by —* —* a(v) = [v'] , where p (v) = 6(v'), then Ker a = Ker p . a is called the suspension.

(4) Relative Transgression. Let F >• E >• B be a fiber space and * £ B C B. Let E = p 1 (B ) E and p: (E, E ) >-(B, B ) . o oo o o Take the pullback of (<$, p ), (11 , H ) . That is we have the diagram:

II

U >• H (E ; G) 0

If. P.B.

* * - *

H (B, Bq; G) • H (E, Eq ; G)

where IT and are projections. Define S (B, Bq; G) = ff^U and

A AAA T (E ; G) = 1IJJ , and maps, a : S (B, B. ; G) • T (E ; G)/Im 6 and o 2 oo oo T q: T (E ; G) >- S (B, Bq ; G)/Ker p to be the ones induced by

-1 -1 ^2^2. ^1^2 respectively.

2.3.5. Tq has the following noteworthy properties:

(1) If we denote the map B >• (B, B ) by j and the inclusion * * F c E (B is non-empty.) by k , then Tk = j T oo J o *

(2) Tq is an H (B)-morphism.

(3) <(>T = T ij> for any primary cohomology operation, , and t|> is its suspension.

(4) Suppose we assume that TT^ (B) operates trivially on

H (F> Z) and that H.(B, B ; Z) = 0 for i < (a-1) and H.(F; Z) = 0 for l.o — • 3 0 < j < b; it is not hard to show that the following sequence is exact. * - T i ... • H1 (E ) —H (B, B ) >• H (E) —^ H (E ) • 0 0 o a+b-1 ,„ . • • • • H (EQ) ,

I+1 I+1 1+1 where is the composition, H (B, BQ) • - • -»• H (E, E ) >• H (E)/ * and i is the homomorphism induced by inclusion i : E C E. (We have dropped the coefficient in the case of integral cohomology). * (5) Letting T = xk , if k : B C B and 0 < t < a+b-1 (a, b, 1 oo — * * * as in 2.3.5(4) ) and if Ker p D Ker k in dimension t and k is onto

in dimension t, then the sequence,

1 T Hfc(E) Ht(E ) — > Ht+1(B) , 0

is exact.

(6) Naturality of 2.3.5(4). Given a fiber space map:

F ===== F

Suppose H_. (F; Z) = 0 for 0 < j < b and fA : BMB^ Z) >• (B; Z) is an isomorphism for o < r < a-1 and an epicmorphism for r = a-1. Then * —* the sequence in 2.3.5(4) is still defined and exact with i = f and

(B, Bq) thought of as (Mf, Bq) where M is the mapping cylinder [58].

Following Thomas [58] , we describe in the next section a means of decomposing a fibration.

§2.4. On Decomposition of a Fibration and

the Lifting Problem

2.4.1. Let (E, p, B, F) be as before a fixed fibration. Take the standard Hurewicz fibration induced by a map to: B > C. We assume that p UJ - * so that we get a lifting q: (E, F) y (E , Qc)

with p^oq = p where p^ is the induced fibration suggested by the

following diagram:

E y* 0)

Pl

/ p E y B y C Obviously the map induced on the fiber, v , is the map q|F: F QC

We say v is geometrically realized by the pair (w, q). Using

the Barratt-Puppe sequence:

[E, QC] >• t(E, F), (E , BC)] —^ t(E, F) , (B, *) ] —[E, C]

* i i [F, fiC] ======[p, nc]

* -1 — — define Eu> = i p^*. [ p ] C [F,fiC] where [ p] is the homotopy class of p: (E, F) •*- (B, *) . It is easily seen that Eto = all homotopy

classes that can be geometrically realized by (w, q) for some lifting q of p. The following gives a characterisation of Eu.

THEOREM, -ato = Ea) , where a is the suspension defined in

2.3.4(3).

2.4.2. We would like to pursue the method outlined in 2.4.1 to decompose q further. We add the assumption that TT^(B) acts trivially * on H (F; G), as is the case when B is simply connected. Throughout this section this assumption is made. 24

Suppose F has non-zero homotopy groups (F) i-n dimensions n(l), n(2), ... , with o < n(l) < n(2) < ••• ; and if n(l) = 1, we shall assume that rr^ (F) is abelian.

Let i e H.n ^ (F; TT ,„ , (F) ) be the fundamental class of F. 1 n(l)

From the exact sequence for the fibration, we see that i is transgressive. Let (u = -T(I^), then we have the following commutative diagram:

-> F •*• QC1 = K(TTN<1) (F) , n(l))

E, E

B K(TT (F) , n(l)+l) n(l)

= C. with = Kt^^j » n(l)+l) and F^ = the fiber of the fibration,

•i : F K^7rn(l)^F^' n^1^ • For bY 2.4.1, we have a lifting, q^, of p to E^ with q^F = . Then by the exact homotopy sequence for the fibration, F^ >• F -> ^C^, and the connectivity of ttC^ we have that

f 0 , r < n(l)

vrrr(F) , r > n(l)

Thus, we have successfully killed one homotopy group of the fiber, n (2) 7r F e t ie Now, let i2 E H (F^; n(2)^ ^ ^ * characteristic class of F^ Again, it is transgressive for the fibration, F^ •> E >• E.^ Let w = -T(I ) c Hn (E , TT (F) ) . Repeating the procedure, we obtain a sequence of cohomology classes and spaces: E , to , E^, u^, •••

••• . The co.'s are referred to as the k-invariants of E. x

To study the k-invariants, we have to look into the variation under two liftings, f^, f^. When can we use the k-invariants to decide the liftability of maps from X to B ? Can they be used at all ? In otherwords,' we really hope that the k-invariants can determine the liftability of a map f: X y B . Questions naturally arise as to how one can calculate with these k-invariants, once it is decided that they indeed can be used.

In the next chapter, we shall describe some classical obstruction theories. To complete this chapter, it is thought suitable to say something about G-spaces since there is an obstruction theory for G-spaces, and because equivariant cohomology can be used to obtain the classical obstruction theory. Before we close this section, we state the following results for future reference.

2.4.3. Recall that fiC is an H-space. So fiC acts on the induced fibration. We have the following result where y: fiC*E^ »• E^ is the group action defined in 2.2.3.

LEMMA. The following diagram is commutative, where p^ is the induced principal fibration and q^ is a lifting of p to E^. Also

TT: fiCxE y E is the obvious projection. ftCXE E,

-> B

Proof. Just checking the definitions and explicit maps, u and

COROLLARY. Suppose F > E y B is a fibration with F

(m-1)-connected, C and E^ as before. Then, by. 2.'3.5 (6) , 2.1. 3, the above lemma together with 2.3.5(4), we have the following exact sequence: * x v

1+1 h1+1(E ) 1+1 ... > HNncxE) ^ H (B, E) > 1 * H (fiCXE) >

H2m(fiCxE) where v = uo(lxq^) .

§2.5. G-spaces

2.5.1. Recall the definition of a G-space:

If G is a topological group, for example a compact Lie group, then by a G-space X, we mean a topological space X together with a

G-action on X defined in terms of a map u: GXX > X which is continuos and which satisfies the following condition (associativity)

g1'(g2*x) = (g g )*x , for all x e X , where • denotes the action. By a G-map between G-spaces we mean a map, f: X > Y, which commutes with the action of G, i.e., the following diagram:

GXX

lxf

GXY i s commut at ive.

Suppose we are given another topological group, H, and a homomorphism, : H y G. Suppose X is a H-space and Y is a G-space.

A map, f: X y Y, is said to be ^-equivariant if the following diagram:

,u . H>

GXY is commutative, where, with abuse of notation, we denote the group action by the same letter.

We can define G-complexes, in similar fashion as G-spaces and

G-subspaces, and talk about equivariant (generalized) cohomology theory.

2.5.2. Remarks. >

(1) One can define a local coefficient system on G-complex, K, as a covariant functor -> Abel , where K is the category of

G-subcomplexes of K and Morphism(A, B) = {g e G; gA C B}, and deduce analogous equivariant cohomology theory with local coefficient system ; and hence one can have equivariant cohomology sequences for pair, triple, etc. .

By Grothendieck*s result (see [47, 4.7, pp.181]) , the local coefficient systems on K form an abelian category

= Funct(K, Abel).

K

(2) One can similarly, in an algebraic way, define equivariant homotopy, and deduce the corresponding exact homotpy sequence for pairs and a Hurewicz type theorem.

2.5.3. Similarly, given p: X >• Y , an equivariant map between two G-spaces, we say that p is a G-fiber map if and only if it has the equivariant lifting property with respect to G-complexes. In this category of G-spaces and G-maps,.we have the following important result.

THEOREM. An equivariant map, p: X >- Y, is a G-fiber map iff H H H H p|X : X y Y is a (Serre) fibration for every H c G, where X denotes the stationary space of X under H.

Proof. Omitted.. (See Bredon [9]).

Remarks.

(1) One can define Eilenberg-MacLane G-complexes or G-spaces and obtain the corresponding classification theorem for them. (2) The suspension functor and loop functor can be defined in a similar' way. Obviously we also have Hopf G-spaces with the • • corresponding Q (Hopf)-space structure.

(3) The results in 2§1 and 2§2 still hold true for G-spaces except for a slight modification for 2.2.5.

(4) We now know we can define a G-spectra so that a theory of equivariant spectral homology and cohomology can be used.

2.5.4. THEOREM. There is a spectral sequence {(EP,q, d )} such that

(1) EP'q^ExtP(H (K, L; Z) , V)) and 2 • -q

(2) EP'q =>H*+q(K, L; D , G where T is a (generic) local coefficient system for K, and (K, L) is pair of G-spaces.

The proof of this theorem is omitted and can be found in [9]

Remarks.

(1) The results in section 3 of chapter 2 still hold true.

(2) Obviously we obtain analogous results as in chapter 2 section 4 for equivariant lifting. CHAPTER.3.. .CLASSICAL

: OBSTRUCTION THEORY

In this chapter, we shall give a discussion of the classical obstruction theories, listing the properties and the main classification theorems. This discussion relies heavily on Steenrod's book 154].

§3.1. Obstruction to Extension

Let L be a subcomplex of K, and suppose E is (n-1)-connected

( n >_ 1 ) ( therefore arcwise connected ) and (n-1)-simple. Suppose xve are given a map, f: L >• E, we ask if there is an extension of f to

K. We shall assume familiarity with the definition of obstruction cochain.

If f is already given on the q-skeleton of K, i.e., on L uKq, then the obstruction to extending f to LWK^ is a cochain cq+"^(f) e q+1 o> , C (K, L; TT (E) ) where TT (E) denotes the local coefficient system q q over E, defined by the homotopy groups {TT (E, e); e e E}. It has the q following properties:

(1) cq+1(f) is a cocycle.

q+1 q+1 (2) fQ - f± => c (f0) = c (£.,_>.

. (3) cq+1(f) = 0 iff f can be extended to LuKq+1. (4) Suppose we have two extensions, f^, :L^Kq > E ,

3-1 q_1 ^ic with fQ|L = f-jL; and suppose that f0|LvK ~ . f^L rel L.

Then the homotopy k induces a map, K: (Kxi)qwLxl ->'"E , defined in

the obvious way by

K| (x, 0) = fQ(x), K(x, 1) = f^x) .

Define dq(f^, k, f^) to be equal to the obstruction to extending K to

q+1 q q+1 (Kxi) w LXI. Now d (f , k, f ) lives in c (KXL, Lxl; y (E)xl) OX. q q ^ which is isomorphic to C (K, L; TT (E) ) . We- also denote the image of q. q q q d (fQ, k, f1) in C (K, L; T? (E)) by d (fQ, k, f.^) and in future we cr cr ^ only talk of d^(fQ, k, f^) as in C^(K, L; TT^(E)). TO be precise,

q q+1 q+1 q+1 d (fQ, k, ^jxi = (-l) {c (K) - c (f0)x0- c^f^xl} where I

'Vq+l q+1 — is regarded as-a 1-cell and c (K) = c (K)xi. Hence,

6dq 6 q (fQ, k, f^jxi = (d (fQ, k, f^xl)

q+1 q+1 = c (f0)xl- c (fl)xl , which implies that

q q+1 1 «d (f0, k, fl) = c (f0) - c^ ^) .

q (5) d (fQ, k, enjoys the following property:

q If fQ, f^, f2 : LuK >• E with f = f = ±2 on L, and

suppose , kl K2 _ q q q fQ|LwK = f |L«-»K « f2|L^K ( rel L) where the homotopies are relative to L, then

q q q d (f0, k2#V f2) - d (fQ/ kx, fx> + d cf;L, k2, f2). q q 1 1 3 1 (6) d (f0, k, f1) = 0 iff k: f0|L^K * f1 L ^K* can be

extended to a homotopy f^L^R"1 - f^|L^Kq.

q K E q TT (7) If fQ: LuK E is a map, let d C (K, L; (E) ) .

Then f^lLw^ 1 may be extended to f^ on L^Kq such that

q d (f0, 1, fx) = d .

(8) Let f: LuK^ y E be as before and suppose cq+"^(f) = 0. cr+2 cr+1' Then { c (f; f an extension of f to LvK } is a single cohomology class. It's vanishing is a necessary and sufficient condition

for f to extend to LuKq+2. Denote this class by c q+2(f). It is sometimes called a secondary obstruction.

(9) Let f: L uKq -—y E be given. Then f|LuKq_1 is extendable to LuKq+1 iff cq+^(f) is a coboundary.

(10) Analogous to (8), we have the following:

If q y q 2 f^, f1 : LwK E are two maps, and k: f_jL'~'K -

f^|LOKq 2 ( rel L ) is extendable to a homotopy k': f^|LvjKq 1 - fjLuK*1"1 ( rel L ) , then dq_1 (f | L u Kq_1, k, fjLuK3"1) = 0 and the secondary difference obstructions {dq(f^, k'', f ) ; all k'} form a single cohomology class, d ^(£Q/ f^)•-The homotopy k: f |L Kq 2 -

3 2 q q f^LuK* " ( rel L ) is extendable to (L^K )xi iff d"- (f0, f±) = 0 .

(11) Using the homotopy extension theorem we hwve the following: -1 I Cf

Let f^, f^ : K y E be two maps such that fQ|L>->K = f^|LvjKq X. Then, there exists a homotopy H: ^ f* rel L«^Kq 2,. where f.jLuj.c'1 = f^|L^Kq, iff the difference cocycle ^(ZQI IF f-^ E

C^(K, L; TI (E)) is a coboundary in K — L . q (12) If f , f.^ : L^Kq •> E are two maps such that

q_1 k q_1 q+1 q+1 q f jLvK f lLWK then c (fn) * c (f.) and 6d (f , k, f ) 01 1' 0 1 o 1

q+1 q+1 = c (f0) - c (f1). ( This follows from 3.1(4) ).

(13) Naturality. Let h: K > K' be a map. It induces a map h: (K, L) > (K', L'). Then,

c(foh) = c(f) = h#c(f) e H*(K, Lj TT (E)),

V_J q whenever c(f) and c(f^) are defined. Similarly if f*r f^ : L' K' —

y E are two maps such that £Q|l' = ^lL' an^ suppose that

d(fQ, k, f ) = d(f^oh, k, f^oh)

= h*d(fj, k', f£) , whenever they are defined.

We next define primary obstructions and secondary obstructions, primary and secondary difference obstructions.

§3.2. Primary Obstruction

We shall assume E to be (n-1)-simple, arcwise connected for the remainder of this chapter. We list the following conditions which we would like E to satisfy.

(A) H-,+1(K, L; v (E)) = 0, for j £ n-1.

(B) H2 (K, L; TT_. (E)) = 0, for j <_n-l.

(C) HD~1(K, L; TT (E)) - 0, for j <_ n-1. 3.2.1. Suppose conditions (A) and (B) are satisfied. Given

f; L y E, by3.1(3) f can be extended to LuKn. Then 3.1(8) applies

to give that the set { cn+"^(f); f any extensions of f to Li~/Kn } is

a single cohomology class, c (f). We call this class the primary obstruction of f. Properties 3.1(2), 3.1(3), 3.1(5), 3.1(6), 3.1(7) n+l and 3.1(13) hold analogously for c (f).

3.2.2. Suppose conditions (B) and (C) are satisfied. If

fQ, f^-: K y E are maps with fQ|L = f-jL, then by 3.1(10) we say

that d n(f_, f,) e Hn(K, L; TT (E)) is the primary difference of f

0 1 n U and f^.

Suppose conditions (A), (B) and (C) are satisfied. We shall

further assume conditions (B) and (C) for L as well as for (K, L).

Then we have that analogous results hold for 3.1(5) and 3.1(13). And

d n(fy/ f-^) satisfies the following property:

N n f |L K" = fjl K iff d (fA, f,) = 0. o1 11 0 1

In place of 3.1(4) we have the following lemma.

LEMMA. Lf fg, f^: L *• E are two maps, then the primary

difference, d n (f , f ) , lives in Hn(L; TT^ (E) ) by conditions (B) and

(C) for L and 3.1(10). Moreover,

n+1 6d n n+1 o lt.) , (fn, f.) = c (f ) - 0 1 o 1

6: n n+1 n+1 where H (L; v (E)) • H (K, L; W (E)) andc^ffj, c" (f1) n n u x are given by 3.2.1. §3.3. Extension Theorems and Classification of Maps

3.3.1. THEOREM. With assumptions (A) and (B) of §3.2 or

i+1 ^

IT_, (E) = 0, for j £ n-1. If furthermore HJ (K, L; n\ (E)) = 0 for n < j < dim (K — L), then a map £: L • E has an extension to K iff c n+1(f) = 0. Proof. Apply the analogue of 3.1.3 twice.

3.3.2. THEOREM. With assunptions (B) and (C) of §3.2. Further- more assume HJ (K, L; IT (E) ) =0 for n < j <_ dim (K — L) . Suppose

f f ! K E arS maps suc tn 0' l *" h at £Q|L - then fQ - f^ rel L

n iff d (fQ, fx) = 0.

Proof. Just apply 3.2.2 twice.

3.3.3. Hence pursuing in this direction we have the following classification theorem.

THEOREM. (With the usual assumption on E, i.e., E is (n-1)- simple and arcwise connected). Suppose conditions (B) and (C) of §3.2

is satisfied. That is,

H^(K, L; TT (E)) = 0 = H-'~1(K, L; TT_. (E)) for j £ n-1 and

,+1 H"' (K, L} 7T\(E)) = 0 = H- (K, L; TT_. (E) ) for j > n.

Let f: K >- E be a map. Then the set of (relative) homotopy classes of maps f':K • E (relative to L) with £'|L = f|L are in one to one n ^ correspondence with H (K, L; TT^ (E)) under the assignment

[fl ydn(f% f) . Proof. The fact that this correspondence is one to one and well defined is 3.3.2 and the addition formula. To show.it is onto, we need the following theorem.

3.3.4. THEOREM. Same hypothesis as for theorem 3.3.3. Let — n ^ f: K y E be a map. Then for each class d e H (K, L; IT^E)) , there exists extension f* of f|L to K such that d n(f, f) = d.

Proof. Take a reoresentative e e d. Then 3.1(7) says that f|LuKn 1 can be extended to g: LUK" > E such that e =

n n n n n+1 n n+1 d (f|LuK , 1, g). But 0 = 6d (f|LuK , 1, g) = C (f|LoK ) - C (g)

n+1

= -c (g) so that by 3.1(3), g can be extended to K; let such an extension be f'. Then 3.1(10) applies to give d n(f, 1") = d.

3.3.5. Suppose E is (n-1)-connected, n > l.If n = 1, we assume that TT^(E) is abelian. Let f, f^: L >- E be maps. Suppose f^

n+1 n+ n is extendable to LuK ) then c ^(f) = 6d (f, f^) .

3.3.6. Using arguments analogous to that in N. Steenrod [54], we can prove the following:

n+1 ^

THEOREM. Suppose HJ (K, L; TT_. (E)) =0 for j > n and the conditions (A), (B), and (C) are satisfied for (K, L) as well as for

L (e.g. TT.(E) = 0 for j < n; if n = 1, assume TT (E) is abelian). Then

n a map f: L *• E is extendable to K iff d (f, fQ|L) is in the image

N T of i*: H (K; ? (E)) > H" (L; TT (E)) , where f : K >• E is a fixed n no — n ^ » . map. If f does extend to K, then for each such d' e H (K; Tr^(E)) wxth n n i (d') = d (f, f^L), there exists extension f of f with d (f, fQ)=d' We shall now proceed to define the primary obstruction to contracting E to a point.

3.3.7. Let E be a (n-1)-connected CW complex (n >_ 1; if n = 1, assume that TT^ (E) is abelian), and e^ a point in E. Let g^: E ——>• E be the constant map given by 9Q(S) = 0Q f°r all e e E. Define the primary obstruction'to contracting E to a point e^ to be

Hn(E; E)) ke = ^"^O' 1E) e V ' kn enjoys all the properties of a difference obstruction. If n > 1, eo then E is arcwise connected and so any two constant maps are homotopic.

This implies that kn is independent of the choice of e . Hence we e 0 o n denote this class by k . In particular, we have the following results.

(1) Given f: L > E and L C K. Then, by 3.3.5 f is extendable

n+1 n+1 n+1 n to LuK iff c (f) = 6fV\ for c (f) = 6d" (g0of, lof) =

n n 6f*d (gQ, 1) = 6f*k .

(2) Corresponding to 3.3.6 we have the following theorem. THEOREM. If H')+1(K, L; TT (E))= 0 for j > n and if TT_, (E) = 0 for j < n (if n = 1, assume TT^(E) is abelian) , then a map f: L > E

— n * n n ^ is extendable to K iff d (lof, gQof) = f k e H (L; TT^E)) is in the

* n 'V' n ^ image of i : H (K; ^(E)) »• H (L; TT^ (E)) which is induced by

_ *_ * n inclusion. If f does extend to K, then for each d' with id = f k there exists an extension f" of f: L *• E to K such that

( d "(lof, gQof') = ) f'V = d" . (3) A similar statement to 3.3(3) holds, where we replace

— n * n d (f, f) by f' k and so the correspondence there now becomes * n [ f ] , < • f k . rel L

3.3.8. G-spaces.

We come back to questions relating G-spaces and extensions.

The technique in obtaining the previous results ('cf. N. Steenrod [54]

) can be carried over to equivariant extensions in G-spaces with slight

modification, and "additional" assumptions.

All the usual assumptions in the previous sections are retained

H

and we require also that E to be j-simple for j < n and for all HCG.

Here E11 denotes the stationary space of E under H.

Analogous results to §3.1, §3.2 and the preceding sections of

§3.3 hold word for word except that everything is now in the category

of G-spaces and equivariant maps and so cohomology becomes equivariant

cohomology and homotopy, equivariant homotopy. (For details see

G. E. Bredon [9]).

3.3.9. It is said that one could have developed the theory of

equivariant extension and then everything could be carried to the

non-equivariant case, simply by considering G = {e}.

Remarks.

(1) We have now come to the end of this chapter; and if we

consider the map f as a cross-section to the bundle (E, p, B, F), then we obtain the classical obstruction theories to extending cross- sections and of course also the equivariant classical obstruction

theory to extending equivariant cross-sections of G-fiber bundle.

(2) We note that in the case of bundles our coefficient system

is chosen to be TT (F) , where F is (n-1)-connected, since we want to n

extend a cross-section to a cross-section, not merely an extension.

Hence, in place of E in the previous sections we insert F and we often

assume F to be n-simple. That is to say all the conditions on E in the previous sections become conditions on F.

In the next chapter we shall discuss a new development of obstruction theory for fiber spaces, utilising a decomposition of

fiber spaces due to Moore-Postnikov. CHAPTER 4. GLOBAL OBSTRUCTION

In this chapter we shall stay in the category of based spaces and based maps. (E, p, B, F) will denote a fiber space in the sense of

Serre. B will always be assumed to be simply connected. The exposition is based on R. Hermann [18].

§4.1. Definitions of Homotopy Obstruction

and Homology Obstruction.

Let (E, p, B, F) be a fiber space. Suppose it has a cross- section, f: B >• E, then from the homotopy exact sequence for the fiber space, we get the following splitting:

ir.(E) = i„(Tr.(F)) © f „ (ir. (B)) for all j, 3 ff 3 # 3 where i: F »- E is the fiber inclusion. This determines an onto- homomorphism, f: TT_. (E) >-»• TT_, (F) , which is the composition,

{i// P'J proj. , (E) == 9 (4.1.1.) TT i„TT. (P)e-f*Tr. (B) ~ ^ TT . (F) TT . (B) »• TT. (F) .

3 # 3 " *? D s 3 3 3

We need the following technical lemma to obtain a similar type of map, H (E) H.. (F) .

4.1.2. LEMMA. Suppose (E, p, B, F) admits a cross-section, f, and suppose B is (n-1)-connected (n >_ 2), and F is (m-1)-connected. Then i • H.(F) »• H.(E) is one-to-one for j < m+n-2, and * 3 3 ... ^(B H.(E). = frtCH ))ei*(H'^(F)) for 1 £ j <_ m+n-2. Proof. By 2.3.3 p^: EL (E, F) y H (B, *) is an isomorphism for j < m+n ; and f : H, (B) + H.(E) is one-to-one for f is a cross- 3 3 section. Consider the following diagram:

-1*

H (E) Hj+1(E, F) -+ H. (F) H. (E) j+1 3 P* P* = H

where p^ is onto since f is a cross-section. Thus j is onto in dimensions greater than 1 and less than or equal to (m+n-1), implying that i.: H.(F) y H. (E) is a monomorphism for 0 < j < m+n-2 by 3 3 — — exactness. Therefore we have the following splitting:

H.(E) = i (H.(F))© f. (H . (B)) for 0 < j < m+n-2 . 3 * 3 * 3 - —

4.1.3. Thus by 4.1.2 we can define f: H.(E) y H.(F) for 3 3 1 < j < m+n-2 to be the composition,

{I * ' P*> pro] , H.(E) = i:H. (F) ©f. H . (B) -> H. (F)©H. (B) y H. (F) 3 3 3

We observe that the naturality of the Hurewicz homomorphism, $L,implies the commutativity of the following diagram:

f TT. (E) -y TT . (F) D 3

it

H. (E) -y H. (F) 3 3 for 1 < j < m+n-2 4.1.4. Let B CB with B f d> . Consider the induced fibration o o

E CZ o

P E,

B C B where E can be taken to be p V ). Suppose f: B y E is a cross- o ooo section of the fiber space (E , pIE , B , F), we shall consider the o 1 o o problem of extending f to a cross-section over B. With 4.1.1 in mind we define the homotopy obstruction to extending f to a cross-section over B to be the following compositions:

P# 8# F TT TT Tr (4.1.5) to (£) : . (B, B ) ^ . (E, E ) —-*•»• rr (E ) y ._1 (P) for j = 2, 3, i.e. , to. (f) = f^1

Suppose we have that H. (B, BQ) = 0 = H_. (Bq) for 1 <_ j <_ m and £ F is (n-1)-connected. Then p : H.(E, E ) y H.(B, B ) is an DO 3 o isomorphism for j <_ m+n-1 (see 2.3.3(B)); and so we can define the homology obstruction to extending f to a cross-section over B to be the following compositions:

p~*-1 .9^ f (4.1.6) V. (f) : H. (B, B ) • H. (E, E ) y H. (E ) ^ H. (F) _1 3 0 3 0 -1 D o £34 D . D for 2 <_ j <_ m+n-2 ; i.e., v.. (f) = f^p*""1 . From definition 4.1.4 we see that the following diagram is commutative.

'"u. (f) -1—> -rr , (F) TT.(B, BQ)

(4.1.7) ft. v.(f) H.(B, B ) J • H. .(F) DO D-l j = 2, ••• , m+n-2.

4.1.8. At this point it is clear how one can obtain the cohomology obstruction to extending f to a cross-section over B.

However, we are more interested at the moment in the homology obstructions, { v_. (f) }, and the homotopy obstructions, { to_. (f) }.

The latter have the following properties.

(1) If f is extendable to B, then to_. (f) = 0 for all j,

(2) Consider the fibration,

fiB, with B c B and f: B • E as before. Then f is a cross-section of o oo the loop space fibering. The following diagram is commutative,

w. (£) TT_, Tr (B, BQ) ;._1(F)

7T. , (fiB, fiB ) 3" • TT. „(flF) 3-1 o 3-2 where a is the loop isomorphism. That is, we have invariance under the loop functor. (3) Naturality. Given two fiber spaces, (E, p, B, P) and

(E', p", B^, F'), and a fiber map, {g, h, k}: . g F >• F'

Suppose B C B and B'cB' are such that k(B .) cB', and that f: B

and f: B" »- E' are cross-sections making the diagram, oo . k B -> B

E -> E' o o commutative. Then {g, h, k} induces a commutative diagram, OK (f)

+ TT (F) J - • TT.(B, BO)

wj(f'} TT. (B', B") Tr. (F") 3 o

for j = 2, 3, • • • .

(4) y If f^, f^: B E are cross-sections such that fx|BQ

^2^Bo ' ^en one can define difference obstructions to deforming f

onto f over B as follows:

We know loop spaces have an H-space structure,

V: RXxfiX y SIX .

We use this structure to define a map, g: (SIB, QBq) * (SIS.,. *) , by g(x) = Slf (x) v Slf^(x "S , where x is the homotopy inverse of x.

With this we define the difference obstruction to be the compositions: - .-1 > (SB, SIB (StE) —^> d. (f , f ) : TT. (B, B ) TT. , ) —^ IT. . 31230 3-I 0 3-I - -1 a (SIF) y TT. , TT. (F) . D-l 3 d.(f , f ) fits in the following diagram:

TT_. (E) —^ —— y TT (F)

(4.1.9)

-> TT w (E, EQ) . (B, B ) 3 0

We would like to consider some of these co.'s as cocycles. This 3 leads us to the following notion of a GCW decomposition due to W. Barcus and R. Hermann. The objective is to obtain a global definition of obstruction.

§4.2. Generalized Cell Complexes and Obstructions

4.2.1. Definitions. A generalized'cellular decomposition of a space, B, is a sequence of subspaces

B(0) ~{b )CB(1)c CB(n)cr ... CB 0 such that (4.2.2) H.(B(n), B(n_1)) = 0 for j f n, n = 1, 2, ••• . 46

A B subspace, q, is said to be adapted to such a decomposition

C abbreviated GCW decomposition ) if the following is satisfied.

(4.2.3) H. (B nB(n), B AB'""1') = 0 for j ^ n, and jo o H.(B wB(n), B WB(n-1)) = 0 for j ^ n . jo o

For example, the skeleton decomposition of a CW complex, X, is a GCW

decomposition and any CW subcomplex, X , is adapted to a skeletal

decomposition.

We shall now proceed to describe an obstruction theory with

respect to such GCW complexes.

4.2.4. Let B(0)= {b }CB(1)c CBW C - dB o

be a GCW decomposition for B, and BOCB be a subspace of B adapted to

the given GCW decomposition for B. Define an algebraic chain complex,

&> =' { C (B, B ) , 9 } as follows: no

Denote by B^ the union B UB'K' and define o

(4.2.5) C (B, B ) = H (B , Bu '), and non 9: C (B, B ) > C AB, B ) , n o n-1 o Mn) ^(n-1) Mn-2). to be the boundary operator, 9*, of the triple, (B , B , B ).

It is clear that 92 = 0 . By usual homology argument we can show that

H (S) C^. H (B, B ) for all n. n no

4.2.6. We can now proceed to see how our previous concept

comes into play in fiber spaces. The main difference from the classical

obstruction theory of fiber bundles is that the definition of obstruction

cochain is defined in terms of the global properties of the homotopy groups of the fiber.

Let (E, p, B, F) be as before. Suppose we are now given a cross-section f of (E , p E , B , F) over B , which is adapted to a o 1 o o o .

GCW decomposition of B. Assume Bq ^ . of course.

Analogous to the classical theory we define the obstruction

n Tr cochain on the complex, defined in 4.2.5, by y (f): C^ ^ n_-L (F) as the homotopy obstruction to extending f over B^; i.e., wn(f) is the composition,

Mn) -(n-1) "n(f) Cn= VB ' B > -n-l(P) ' where the isomorphism is given by the relative Hurewicz theorem.

We shall proceed to show that un(f) is indeed a cocycle and list its properties to justify its name.

4.2.8. (1) 6co(f) = 0.

/^x ^ * -(n-1) -(n-1)

(2) Suppose £ , f2: B >• E are cross- J -(n-1) , ^ _ i-(n-2) _ i-(n-2) , sections already given on B such that f^|B = f2|B . (

-(n-1) = p-l-(n-l)K) We can then define the difference obstruction cochain by ,n-l._ _ ...... _. ,-(n-l) -(n-2). d (f, , f„) = the composition C , TT . (B , B ) - 1 2 n-1 n-1 d (f , f ) -3 * TT (F) . n-1

1 = Af^ - (Af,,) • Then Sd"" ^,

Proof. We shall only prove (1)> (2) follows almost trivially by checking the definition of dn 1 (£^r an,3 diagram 4.1.9. Consider the following diagram:

n £5 n ' s n n-1 n

•i# L2# "3#

p. Mnn>) ^ Mn) Mn) & ,*)< W ^2% 1 3# H TT (B , *) rr (E "', F) * TT .(F) n, n •—f n n-1

1# i_ Hi.- c «_2 /^n+1) Mn) ^3# Mn+1) Mn)^l ^(n)

for n _> 2 . 0 Observe that it is commutative. First 9„.ui.,, = by exactness. 3# 4#

.0 13 6 (f) = tt(f>3 - fa^ii^a - «2#( Vi#>"

3 *fi3#83# 4#P3#*21

i = ?i3#(33# 4#>35#P3#*t21 " °'

4.2.9. Now, we shall pass on to describe the corresponding

Mn-1) homology obstruction to extending f: B • E to a cross-section Mn) Mn-1) over B . We assume that F is (m-1)-connected and B is (q-1)- connected, and n <_ m+q-2, so that we can apply 4.1.2 and therefore

4.1.6 makes sense. Using the commutativity of diagram 4.1.7, we define

vn(f) : C (B, B ) tZ-M'rr' (F) * H _ (F) n o n-1 n-1 to be the map shown above; i.e., it is the map making the diagram, 49

w (f) -*• TT .(F) C (B, B ) n o n-1

v"(f)

C (B, B ) **" H (F) n o n-1 commutative,

4.2.10. vn(f) is clearly a cocycle by 4.2.8; and by the commutativity of diagram 4.1.7 we observe that vn(f) is also the composition,

1 p* -(n) -(n-1) (4.2.11) C (B, B ) -rr-> H (EW, E( ' ) »• H . (E(n 1]) • H , (F). n o = n n-1 n-1

4.2.12. THEOREM. In addition to the hypothesis of 4.2.9, suppose B is (p-l)-connected and n < m+q-2, m+p-2. Then under the o —

K n canonical homomorphism, : H (B, B ; H n (F)) y Horn(H (B, B ); H (F)), o n-1 n o n-i K(V n(f)) is the following composition,

(4.2.13) H (B, B ) y H (E, E ) »• H .(E) y H . (F) no ^ no n-1 o n-1

Proof. The canonical homomorphism, K, is given by K(Y) = Y

— n — for any y e H (B, B ; H ,(F)), where y is constructed by picking a o n-1 representative y of y such that the following diagram is commutative,

(0N(B, BQ) -*• 2,(B, B ) -y H .(F) n o n-1

/ Y H (B, B ) n o 50

Y is unique upto isomorphism. Nov;, we only need to show that if we — — n n ^ ^n take y = v (f) and y = v (f) , then y = \> (f) will be equal to the

composition , '

H (B, B ) = H (E, E ) > H .(E) • H (F) no n o n-1 o n-1

Consider the following diagram:

(4.2.14) v (f) PT* (n) (S , S (n> g (n 1> y C (B, B ) = H ^^H (E , ^^Jl^H 1(^ - )-^H , (P) n o n = n ' n-1 . n-1

> n Mn-2h 2* ,Mn Mn-2)4 2* ,Mn-2L Z (B, B ) == H • (B , BV -^T— H (E , E ) > H (E ) non = n , n-1.

MnVn); MOlU;) 3* MnW ) MOl ) 3* MO) .(E) H (B , B ) -fr-^ H (E , E ') —^-»- H (E ) = H n n n-1 n-1 o

MO) P4* MD) 4* MO) H (B, B ) = H (B, B ) •< - H (E, E ) • H ,(E^;)=H .(E) n o n s n n-1 n-1 o

All maps are defined here since n < m+p-2, m+q-2. p is an isomorphism

since n <_ n+m-1. p^A is an isomorphism since n <_ (n-1) + (m-1) for Mn) MO) m > 2. (B , B ) is Min(n-1, p-1, q-1)-connected; and so p is an — 3* isomorphism. Similarly p is an isomorphism. We remark that we may take fe> ,(B, B ) = C (B, B ) , (B, B ) = B (B, B ), and n+X o n+x o n o n o

z B an "SJ^fB, BQ) = n( / BQ) /* <3 by assuming the commutativity of the diagram we see that K (v (f)) = fS.J?... . We observe that the diagram 4.2.14 .4 4. is commutative except for the following portion . (n 1) h ,(F) H 1(^ - )" ~ H n-1 n-1

H _ (E ) n-1 o

-(n-1)

We know is (p-l)-connected and B is (q-1)-connected. Then we have the following splittings and maps. ( See 4.1.2. ) {i*1' P*} proj. H . (E ) > H . (F) © H, (B ) >• H , (F) 3 0 eg 3 30 3

1 r.-l , 1 p pro: • ft(n-l), . *__'_- *f „ ,„s^„ ,-(n-lN ) H (Ev" ') • Hj (F) e Hj (B " """ ) > Hj (F) j <_ n + Min (p, q) - 2.

The above diagram is obviously commutative. This completes the proof of 4.2.12.

4.2.15. Remark. 4.2.12 shows that <(v(f)) is independent of the GCW decomposition. It is not easy to see or show how the higher obstructions can be shown to be independent of the GCW decomposition.

§4.3. Extension Theorems

In this section we collect together the main results concerning onstructins to extending cross-sections to a fiber space, (E, p, B, F), and give a characterisation, analogous to that of the classical obstruction defined on a fiber bundle using its local homotopy properties. 4.3.1. Homogeneity Property.

Let (E, p, B, F) be a fiber space. Assume B to be a CW complex

N B C and take for irs GCW decomposition the CW skeletons, { B }. Let 0 B

be a subcomplex. Then Bq is adapted to such a GCW decomposition. Suppos

N U N n f: B ^ Bq »- E (n >_ 2) is a cross-section over B Consider w (f) e cN(B, B ; TT , (F)) as defined in 4.2.7. Then o n-1

(1) ojn(f) = 0 =£• f can be extended to a cross-section over a. n B .

N ; TT (2) If d e c (B, BQ ^ (F)) is another cocycle such that d

n i—n—2 is cohomologous to w (f) , then f|B •> E can be extended to- a cross-section, f^: BN 1 >• E, such that wn(f) = d.

4.3.2. THEOREM. Let (E, p, B, F) be a fiber space, B, a CW complex with a GCW decomposition, B'^CB'^C '** CB^'c" C B, where B^ are subcomplexes of B. Assume F is j-simple for j <_ n-1 .

(n—1)

Then a cross-section f: B >• E can be extended to a cross- section over B^N^ when its homotopy obstruction, wn(f)> is zero.

Proof. Take the skeletal decomposition of B^. Then we know (n—1) B is adapted to such a GCW decomposition. Apply 4.3.1 to get the existence of an extension, f., to f, ,: B^'3 ^UB'" > E. Since 3 3-1 tip' (f ^) E H^CB^, B^N ^ ) = 0 for j ^ n, the inductive application

to n X n 0 of 4.3.1 gives an extension, %n_^r °f f B^' . Now, io (f) = implies wn(f ) = 0 ( see R. Hermann [18] ); and another application n—1 of 4.3.1 completes the proof. §4.4. n-systems

We shall proceed to show that we can define secondary and higher obstructions by utilising a notion due to Moore-Postnikov of decomposition of a fibration, and to show that the obstructions are somehow related to the k-invariants of the fibration under suitable conditions. In this section (E, p, B, F) is a fixed fiber space with a connected fiber and with B being a simply connected CW complex.

4.4.1. Definitions. We shall denote by B, the skeletal k

GCW decomposition of B.

We have seen the idea of killing homotopy groups of the fiber in §2.4. We now generalize" this idea. (cf. [43]). For a given space, F, the pair, (F g ) , where F is a space and g a. fiber map, n n n n F" > F > F , will be called a n-Postnikov system for F if n n

(4.4.2) gn#: ^ (F) • ^(Fj is an isomorphism for k <_ n and ir^ (F ) = 0 for k > n. It then follows that

ro , if k < n u, (F') k n (F) , if k > n

We shall denote such a Postnikov system for F by (F, g , F^, F^) when there is no ambiguity, (cf. §2.4).' 4.4.3. The triples, (E , F , h ), which denote a family indexed c n n n by n, where

F E -> B and

n F' •+ E -+ E n . n are fiber spaces such that the following diagram: 9*. n -> F n

F -y E -> E

is commutative, and where (F, h F, F , F') is a n-Postnikov system n1 n n for F, are called a n-Postnikov system for E. This is the essence of

Moore-Postnikov decomposition of fibration.

4.4.4. Application of the homotopy exact sequence to the fibration, F' y E >• E , together with 4.4.2 gives that n n h : TT (E) > TT (E ) is an isomorphism for k < n. n k k n —

Call the characteristic class of the n-Postnikov system,

gn F' y F y F , the n-Postnikov invariant of F, sometimes also n n n+2 n+ TT called the k-invariant of F. Denote it by k e H ^ v(F , (F)) . . jr n n+1

That is, kn+2 = T(I ,) , where a ,, e Hn+1(F''; TT ,. (F)) is the n+1 n+1 n n+1 fundamental class of F" and T is the transgression of the fibration,

gn n F' -y F • • •> F . Similarly, call the characteristic class of the n n h n-Postnikov system, F" >• E —E , ic^2 e Hn+2 (E , TT . (F)) , the n-Postnikov invariann t for E. n n n+1 4.4.5. Since h F = g we see that k , restricted to F, is n1 • n , n+2 . . _ *-n+2. . , -n+2 .

k . -invariant for F; h^(k ) is zero, because k is in the image

of the transgression and this follows either by Serre exact sequence

or by argument in § 2.4.

4.4.6. Thus, if we are given a cross-section, f: B y E

Oi i a* n+2 n+2 E B TT of (En+1, P| n+1' n+1> E) , let to (f) e H (B; ^+^ (F)) be the

obstruction to extending f over B _ . h of: B , >• E will be a • n+2 n n+l n cross-section to the fibration, E ——> B ; and since rr, (F ) =0 for n k n k > n , h f can be extended to a section f: B > E by 4.3.1(1). n n We shall show in the next section that

^*,-n+2. to(f) = f (k ) .

§4.5. Secondary Obstruction and Construction

of a Postnikov System

In this section, F ——>• E -P-+- B, is a fibration with F

satisfying n, (F) = 0 for m

exact sequence,.we can deduce the following technical lemma.

4.5.1. LEMMA ( cf. 128, §10.7] ) . Let f: B , * E be a n+J.

cross-section ( assuming it exists ) and let G be an abelian group.

If a e Hm(F; G), then there is an unique class, a, in Hm(E; G) such

that * * i (a) = a and f (a) = 0. This class is the same for any two cross-sections that agree on B^ . 4.5.2. Construction of a Postnikov System.

Let a(f) be the unique class given by 4.5.1 such that * * rn m m f (a(f)) = 0 and i (a(f)) = i e H (F; rr (F)) . ( x = fundamental m

1 class of F ) . Let v" e Hm(K(Tr (F) , m) ,* TT (F) ) be the fundamental clas m m • * ^ Consider a map, g: E > Ktir^CF), m) , suchthat g (i ) = a(f). g

exists by the well known classification:

HM(E, Tr (F)) < y [E, K(TT (F) , m) ] m m * Mil given by the correspondence, g (i ) •* >• [g] . Then g, restricted to

F, is an ri-Postnikov map for F. (If it is not a fibration, we can

always convert it into one; however, it is always a fibration in this

case.) Thus we have the following situation:

(4.5.3) m -y F = K(TT (F) , m) m m

gxp = h E > K(TT (F) , m) xB = E m . n

pro3.

•y K(TT (F) , m) m

This is precisely a n-Postnikov system for E by 4.4.3.

Mi+2 n+2 Let k e H (E ; TT (F)) be the corresponding Postnikov n n+1 invariant for E. Let X: B '-y E be the inclusion map and let n -> E be an extension of h f: B , -y E. Denote by pro j . : n n n+1

E y K(TT (F) , m) the obvious projection on the first factor. n m Then (proj.of") (i ) = 0 by 4.5.3, i.e., f - A . Thus we have the lemma.

* o.h+2 * 'Vn+2 4.5.4. LEMMA. A (k ) ( - f Ck ) ) = 0)(f) .

§4.6. A Postnikov Decomposition

7! an< Let (E, p, B, F) be a fiber space. Assume n^^ ^ 0 3

TT (F) = 0 for k fi { n(j) }, where n(l) < n(2) < n(3) < If n(l) = 1 , we assume TT^ (F) to be abelian. Consider the following decomposition, outlined in chapter two :

, n(i))

(F), n(i-l))

, n(l)) where h., restricted to F, is g for i = 1, 2, 3, ••• ; and i i (F, g., F. = K(TT ...(F), n(i)), FT) is a Postnikov system for F. Let ii n (l) i p^: E^ >• B be the fibration, determined by the decomposition, which has F. as fiber, l

4.6.1. Consider a CW decomposition, B^ C B^ C C B .

Suppose f: B ... > E is a cross-section. Then the obstruction n(})

n +1 7r F If = tne cohomology class, u)(f), lies in H ^ (B; n(j)^ ^* J 1/

first obstruction class, a/1 ^+X(f), is simply the characteristic class of the fiber space, p: E »- B.

Consider the diagram,

h j-1 E J > E. y1 I

I I

B ... r- B We have an extension,n (3f) t< •B >- E. , , of h. J: B ... >• E. , , D-l 3-1 n(3) 3-I since h^_^f has no obstruction. We wish to relate the secondary and higher obstructions to the first obstructions of some n-Postnikov maps of type F' *• E , »• E . * n n+l n

4.6.2. Conversely given f: B > Ej_-j_' a cross-section of

(E. , p. , B, F. • ), let £.. denote f"|B Then f., can be lifted 3-1 3-1 3-1 1 n(3) 1 to E, since it has no obstruction to lifting. 4.6.3. Denote by kj_1(E) e Hn(j)+1(E. . ; TT ...(F)) the j-l n(]) characteristic class of the Postnikov system,

K(rr ...(F), n(j)) >• E. »• E. n(;j) J j j-l . We have the following proposition relating the obstruction with the k-invariant of E.

4.6.4. PROPOSITION, u) (f) = f "* (k3"1 (E) ) , where f." is a cross-section of (E , p. ,, B, F._ ) and also an j x j A j i extension of h. ,f: B ... >• E. , given as in 4.6.1. 3-1 n(3) 3-1 Proof. See R. Hermann [19].

^1 4.6.5. We shall proceed to give a characterisation of k (E) e

Hn (BXK(TT .(F), n(l)), TT ..(F)). With the previous assumptions, n i A; Yi \ Z) suppose there is a cross-section, f: B y E, over B ... By n(l) n(l) lemma 4.5.1,

n (1)

(4.6.6) there exists unique a(f) e H (E; ^(u^)) such that

(1) f*(a(f)) = 0, and

(2) i*(a(f)) = in(1) e Hn(1)(F; rr (F)) . n(l) * r\j ^ Take g_^: E y K^7^^ (F) > n(l)) such that ^(i-^) = a^f^' where is the fundamental class of K(TT (F) , n(l)). Define E, to be the product n (1) 1 BXK(TT (F) , n(l)) as we have done before in 4.5.3 and obtain the nil; fibration given by h^ = 9^XP= E *" E^, and the corresponding k- 'vl n(2)+l invariant k (E) e H (BXK(TT ,,, (F) , n(l)), TT (F)) of the , n(l) n (2) hl fibration, F^ • E y E^. -1 We have the following useful characterisation of k (E), if

7r /0\ W ===' Z or Z £ a positive prime.

4.6.7. THEOREM. With the above hypothesis of 4.6.5 and 4.6.6 <\,1 v/e have that k (E) is uniquely determined by

(1) h^ic^E)) = 0 , and —1 (2) k (E) , restricted to K(TT .,. (F) , n(l)), is the Eilenberg- n (1) MacLane invariant of F, provided it does not vanish.

4.6.8. Let X: B > E^ be the injection. Then, by 4.5.3, we

* 'Vl

see that X (k (E)) = u(f). Recall the following fact that the extension,

f, of h of satisfies f'*(k1(E)) = io(£). (4.6.4)

We look at the following example for a spherical fibration, n (1)

F = S >• E • B, n(l) > 2 . S. D. Liao [28] gave a formula for a spherical fiber bundle with structural group, the orthogonal group, as follows:

p*(u(f)) = Sq2(a(f)) + p*(y(Sq2a(f)))^a(f) , where ¥: Hn^+2(E; Z^) >• H2(B; Z^) is the "integration over the

fiber" homomorphism. Apply 4.6.7 to x = Sq2(a(f)) + p ¥(Sq2a(f))a(f). * n(l)+2 Using the Gysin sequence we get x = p (b) for some b e H (B; Z^),

n +2 n +3 E z because 6: H ^ (E; Z^) > H ^ (E, n^1j/' 2^ takes x to zero.

Thus, h*{ proj.*Sq2 (i ) + ¥ (Sq2a (f j) + pro j. *b } = 0 in By

4.6.7 we get

p (u(f)) = p (X kX(E)) = p (b) = x . 61

Example 4.6.8 is taken from R. Hermann [18]; the reason is that we can compare the way, S.D. Liao [28] obtains this formula, which is essentially being used in theorem 4.6.7, with the ease that

4.6.8 obtains the same formula. However, Liao gave several characterisations, in particular, an explicit description of

¥(Sq2a(f)) [28, §19.3] .

§4.7. Localisation

The above example leads us to the method of using reduction mod SL (SL a prime) . There has been a lot of work done in this direction, especially that of Serre, and studies made, in the relation of the

Steenrod algebra to reduction mod £ homotopy with the use of the

"natural homomorphism of the second kind" from homotopy groups to homology groups, and characterisation of certain spherical homology classes.

( Chou [13], Mahowald [30, 31], Thomas [59], Yo [62], Yu [63] ) The latter is important to a later- chapter and also to applications to question of the realizability of a spherical class as an embedding.

We shall only sketch the idea of reduction mod SL in obstruction theory.

4.7.1. Notation as before. (E, p, B, F) is a fixed fiber space.

B E Suppose f: ^+^ > is a cross-section and i a prime number. Let

w(f)^ denote the obstruction class reduced mod SL corresponding to w(f) in Hn+2(B; TT (F))Z„. First a technical lemma. 4.7.2. LEMMA. Suppose TT . (F) Z = 0 for j <_ n, i.e., rr. (F) has

no £-torsion for j <_ n. Then to(f)^ is transgressive. In fact, w(f)^ =

T(.I^) where \^ is the reduction modulo I of the fundamental class, \,

of Hn+1(F; TT . CF) ) . n+I

Hence i represents a homotopy class of map F > TT (F) ©Z

Thus we can employ the method of §4.6 to the secondary obstruction

mod £ to the case when TT , (F) ®Z = 0 for j

m P case of a spherical fibration, F = S y E y B, m >_ 2, we can

show,.similarly as in 4.6.8, that -if f: B , y E is a cross-section n+1 then we have Liao's formula:

(4.7.3) p*(03(f)£) =

§4.8. Secondary Difference Obstruction

We want to show how our previous discussions can help to

describe difference obstructions and their properties.

4.8.1. Notations and assumptions as in §4.6. Let (E, p, B, F)

B y E are two be a fixed fiber space. Suppose f^, f^ : n+1 # cross- sections. Assume F is (m-1)-connected and TT . (F) = 0 for m< j < n . D — Then, by 4.6.6, we have the following fibrations:

E BXK TT h = pxgi: y E = ( (F), m) , and n 1 n m = Pxg„: E y E = B*K(Tr (F , m) , n 2 n m — . ^n+2 Mi+2 n+2 and corresponding k-invariants, k, (E) , k„ (E) e H (E ; TT , (F)) . 12 n n+1 * rn Tn a(f±) Thus, by 4.6.6, i (a(f2) -a(f )) = i - i =0. Hence a(f ) - p (b) for some b e TT (F)) by exactness. Let i E m m M H (K(iT (F) , m) , TT (F)) be the fundamental class and let a: E mm n • y K(TT (F) , m) be the projection; then (aoh') \ = a(f„). Thus m n m 2.

b = f*p*(b) = f*(a(f2) - aC^)) = f*a(f2)

* * -ki\j = 'f.H' a T In m

Let Y, : B *• K(TT (F) , m) corresponds to b under the isomorphism, b m Hm(B; TT (F)) < > [B, K(ir (F) , m) ] , given by X 7 < > X ; that is, m m m Y, i = b . Then y, \ = (ah'f_) i implies that bm bm nlm B (4.8.2) YKI1 ^. - oh'f b n+1 n 1

G : B Consider the graph of / (Yj;)) *" , given by

G(YB)(b') = (b', Yb(b')) for all b' s B. Then, by 4.8.2, 4.6.2 and

4.5.4, we have the following lemma.

* ^n+2 4.8.3. LEMMA. G (Y, ) (k' (E)) = to(f,) . b 2 1

TT 4.8.4. From here we put additional conditions on N (F) . We n+1 stipulate that TT (F) has a pairing, TT , (F) ® TT (F) * TT (F) , n+l n+l n+1 n+l so that we can apply the Kunneth theorem. Let w denotes its cup- product with respect to this pairing. We assume also that N "vr\+2 '' n * * n+2 n+ (4.8.5) k2 (E) = I pn(b )wv (6 ) £ H ^(En; ir (F)) , j=0 2 2 * where V: E >• K(TT (F) , m) is the projection, b. E H (B; TT . (F)) , n m j n+l IT 9. e H*(K(f (F) , m) , ,, (F)) , dim(bA) = 0 and 6, = n+2 . 3 m n+1 0 N N * —n+2 r * 4.8.6. PROPOSITION. G(y.) (k (E)) = I b.v/y, (6.) , and ^ j=0 3 3

N * •

o)(f ) - w(f) = I bvy.ie ) . j=i ° J Proof. The first statement follows from the definition of

Gty. ). The second comes from the first and the fact that b * 'Vn + O * Vn+?

* rhn+"?

Oi(f±) = G(Yb) (k2 (E)) (4.8.3). . * "Vn+? * * * * * *

For X (k^ (E)) = X p (bQ)^X V (0Q) = b^X V (6Q)

= Vf2V(V

= b0^Y^(90) bY 4.8.2 ;

it is then easily seen that the second statement is true. This

completes the proof.

We are now ready to state the main result of this section.

' * 4.8.7. Every class 0 e H (K(TT (F) , m) , TT ...(F)) determines a m n+l primary cohomology operation. ( See for example [44].) Denote it by

m+3 the same letter, 0: Hm(X; TT (F)) >• E (X; TT (F)) . Letting X = B , m n+l * we see that YMOJ) corresponds to 0.. (b) . Thus, with previous notations and assumptions, we have the following theorem. —n+2

THEOREM. Suppose k (E) satisfies 4.8.5. For a fixed cross- section, f: B , y E. there is a cross-section over B _ iff there n+l n+2 is a cohomology class, b e Hm(B; ^(E)), such that

(4.8.8) a)(f) + I (b) = 0

j=l 3 D Proof. ( => ) By * % 4.8.6. l (<= ) Let y corresponds to b = y ( m) • Then

Y: B y KCTT IF), m) gives a graph, G(y): B y E . Since m n h^: E y E^ induces isomorphisms for homotopy groups in dimension less than or equal to n, G(y) can be lifted to a cross-section,

B E wit h G Y B fl: n+1 ' ^ ^l n+l = hnfl' T]ie:n 4-8'6 and 4.8.8 apply to give w(f.) = 0 ; and so it has a cross-section over B ... 1 n+2

We have so far not consider any obstruction for principal fiber bundle, nor did we use any theory of classifying spaces and classifying bundles to a large extent. We shall proceed to the next chapter describing obstructions to extending cross-sections of principal fiber bundles. The technique follows more or less the

Moore-Postnikov type decomposition. CHAPTER 5. PRINCIPAL.FIBER.BUNDLES

In this chapter we shall discuss an obstruction theory utilising the idea of a Postnikov resolution which has now become the most powerful tool in algebraic topology. Among the principal fiber bundles discussed are U(n), SO(n)-and SP(n) bundles. We shall more or less state the classification theorem for principal G-bundle; and much is assumed about the construction of universal bundles and classifying spaces [41].

§5.1. Definitions

A principal fiber bundle, £ = (E, p, B, G), in the sense of

Steenrod, is a fiber bundle with the fiber, G, a topological group, acting continuously on E and compatibly with the co-ordinate neighbourhood system of E; that is, the local trivialisations,

{ (Ji^: U x G •> p 1 (U) }^ , satisfy the folloing conditions:

If b E D, and g,, g e G, then 1 2

tb g )= (b g g ) CD 32**u ' l *u ' 2* l " (2) g«x = x for all x e E g = e .

C3) g2«(g1*x) = (g2'g1)x for all g , g2 e G and any x e E

We shall always make the following assumptions in this chapter unless otherwise stated. If £ = (E, p, B, F) is a fiber bundle, the base space, B, is always a locally finite connected simplicial polyhedron. Bn will denote the n-skeleton of B. We shall also assume 67

that the fiber, F, is connected and is n-simple in all dimensions.

We shall be dealing with fiber bundles in the sense of Steenrod and so we need some facts about pair of fiber bundles over B.

§5.2. Preliminaries

5.2.1. COVERING HOMOTOPY THEOREM. Let (£, where £ =

(E, p, B, F) and = (E', p% B, F') , be a pair of fiber bundles over the base space, B. Let X be locally compact and paracompact, and suppose X is closed in X. If f: (X, X ) y (E, E') is a lifting of o o X > h|xx{0} : X > B, then there exists a map, H: (Xxi, QXI) (E, E') , such that H|XX{0} = f and poH = h .

Proof. Apply the usual covering homotopy theorem and the covering homotopy theorem, given as in chapter one.

5.2.2. LEMMA. Let (£, %,') be a pair of fiber bundles over B; and suppose B^ is a subspace of B. Let k:B^C B denote the inclusion map. Then, for m >_ 2 , the following induced map:

•*• TT (E, E') is an isomorphism and we have the splitting:

TT m (E, E'') © TT (Bm, Bo ) . Proof. Consider the diagram:

TT (E, E") o m

k_ rr (E, E' ) m B o

TT (E , E" ) m B m B B o o o

j. is an isomorphism for m > 1 , since rr (E.E ) ^ TT (B. B ) ~ * — m B o '— mo —

Trm(E, Eg ). This implies is onto and j# is a monomorphism. So for o

2 m >_ this gives the two short exact sequences shown. kA is onto by the butterfly lemma and the fact that k^ is onto, k^ is also a mono• morphism. Because if k (x) = 0, then k'k (x) = 0. So by exactness TT 0 there.exists y e (E', E' ) such that k. (x) = jA(y). = j'k (x) m j*j*(y) = 0 implies y = 0, since j is an isomorphism. Thus k^(x) = 0,

0. 0 i.e., x = Hence Ker( k#) = and k^ is a monomorphism.

Thus we can construct a split exact sequence,

0 »- TT (E', E" ) -> TT (E, E' ) -> TT (E, E') -> 0 m B m B m o o and so we have the splitting,

TT (E, E" )^TT (E, E') 9 u (B, B ) m B m m o o 5.2.3. COROLLARY. Hypothesis as in 5.2.2. Taking Bq = *, we see that for m > 2 we have the following isomorphism:

k. : TT (F, F') 5- rr (E, E") . * m m

With. 5.2.2 and 5.2.3, one can easily prove the following characterisation of the simplicity of the pair, (E, E').

5.2.4. THEOREM. (1) Suppose (F, F") is m-simple. Then TT^ (B) acts trivially on TT (F. F') iff (E, E') is m-simole. m C2) If CF/F') is (m-1)-connected and (F, F") is m-simple, then the induced map, k.: H (F, F") > H (E, E') is an • * m m isomorphism if and only if TT^ (B) operates trivially on 17 (F* F')»

The proof of 5.2.4 relies on the following proposition, which we state here for future use.

5.2.5. PROPOSITION. Suppose (F, F') is m-simple. Let to e

F TT (X E TTM( / CE', x) , P Q) = bQ, o F') and k: (F, F') >- (E, E') be the inclusion. Then

k* (p^ ( to ) • ct ) = to © k# ( a ) ,

TT X TT where " © " denotes the usual operation of . (E'7 ) on (E, E', x ) 1 o mo C see Steenrod 154 ] ) .

Proof. See Lundell [29] .

§5.3. Obstructions for Pair of Fiber Spaces In chapter 3 we have discussed an obstruction theory to the existence of cross-sections for fiber spaces. In an analogous way

( N. Steenrod [54] ), if given a pair of fiber spaces over a simplicial

a. complex, one can define a local coefficient system, t tF, F-*) , over B and develop an obstruction theory for pair of fiber spaces.

Let (.£, £') be a pair of fiber bundles over a simplicial complex, B, such that the pair, (F, F") is (n+l)-simple. Then, if

n f: (B, B ^Bo) *• (E, E') is a cross-section, there is a deformation cochain (. in fact a cocycle ) dn+1(.f) e cn+1(B, B ; TT (F, F')) o n+l [29, 54] having the following properties.

5.3.1. (1) dn+^" (f) = 0 iff there is a cross-section fB »- E satisfying f'(Bn+1UE ) C E' and f - f rel (Bn *J B ). o o

n+1 n+1 (2) f =- g rel Bq =>d (f) = d (g) .

(3) 6dn+1(f) =0. — n+l

(4) d (f) is a topological invariant.

(5) If d (f) = 0, then f can be deformed relative to

n n+1 B \, Bq to a cross-section, f: B > E, such that f(B (jB ) d. E".

( See chapter 3, 3.2.1 or [54].)

We next relate the connection between the deformation cochain n+l and the fundamental class, i , of (F, F') in case (F/F') is n- connected.

Consider the boundary operator, 9„: TT (F, F') >• TT (F') . # n+l n In a natural way, it induces the following map of local systems:

rr , (F, F") > TT (F') and hence a map, n+1 n n+1 'v n+1 ^ N+± » 9#: C (B, BQ; ir^tF, F") ) c" ^(B, BQ; IT^F')), also denoted by 5 . tr

We have the following proposition connecting the deformation cochain with the obstruction cochain.

5.3.2. PROPOSITION. Suppose f: (B, BN O B ) >- (E, E') is a o ~, ^ ,n+l . „. n+1 , _ I _n „ . cross-section. Then 9,.#d (f) = c (f ' B <->Bo ) .

Proof. Straightforward. Just check the definition, of c (f|B u B ).

From now on, we assume (F/F') is n-connected and (F, F') is

(n+1)-simple. Then 5.2.4 tells us that TT^(B) acts trivially on

TT , (F, F') if and only if (E, E') is (n+1)-simple. So we assume TT. (B) n+l l operates trivially on TT (F, F") , so that 5.2.4(1) and 5.2.4(2) hold; n+l i.e., (1) TT . (E, E") is simple; and (2) H.(F, F') >-H.(E, E') is n+1 j j an isomorohism for j £ n+1.

n+1 5.3.3. LEMMA. Let i be the fundamental class of (F, F"). — n+1 * *—*—1 n+1 n+1 Then d (f) = f k i (i ) e H (E, E' ; TT (F, F )) and B n+1 o — n+1 * *—*-1 n+1 n+1 c (f) = f (3„(k i -"(i" X))) e H (B, B ; TT (F')), where # on

1 n+1 n+1 i*" : H (F, F'; *N+1&, ) — • H (E, E'; ) / k*: Hn+1(E, E'.- TT (F, F\)) + H^CE, E' ; TT ^ (F, F')) and n+1 • — B n+l * n+1 n+1 0 f. : Hn+±(E, E' ; TT (F, F')> »• H" X(B, B ; TT (F, F')>. B n+1 o n+1 o Proof. See [29]

n+l *—*—1 n+l If we define A (£,C) = k i (l ), then 5.3.3 gives — n+l * n+l — n+l * n+l n X LK, V n n+J r). (5.3.4) tf (f) = f X )i c (f) = f 9#A -(£,

§5.4. Applications

In this section, we shall talk about principal fiber bundles, state the well-known classification theorem and describe the application of the last sections to principal G-bundles.

5.4.1. THEOREM. (1) A principal bundle, (E, p, B, G, G), is equivalent to the product bundle iff it admits a cross-section.

(2) (Classification) There is a one-one correspondence between cross-sections of a principal fiber bundle,

(E, p, B, G) , and maps,' { h: E > G; h(g.x) = g.h(x) for.all x e E and for all g e G }•

We assume the definition of a principal map between principal bundles due to Steenrod; i.e., it is a triple (, k, I) such that the diagrams: . u G - G G x E - > E

xk

-> E' G' x E' -> E'

B -y B are commutative. Then the map, h^: E. >• G, corresponding to a cross- section as in (2) of theorem 5.4.1, is given by

h^(x)'fp(x) = x for all x e E

[7, 54]. With, this classification, we have the following immediate consequence.

5.4.2. COROLLARY. Suppose B = B", £ = 1 and notations are as B in 5.4.1. £) is a pair of principal fiber bundles over B. If £ has a cross-section, f: B > E, then .f = kf is a cross-section and

hf = h Jc.

5.4.3. With, this corollary, one can define a deformation cochain for principal bundles as a first step towards a classification of cross-sections, {f: B >• E}, which agree on a subspace or sub-

complex, BQC B.

Suppose f^, f^: B • E are cross-sections of £ = (E, p, B,

f B Let G, G) such that fQ|B = xl * hf , hf : E >• G be given by 5.4.1. o 1

Then define h^ f : B *• G to be the composition, h^ of^. The o 1 o following is a classification theorem.

THEOREM, f - f, rel B h, ^ - * rel B . o 1 o f f, o o 1 Proof. See [29, pp.170].

Thus, with this theorem, one can define h : (Bn u B , Bn "*"UB 0 1 ° ° >• (G, e) if one is given cross-sections, f , f, : B uB > E o 1 o n—1 uB which agree on B 0* With this map, one can define a cochain, n n — d (f , f ) e C (B, B ; TT (G, e)) unambiguously by ox on n — i • by d (f , f )(a) = h f |(a, a) for all a. This definition coincides ° 0 1 with that of the difference obstruction due to Steenrod; it has the properties given in chapter 3, namely 3.1(4), 3.1(5), 3.1(6), 3.1(7) and 3.1 (10).

The following proposition relates the fundamental class of G with the difference cocycle. The proof is analogous to that of lemma

5.3.3.

5.4.4. PROPOSITION. Let (E, p, B, G, G) be a principal fiber

n+l bundle with the fiber, G, (n-1)-connected. Suppose f , f, : B >• o 1

I I 1 n+1 n are cross-sections such that f B" = f, B" . Then hr r : (B , B o1 11 f f, * _ 01 n n f ) n TT • (G, e) is defined and hf f (i (G)) = d (fQ, e H (B; (G)) . o 1

§5.5. Decomposition of Principal Fiber Bundles

and Moore-Postnikov Invariants

In this section, we shall begin with the following results about universal bundles. Throughout £ = (E, p, B, G, G) denotes a principal bundle with fiber and structural group, G. G is assumed to be a connected countable CW-group in the sense of Milnor [41].

5.5.1. THEOREM. Any countable CW-group, G, has an associated universal bundle, ^ = (EG, p, BG, G, G), where BG is a countable CW- complex and EG is contractible. ( Milnor 141, II§5] ) 75

We remark that there is also an universal spectral sequence by which one can relate the group, G, and its classifying space, BG ( See

[7] or 141]. )

Our method will depend heavily on Postnikov decompositions.

The following proposition guarantees the existence of a Postnikov system for G; thus we can then use this to obtain the desired decomposition and hence to describe an obstruction theory for principal bundles with fiber and structural group, G.

5.5.2. PROPOSITION. Let G be a connected countable CW-group.

For any integer N ^_ 1 there is a finite CW-group sequence,

^ YN „ YN-1 „ „ Yl G = G . y G y G , , >• »•• G„ y G, , N+l N N-1 2 1

ls an which is a Postnikov system for G, where each Yn embedding and

G is homotopically equivalent to G = G ^*

Proof. See Lundell [29].

5.5.3. In particular, if we are given a principal bundle, £, over a CW-complex, B, with fiber and structural group, G, then we have a Postnikov decomposition for £, which we may assume to have been induced from the Postnikov system for G, G C GN+1 ° GN CVlC CG1 * By 5.5.2 we may even assume that { (£ . , £ ) } are pairs of principal n+1 n fiber bundles over B, where £ = (E , p , B, G , G ) in the Postnikov n n n n n decomposition [ Chapter 4 ]. In the terminology of chapter 4, (E , G , g ) is a Postnikov system for E; i.e., the diagram, n n n

> G n-1

> E n-1

is commutative, in particular,

0 TT . (E . , E ) = for i j4 n+l ; 1 n-l n

TT . (E , E ) = TT (G) n+l n-1 n n This finite decomposition is abbreviated (£, N, {£ }, {(y , g , 1)}), m mm a decomposition of length N.

We would like some kind of compatibility condition on this decomposition. Kahn [25] has shown that given two Postnikov decompositions on two fibrations, then if there is a map from one to the other, it induces a map between the Postnikov systems. Although this construction is not functorial, we are still contented since we only need to know that if we have a map from one system to the other, we are guaranteed a relation between the k-invariants.

5.5.4. LEMMA [25; 29]. Let OS, N, {£ }, {(y , g , 1)}) be a n n n decomposition for £ = (E, p, B, G, G) . Suppose f: B' >• B is a map.

Let f (?) be the induced bundle. Then f." ) are the bundles in a n * decomposition of f (.£) . This lemma tells us that IE, N, {E }, {(Y , g , 1)}) is induced n n n by a classifying map,

So from now on we assume we already have a fixed decomposition for the universal bundle over BG and any decomposition over £ = (E, p,

B, G, G) is induced by its classifying map, ti : B * BG.

5.5.5. PROPOSITION. Let E = (E, p, B, G, G) be a principal bundle over a CW-complex, B. Then there exists a cross-section, f: Bn >• E iff there is a cross-section, f: B ——> E , , of £ , = n-1 n-1

(E »,p n,B,G »,G _) which is a term in the decomposition of E n-l n-l n-1 n—1 of length N >_ n-1.

Proof. See Hermann [18] or 4.6.1.

The footpath has been set; the rest of the work is analogous to that used by R. Hermann to calculate the obstruction.

5.5.6. Since G _/G is n-connected, we see that (G ., G) is n-1 n-1 (n+1)-simple and TT (B) operates trivially on TT (G , , G) . Hence l n+1 n-1 —* n+1 n+1 i : H (E -,.E; * (G , G) ) > B. [& , G; TT , (G , G)) is n-1 n+1 n-1 n-1 n+1 n-1

an isomorphism. Suppose Bq is a subcomplex of B. Define as in 5.3.4,

.n+l,,. ... *-r*-l, n+lv n+1, . .% K (5 . , K) = k i (x ) e H (E _ , E ; TT CG . , G)) , where B n-1 n-1 o n+1 n-1 *° n+1 v n+1 k : H (E , E; ir . CG . , G)) H (E , E ; TT . (G , G)) is n-1 n+1 n-1 n-1 o n+1 n-^1

induced by inclusion, and i is the fundamental class of (G ,# .G). n-l

Since 3„: TT IN (G , , G) >• TT CG) is an isomorphism ( 5.5.2 or 4.4.2 ), # n+1 n-1 n define X*+1(£) = 3„x"+1(E . , E) z Hn+^" (E E : TT (G)) . B # B n-1 n-l o n o o 78

The A's enjoy the following important properties:

(1) Let h , : . (E, E ) -> (E , , E ) be the map given in the n-1 o n~l o * n+l decomposition. Then h ,(A„ (£)) = 0 . n-1 B o

(2) If £ has a cross-section, f: B \j Bq >• E, let f' be an extension of h .f: BN u B • E . , then n-1 o n-1

— n+l,... ,^*,n+l.., , n+l • • ,„\v c (f) = f A (?) e H (B, B ; rr (G)). B on o Proof. See 5.3.4.

5.5.7. In a similar way, we can deduce the following connection between the k-invariants of G and the A's. Since (G . , G) is (n+1)- n-1 simple, the k-invariant of G, kn+"^ (G) , is defined, and equals

T(T.n+1(G , G)) e Hn+1 (G TT (G)) , where T is the transgression in n-1 n—1 n the fiber space, G' »- G >• G . kn+"^ (G) is also the image of n n in+^(G ,, G) under the composition, n-1

; G)) ,G G)) Hn+1(G G (G 7— Hn+1(G ; n-l' Vi n-l' n-l Vl n-l' ~ ' .- n+1 Hn+± (G ; rc (G)) . n-1 n

Let An+1(£) denote the corresponding element to An+1(5) when B =0. B 0 0 Then we have the following:

PROPOSITION. (1) i* _(An+1(?)) = kn+1(G), where I : G C E n-1 n-1 n-1 n-1 is the inclusion of the fiber.

(2) Naturality. If : B' >• B is a map and * ( ,,<(>): K , • ? , is the induced map from the induced n-1 n-1 n-1 * * n+l decomposotion on £ by a decomposition on £, then ^.jtx (?)) = n+1 A <**<0> e Hn+1(E' .; TT (G)). n-1 n Thus this proposition together with 5.5.6 gives the following

results about the decomposition on the universal G-bundle.

PROPOSITION. Let C = (EG, p, BG, G, G) be the universal bundle

'Xj • ^'Xj • 'Xj 'XJ - for G. Suppose it is given a decomposition, (.£, N, {£ }, { (y , g , 1)/) m mm of length N. Then

for i < n+2,

(1) TTi(En) =* {

rr. , (G) for i > n+2; i-l —

for 0 < i < n+l,

(2) H. (E ) = { l n rr , (G) for i = n+2; n+l i — n+2 "\J (3) H (E ) = 0 for 0 < i < n+l, and H (E ; TT ,(G)) is a n — n n+l n+2 — cyclic Horn (TT (G) , rr (G)) -module generated by X (5(G)) n+l n+l

The proof of this proposition is straightforward, by checking on.(1) and applying the Hurewicz theorem and universal coefficient theorem to get (2) and (3). ( See [29].)

Collected in the next section are the main theorems for principal G-bundles, which we shall use to obtain information about

U(n), SO(n) and SP(n) bundles.

§5.6. The Main Theorems

In this section, we shall always assume that G is a connected countable CW-group. Most of the following can be proved by explicitly looking at the Postnikov decomposition as in chapter 4. We shall assume * n+1 n+1 that p : H (B;'ir (.G)) »- H (E; TT^ (G)) is a monomorphism. Then the following theorem determines the obstruction completely.

5.6.1. THEOREM. Suppose (E, N, {E }, { (y , g , D>) is a m mm decomposition of length N >_ n.

Cl) Let f: BN E be a cross-section of E over the n-- skeleton. Suppose f': B > E , extends h , f: BN > E , so that, n-1 n-1 n-1 by our classification theorem, h„.-:E , G , is defined. Then f n-1 n-1 * — n+1 * * n+1 p (cn+1 (f)) + h .h_.(kn+J-(G)) = 0. n-l t (2) Suppose f^, f^: B" ^ E are cross-sections and f^, f^ :

BN E , are the extensions of h . f and h f to B respectively n-1 n-1 0 n-1 1 so that h_^.^ : B *• G , is defined. Then fofi n-x

N+1 n+1 N+1 CFL) (G)). o - c (fn) = h* (k 0 1 C See Lundell 129,.theorems 6.1, 6.2].)

Notice that (2) says that if k (G) = 0 and the (n+1)- dimensional obstruction is defined, then it is a single cohomology class ( cf. 3.1.8 ).

Let the primary cohomology operation, determined by kq+"^ (G) e

HQ+1(G ,; TT CG)) be denoted by i^*"1". Here, of course, we assume that q-1 q

TT^CG) = 0 for i <_ n-1 or n < i < q. Thus, suppose f: BQ > E is a cross-section to E - (E, p, B, G, G) over the q-skeleton and a n(f) E

H"CE; TT (G)) is the unique element determined by lemma 4.5.1, then, n using theorem 5.6.1, we can easily prove the following: 5.6.2. THEOREM. Hypothesis as above.

(1) P*C c q+1(f)) - ^q+1( a n(f));

q (2) If fQf flS B —y E are two cross-sections, then

q+1 q+ 1 q+1 n c (fx) - o cf0) = ^ c a Cf0, f±)).

An immediate cosequence is the following corollary.

5.6.3. COROLLARY. With the same hypothesis as 5.6.2, suppose f: Bq -> E is a cross-section. Then f: Bq+1 >• E is a cross- • section iff c q+1(f) + ^q+1( d n(f, f )) = 0.

For details of the proof of theorem 5.6.2, we refer to [29].

§5.7. Obstructions and Characteristic

Classes

In this section, we give an application of §5.6 to U(n), SO(n) and SP(n) bundles. In particular, we indicate how one can consider some characteristic classes as obstructions.

For the homotopy groups of the classical groups, at least the stable homotopy groups, see R. Bott [8]; and for the cohomology rings of the classical groups, see Borel [7], Milnor [40] or Epstein and

Steenrod [16]. For the cohomology of classifying spaces of the classical groups, see [7] or [40], We shall make use of the universal Chem classes to define the Chern classes of a principal bundle via the classifying map, similarly for Stiefel-Whitney classes, Pontrjagin classes and Euler-Poincare classes ( Hirzebruch [23], Milnor 140] ). 5.7.1. Let E = (E, p, B, U(n), U(n)) be a principal U(n)-bundle * a, and E = (EU(n) , p, BU(n) , U(n), U(n)) be the universal bundle. Assume

(tr N/ xE )/ {(Y / 9 ' 1)}) is a fixed decomposition of E of length m m m N >_ 2n. Then we have the following :

U ( (n) }} = H (E Z) THEOREM. If X e H ^2k-2' 2k-1 " 2k-2'" for n > k is as defined previously, then a.*

Note that we have the result that n\ (U(n))== Z for i B 1 (mod 2) and i < 2n-l; TT. (U(n)) = 0 for i = 0 (mod 2) and i < 2n-l; TT„ (U(n)) £=; — l — 2n

n!

For the proof of the theorem we refer to [29]. With notations as before we obtain the following:

2k-l

5.7.2. COROLLARY. Suppose f: B • E is a cross-section for E = (E, p, B, U(n), U(n)) over the (2k-l)-skeleton of B. If n > k, the naturality of Chern classes gives

2k ckU) = ± (k-l)!c (f).

Proof. Let $£.: B BU(n) be a classifying map for £. Then * % * 2k-2*^* 2k-2 Ck = ^£^ck^ = f^ ^E P2k-2^°k^' where f' is an extension of h f, * 2k-2* 2k 'Xt

•=± (k-l)lf' ^ T (5) by theorem 5.7.1,

If' k ± '{k-1) V (**(£)) by 5.5.7,

=± (k-l)!c (f) by 5.5.6(2).

This completes the proof of 5.7.2. 5.7.2 has the following easy immediate application.

5.7.3. LEMMA. Suppose M2n is an almost complex manifold such

that the cohomology torsion coefficients in dimension 2k are relatively prime to (k-1)! for k = 1, 2, ••• , n. Then M2n is parallelizable iff

all the Chern classes of its tangent bundle vanish.

Proof. By 5.7.2, 5.4.1 and the definition of parallelisability

[40].

C For the relation between the Pontrjagin classes of M and the

Chern classes of M see Hirzebruch [23, Theorem 4.6.1].)

The following theorem summarizes the characteristic classes as obstructions.

5.7.4. THEOREM. Let S = (E, p, B, U(n), U(n)) be a principal

U(n)-bundle with classifying map, ^: B »• BU(n). — 2

(1) The primary obstruction to a cross-section is c (f) = c^(C). This is the only obstruction if n = 1, since U(l) = S1 (2) Assume n > 1. Then, if c^(£) = 0, the secondary obstruction 4 is c2 (?) e H (B; Z) . c = (3) Suppose c^(S) = 2^ ^' then there is a cross-section, 4 5 4 fQ: B ~—•> E ( f •: B »- E, if n > 2 ) and if f^: B >• E ( 5 respectively f. : B ——>• E ) is another cross-section, let ff." : B — 1 o 1

f -> E3 be the extensions of h^f^ and h^f^ ( respectively h^f^ and h^ x Then,

5 5 2 3 f±) (i) if n = 2, c (fx) - c (fQ) = Sq y (fQ, e

5 IT 5 7 3 3 3 H (B; (D(n))) = H (B; Z2) , where (fQr f±) = h*^(u ), u is a

3 * 3 °31 generator of H (U(n)j Z) and h , H (G ; Z) y H (B; Z); 0 1 3 6 6 2 Y (fQ, , (ii) if n > 2, c (f'1) - c (fQ) « A2Sq where

A2 is the Bockstein operator associated with the sequence, •2

0 ——>• Z > Z ——y Z2 y 0 .

3 w Hence by the homogeneity of Y (fQ' f-^) » ® have 5 (4) There is a cross-section, fi B > E when n = 2 ( respectively f: B •——*- E when n > 2 ) iff c (f) = Sq2(Y3) for some 3 . ^ ^ 3 2 3 3 Y e H (B; Z) ( respectively c (f) = A2Sq (Y ) for some Y e H (B; Z) )

= ae r< In fact, if n = 3 and c^(£) ' = c2 (£) = ®> ^ thi 3 obstruction i

2 6 2 3 given by (A2Sq ) (c"^ eH (B; Z)/A2Sq H (B; Z) , where (A Sq ) is the functional cohomology operation, determined by

2 4>£ and A^Sq , and A2 is the Bockstein operator associated with the

• 2 sequence, 0 > Z >• Z »• Z;> • 0 [49].

5.7.5. Definitions and Notations. Let SO(n) denote the special % i

orthogonal group of GL(n, R) . Let to_^ e H (BSO(n); Z2) for i = 2, S,'" , th — 4i n denote the i universal Stiefel-Whitney class, e H (BSO(n)) for th i = 1, 2, ••• , [n/2] denote the i universal Pontrjagin class. — n

Xn e H (BSO(n)), n even denotes the universal Euler-Poincare class.

ar In particular, and x © of infinite order and H (BSO(n); Z2) =

Z0$„, , u j '( Borel [7] ) . Let E = (E, p, B, SO(n), SO(n)) be a SO (n)-bundle with classifying map, B ——> BSO(n). Define the Stifel-Whitney class,

Pontrjagin class and Euler-Poincare" class as follows:

u. (E), = '•?$.), 1 S -1 oAE) = ^(P^/

xn(o = ^.cxn).

By 5.5.2 it is legitimate to make the following assumption: 'v. 'Xi

(5.7.6) E = (ESO(n), p, BSO(n), SO(n), SO(n)) has a'decomposition of length N >_ 4 [n/2] ...

Below we list the homotopy groups of SO(n) [8], which we need, and state the corresponding theorem linking the A's with the Pontrjagin classes of the universal bundle and hence linking the Pontrjagin classes with the obstructions.

(SO(n)) £=: 5.7.7. 'TT z2 ;

T7 Z2 ; 4(SO(5))fi^

S0(8)) ^ Z + Z ; TT7(

IT . ' (SO(n)) ^ Z if n > 4k+l. 4k-l —

4k ^ 4k ^ 4k ^ THEOREM. Let A (?) e H (E4k_2; ir4 (SO (n))) = H (E4k_2) if n >, 4k+l, be as defined in 5.5.7. Then e 4k ^ r ± 2(2k-l)!A (E) if k is odd;

P4k-2(pk) =

± (2k-l)l\Ak(t) if k is even. 4k-l Hence, if f: B > E is a cross-section to £ = (E, p, B, SO(n),

SO(n)) over the (4k-l)-skeleton, then

r — 4k ' . ± 2(2k-l)!c (f) if k is odd;

— 4k ± C2k-l)!c (f) if k is even.

The proof of this theorem is similar to that of 5.7.1 and 5.7.2.

Thus with this theorem one can deduce with ease the following result.

5.7.8. COROLLARY. Suppose CD H8k+2(B; 7.^ = H8k+1(B; Z) = 0,

8k

(2) the torsion of H (B) is relatively prime to (4k-l)! and the torsion of H8k+4(B) is relatively prime to 2(4k-l)! , for n >_4k+l. Then

£ = (E, p, B, SO(n), SO(n)) has a cross-section, f: Bn 1 > E iff

Pl(5) = P2(5) = V - PI(n_1)/41«) = 0 •

5.7.9. In this subsection we shall assume n >_ 5, and the . , following fact.

(A) Suppose n >_ 5, ^: B > BSO(n) is a classifying map for * <\j

0 = Then the £ = (E, p, B, SO(n), SOCn)) and that ®2(V = "J*^ ^ °* 3 generator u^ e H (SO(n)) = Z is transgressive for £. In particular,

p^CO = ± 2TCU3) [29] .

(1) The following is a table of some cohomology groups of

SO(n) and BSO(n) for n > 5. * Dimension H (SO(n) ; Z) H*(BSO(n); Z)

1 0 . 0

2 0 Z2

3 2 Z^ (generator w^)

4 Z (generator p^) Z2

(2) The following is a table of some k-invariants of SO(n) for n >_ 5.

(a) k4(SO(n)) =' 0;

5 2 5 TT 5 (b) k (SO(5)) = Sq 63 e H (G3; 4(SO(5))) = H (G3; Z2) ;

6 2 6 6 (c) k (SO(6)) = A2Sq 93 £ H (G4; ir5(SO(6))) = H (G4? Z)

k8(SO(7)) H ; Z);

8 7 7 8 (e) k (S0(8)) = A/303 + A3C^e3 + A2ex + A^ £ H (G6; Z+Z) ;

8 8 (f) k (SO (n)) = A B^e e H (G6; Z) for n >_ 9 ; here 6^ 3 a generator is a generator of H (GJ ^ Z for i = 3, 4, 6; 8^ is g of H^G L; Z2) for

i = 3, 4 and 6 respectively; H (G&; Z+Z) H (G,.) + H (G^) and denotes decomposition with respect to the above 6 6 direct summand; A is the Bockstein operator associated with the exact n •n 1 sequence, 0 »• Z »• Z > Z^ »• 0, is the Steenrod reduced 7 cube, and of course 6^ denotes the seven fold cup product of 6^ with itself. For the calculations, we refer to Lundell [29].

(3) PROPOSITION. Suppose £ = (E, p, B, SO(n), S0(n)) has a 4 4 3 cross-section, fQ: B • E. Then, if d E H (B; Z2) and y e H (B; Z), 4 there is a cross-section, f : B • E, such that (f f ) = and a 0# a,

3Cf = y, y Q7 f±) — 3 * where (f f^) is defined1 to be h ^ ..(.u ) and is a generator of y Q/ f f 3

3 01 H (SO(n)) ~ Z.

The proof of this proposition is analogous to that of the Un• bundle case.

Thus with this proposition we get the following main theorem for SO (n) -bundles, n >_ 5 .

(4) THEOREM, (i) The primary obstruction to a cross-section

2tO 2 is u e H (B; Z2) .

(ii) If ^2(£) = 0, the secondary obstruction is

T T (u3) where is the transgression in £ and u3 is a generator of

H3(S0(n)). In particular, if c 4 denotes the obstruction, then

4 Pi(£) = ± 2c by 5.7.9(A) . 4 (iii) Let f: B *" E be a cross-section to K = — 4

w = c = (E, p, B, SO(n) , S0(n)) over the 4-skeleton; i.e., 2(£) °*

Then the application of proposition (3) above and theorem 5.6.1 gives the following criterion: 5—5 o If n = 5, £ admits a cross-section over B iff c (f) = Sq Y 3 for some y e H (B; Z).

2 If n = 6, £ admits a cross-section over B^ iff c ^(f) = A2Sq y for some y e H3(B; Z) . 89

If n = 7, £ admits a cross-section over B if and only if

8 1*7 3 1 c Cf) = AFy + Ad , for some Y e H (B; Z) and d e H (B; Z ). 3 3 2 8 If n = 8, £ admits a cross-section over B if and only if

- 8 1 1 7 7 3 c (f) = A. P Y + A (? y" + Ad + id' , for some y, y' e H (B; Z) and J o J O ^ some d, d' E H (B; Z^). g g T_ If n > 9, C admits a cross-section over B iff c (f) = A^ff^Y 3 for some Y e H (B; Z).

In an analogous way, one deduces the main theorems for SP(n)- bundles.

5.7.10. (A) We list some homotopy groups of SP(n)

Dimension i < 4n+l TT#(SP (n))

i = 3 ( mod 4 ) Z

i H 4, 5 ( mod 8 ) Z2

i = 0, 1, 2, 6 ( mod 8 ) 0

i < 3 0

i = 3 z

i = 4 Z2

(B) Notations. Let e^ E H (BSP(n); Z) denote the symplectic Pontrjagin classes of the universal bundle, £(SP(n)). Suppose

£ = (E, p, B, SP(n), SP(n)) is a SP(n)-bundle with classifying map, * % 4 j_

<(>,_: B >• BSP(n). Then let e. C£) =

Pontrjagin class of £ . We assume that £ has a decomposition of length

N > 4n. As in 5.7.9 we have the following theorem.

4k — 4k — ir (C) THEOREM. Let X . (?) e H (E4k_2; (SP (n)))

4k — E Z or k n C = H ( 4]c_2' ^ ^ — ^ be as defined previously. Then 4k 'v ± (2k-l) ! A (£) for k odd;

P4k-2(\) =

4k 'Xi ± 2C2k-l)!A (£) for k even. 4k-l

Thus, if £ = (E, p, B, SP(n), SP(n)) has a cross-section, f: B

for k < n over its (4k-l)-skeleton— 4k , then ± (2k-l)!c (f) for k odd;

eK(0 — 4k ± 2(2k-l)!c (f) for k even.

(D) Analogous to 5.7.3, in the case of an almost

4n quaternionic (4n)-manifold, M , we have the following immediate result by (C). 4n i 4n THEOREM. Suppose M is as above, H (M ; Z^) = 0 for i = 5, 6

t mod 8 ) and that the cohomology torsion coefficients are relatively prime to 2 (4i-l)! in dimension 8i and are relatively prime to (4i-l)! 4n in dimension (8i + 4). Then M is parallelizable iff e^C?) = e2(£) = 4n e e3(.£) = ••• = n(-£) = 0 / where £ is the tangent bundle of M

(E) The Main Theorem.

Suppose £ = (E, p, B, SP(n), SP(n)) is a SP (n)-bundle.

(1) The primary obstruction to a cross-section is e^(£) . 91

4 (2) Suppose e^(5) = 0 and f: B ——*• E is a cross-section. 5 Then there is a cross-section, f": B —:—* E iff there is an element 3 — 5 o d e H (B; Z) such that c (f) = Sq^d.

Proof. (1) follows from theorem (C).

(2) follows from the fact that k (SP(n)) = Sqz0 , where 0^ is the fundamental class of = K(Z, 3), and theorem 5.6.1.

§5.8. Conclusion

We have seen how one can use the idea of Moore-Postnikov decomposition in computing the higher obstructions. The main difficulties lie in computing the k-invariants. In the more general case, one considers principal fibrations and thus one needs to know some higher

cohomology operations usually with twisted coefficients, determined by some k-invariants. Of course, this means one is reduced to looking

at cohomology with twisted coefficients.

In this direction, one has some deep connections with the old problem of determining the solution to the lifting problem:

Y

H

£

x > B One may assume H to be a fibration. Let to be a cohomology class in * KerCH ). Then, taking o> as a classifying map, one obtains a principal fibration,i{»: E > B, over B and H automatically lifts to ty with to 92

* a lifting, q: Y >• E . Let 3c be any fixed class in Ker (q ) . Then one

can define k(E) = LJf k , where the union i.s taken over all f: X —:—»• E to *

such that ifiof = E . For k(E) to be non-empty' one must have £ to = 0 .

If E to = 0, i.e. k(.£) 0, then one can show that k(£) is a coset in

the image of some (twisted) cohomology operation, determined by some

relation 159; 38]. The result appears a bit cumbersome (even in the

orientable case), even more so if we generalize to the case of two * classes, k , k e Ker(q )• McClendon [39] considers the tower with K-principal fibrations:

f Ik X >- B —> C where the C.i is a product of L,, n)'s and , n) is the generalized Eilenberg-MacLane complex, a fiber space, K(G, n) >- L (G, n) >• K =

1) Ktir^B), and G is considered as a ir^ (B)-module via a homomorphism7 a): TT, (B) > Aut (G) 117]. Thus the k.'s are twisted. Let k. (f) be as 1 ii defined before. He is able to prove the following theorem. THEOREM. Suppose there is a positive integer, N, such that all the C.'s are products of L. (.G, n)'s for N+l < n < 2N. Then there is a stable B-operation, $ = $" ; IX, ftC ] • •>• IX, C.J C an additive .x, f IK i K relation ) such that k.(f) is a coset of the subgroup Im($) C[x, C.] 94

. CHAPTER.6. MODIFIED POSTNIKOV TOWERS AND

. .OBSTRUCTIONS TO LIFTINGS

In this chapter we give a modest account of a modified Postnikov tower which, has principal fibrations for each stage and we use higher order cohomology operations without twisting to describe the k-invariants for a lifting. For this reason we shall assume the fiber bundles we start with, are orientable. This presents no restriction to the technique . used; in fact, one can generalize this to arbitrary fiber bundles and this would have involved higher order twisted cohomology operations [

McClendon, 37,38; E. Thomas, 59]. We shall give only the description as far as orientable fiber bundles, leaving the not-so-obvious generali• sation to the excellent accounts by McClendon and Thomas.

§6.1. Modified Postnikov Towers

The main idea is to replace the usual Postnikov tower by one i i-l where the fiber space, p^: E E , has as its fiber a product of

Eilenberg-MacLane spaces and the fiber of h.: E > E1 satisfies some nice properties. In addition we require some algebraic conditions on h^ which are a generalisation of the construction in chapter 4, via the idea of spherical classes.

6.1.1. Definition. Suppose X is a complex and J = Z or Z^ ( q some positive prime ). Let a E H (X; J). a is said to be (J) spherical 95

k * if there exists a map, f: S —:—> X, such that f (a) is a generator of

k k H (S ; J). We want to work slowly towards the conditions required of the

fiber of h. : E > E1. i

6.1.2. Definition. Let X be an (n-1)-connected space for n >_ 2

such that TT (X) is finitely generated. For k < 2n - 1 define ^ (x; Z ) K • — q to be the subset of H (X; Z ) satisfying the following conditions:

(1) Each a e ij (X; Z ) is (Z ) spherical. q q k * (2) If f : S >• X is such that f (a) is a generator of a a k k H (S ; Z), then the collection of homotopy classes of such f's,

{[f ]} «k,„ „. is a linearly independent set in TT, (X) considered a a e y (x? Z) . k

as a Z-module, and { [ fj }& e tf *(x. z > " { [ fj >a e .^(X; z) is linearly independent over Z , where [ f ] is the image of [f ] under q a a the natural homomorphism, TT, (X) * TT, (X) 0 Z k k q

(3) It is maximal with respect to (1) and (2).

Call ^k(X; Z ) a regular spherical set.

6.1.3. Our aim is to define a modified Posinikov tower ( abbreviated MPT ) where the fiber of each h^: E —:—>• E1 is t-regular ( see definition later ). To have this at all we need the existence of a regular spherical set. The following theorem guarantees this.

Tr THEOREM. Any (n-1)-connected space, X, (n >_ 2) with k(X) finitely generated in the stable range ( k <_ 2n-.l ) has for each prime

k q and for q = 0, a regular spherical set, ^ (X; Z ) with the convention that Z = Z , if q =0 . q 96

Proof. See M. Mahowald 130].

6.1.4. Let (E, p, B, F) be a fiber bundle with F (n-1)-connected

(n >_ 2) . Then, by 2.1.3, we see that we have the following commutative

diagram:

E P Wo - ' M C B where X is a homotopy equivalence. Thus consider the sequence for the induced bundle, (E , p E" , M, F), over M: 0 . Ho * k k -S-> k+1 H (Eu ) > H (F) H (Eu , F) . ° k ° * k We say F is t-regular if (F; Z ) ^ 0 and

6.1.5. Suppose (E, p, BG, F) is an universal bundle with BG a classifying space for an abelian group G and with F being a G-space.

Then it can be easily seen that (E , P|E , M, F) ( notation as above ) is an universal object for all F-bundles with cross-section. We shall often assume this without mentioning it.

6.1.6. Define t5, (F) = U W^tF; z ) . A representation t q k=l q for t5 (F) is a map, h: F >• II II II K (Z , k) = Y fc q k=l a £ t5k(F; Z ) * q q * such that h (oc ) = a where a is the image of the characteristic class a a of K (Z , k) in Y and K (Z , k) is an Eilenberg-MacLane space of type a q a q 97

(Z , k). q

Thus a modified Postnikov tower of dimension t for (E, p, B, F) is a tower

si-l X i-1

h. . i-i E -> E

B

1 1 such that (1) p^: E *• E 1 is a fiber space with fiber x1 = n K (J , k ) a a a aeA where A is a finite index set, J is either Z or Z (some prime q) ; a q

1 (2) h. , : E *• E 1 is a fiber space with its fiber X1 t.- l-l l regular for some t. < t , where t. > n. = dimension of the first non- * l — i—i zero homotopy group of X1 ; and

(3) h.Ix1: X1 •> X1 is a representation of t5, (X1) . l1 t. l

6.1.7. Remarks. (1) If t^ = n(i) for all i, then the MPT reduces to the tower described in chapter 4.

(2) There is a theorem guaranteeing the exist• ence of a MPT of dimension t for a G-universal bundle, £, where G is a arcwise connected topological group, and the fiber, F, is (n-1)-connected 98

(n > 2) with TT_, (F) finitely generated for j <_ .t < 2n . Hence it also guarantees the existence of a MPT for any (principal) G-bundle with F as fiber 130].

The next section describes some properties of Postnikov systems and how the k-invariants are related naturally. We shall then come back to the question of computing k-invariants for liftings. This has been an area of active research [ Thomas, 59; McClendon, 37, 39; Maunder,

33, 34, 35, 36; Mahowald and Tangora, 32].

§6.2. Construction of Induced Maps Between

Postnikov Systems

We shall stay in the category of countable 1-connected CW complexes

(with base point) and based maps. The reason for us to consider this " . category is that it is closed with respect to this construction.

Consider the Postnikov decomposition we have given in §4.6. If we replace E by F and B by K(TT (F) , n(l)) where TT (F) is the first nil) n(l; non-zero homotopy group of F, that is, F is (n (1)-1)-connected. We then have the Postnikov system for F.

6.2.1. Definition [43]. If F is (n-1)-connected for some n >_ 2, a Postnikov' system' for F is a family of n-Postnikov systems, {(F^, 9^)^' and morphisms, { n. .: F. y F. }. . , directed in such a way that if i/D i 3 X>3. i > j > h then

TVi. ,k. = n u,. ",k on 1. ,'3. • Furthermore n. , . : F. , —:—* F. is a fiber space with fiber i+l 1 e KC'T. , (F) , i+l) and n. , .og. , - g. . For 0 < i '< n-1 the F. consists i+l i+l,i i+l • i — — l of a single point, the base point. Therefore K(HN (F), i) for 1 <_ i <_ n-1 is taken to be a single point by convention.

6.2.2. Suppose we are given two spaces, F and F', with their

Postnikov systems, {F., g., n. ,} and {FT, g7, .} and a map, f: F — 111,3 l "l 1,3 y F". Then by a map between the Postnikov systems induced by f, we mean a family of maps, { f^: F^ y Fr } satisfying = ru, (1) f.on . .of. , and l i+l,i i+l,i i+l (2) gfof * f.og. . l ii Our goal is to construct such an induced map.

6.2.3. Suppose we are given two spaces, F and G; suppose also that F is (n-1)-connected, G is (m-1)-connected and n, m >_ 2 . If f: F y G is a map, then the following diagram is homotopy commutative f

F y G

p(F) P(G)

f# K(TT (F), p) — -> K(TT (G), p) P P where is the map induced by f: F > G, p = Min( m, n ) and p(F) represents either the fundamental class i (F) ( p = n ) or the zero class if (n-1) >_ p . The case that p < n or p < m is easily seen by standard homotopy argument the other case for p = n = m follows by using the

Hurewicz theorem and duality. By the natural construction of 2.1.4 we may replace the above diagram by an equivalent diagram ( Covering Homotopy Theorem ):

-> G'

P(F) P(G) .. A. . .

K(rr (F), p) * K(TT (G), p) p p which is commutative and where P(F)' and P(G)' are fiber spaces. This diagram tells us by the homotopy exact sequences of the two fibrations that the fibers, F" and G" of p(F)' and p(G)' respectively are p- connected.

P+1 P+1 P+1 P+1 Let i (F") e H (F"; TT IN(F")) and * (G~) e H (G~; p+1 ^^^(G"")) be the fundamental classes of F" and G"" respectively. Let

P+1 kP+2(F) = T(I (F"))E HP+2(K(TT (F), p); TT (F)) and kP+2 (G) = P P+1 T(IP+1(G")) e HP+2(K(TT (G) , p) ; rr (G)) be the first k-invariants of p p+1 F and G respectively. ( See 4.4.4. )

ir F 6.2.4. Look at the diagram in 6.2.3; if we identify p+1 ( ")

- with TT , (F) and TT , (G"" ) with TT , CG) r then p+1 p+1 p+1 (f|F~) „: TT CF") -y TT CG") 1 # p+1 p+1 is identified with f„ .

THEOREM. Suppose we let f^ denote the coefficient homomorphism induced by f„: TT ' CF) •> TT ,. CG) . Then f*kP+2CF) = Cf») *kP+2 CG) . # p+1 p+1 c ft Proof. See [25]. 101

6.2.5. By the above theorem, replacing maps if necessary by equivalent fibrations, we have the following commutative diagram: K K(TT (P) , p) (TT (G) , p) P

-p+2 P+2 (F) k (G) a. f,

K(rrp+1(F) , p+2)- **"K( ^p+1(G) ' P+2)

C As before we use the ACHP property if necessary to replace the homotopy commutative diagram by a commutative diagram. )

Taking fibers of lcP+2 (F) , )cP+2 (G) , we get a commutative diagram of fiber spaces:

p+1 -> G 'p+1 p+1

n. P+l/P p+i/P

K(rr (F), p) —£ > K'(ir (G), p)

kP+2(F) ^P+2(G)

> K(TTp+1(G), p+2) K(lTp+l(F) ' P+2)

F G where (F » n K(TT (F) , p)) and (G , n , K(TT (G) , p)) are p+1 P+1,P P .:- p+1 P+l/P P principal fibrations and f = f,, . These are the second stages of the p #

Postnikov towers and f^+x is a map between them.

6.2.6. Since F , and G , are obtained by using the k-invariants p+1 P+1

F as classifying maps, as in chapter 4 we see that P(F)' then lifts to p+X

G and p(G)" lifts to p+1 ( Lemma 2.2.2 ). Thus we have the following diagram: *• G p(F) p+1

p+l,p n. p+i/P

(TT F . = K (TT (F) , p ) * K (G) , p ) = G f P P where p CF) is a lifting of p(F) and P(G) is a lifting of p(G) The main idea is to alter f , such that the diagram: p+1

~> G'

p(F) p(G)

•+ G P+1 p+1 P+1 is homotopy commutative and that we still have f on , =11,. of ,., . P P+1,P P+l/P P+1

Let us pause for the moment, assuming that we know we can alter f , as above. Then we can extend the technique to the whole system stage p+1 by stage and the proof works by induction. This is the essence of the proof of the following:

THEOREM. Let F and G be as above. Suppose f: F > G is a map. F F ~ G G ' Then there exist Postnikov systems, {F., g., r\. , .} and (G. , g.i n.,, .}, 11 i+l,i l l i+l,i and an induced map { f.: F. y G. }; i.e., a diagram: ii l f y K(TT (F), p) = F 2- G = K(TT (G), p) P P P P where all the "rectangles" in the front of the diagram commute and the rest of the diagram homotopy commutes. Also, for n >_ p+2 , * n * n f k

6.2.7. With little difficulty we can then use this standard construction to obtain an induced map between two given arbitrary Postnikov * n * n systems for F and for G. Also, f k (F) = f k (G) for n > p+2 . c n-2 — Proof. See Kahn 125] for the proof of theorem 6.2.6 and 6.2.7.

6.2.8. Remark. Application to Hopf-space structures.

Consider the following situation. Suppose X is such that n^.jtX) •+ 0, TT_. (X) =0 for j ^ n(i) and n(l) < nC2) < n(3)' < Then take the Postnikov system for X and consider the following diagram

p+1 •VI,PXVI,P yp Vl ,P W P+1 P+1 E^xF? P TT E xE • >• E ^K(Tf (X) , p) S K( (X) , p+2) P P+1 104

^ p+1 Let be the' H-multiplication. Then E has a H-rnultiplication iff

P+1 P+2 y o(n xn ) lifts to E iff k (X) oy o (n _ xn )• * * . p p+l,p p+l,p p P+1,P P+l/P

P+2 Tr x _ which, when appropriately interpreted [25] ,means k (X) is p+1( ) primitive. Thus we have the following theorem.

THEOREM. Hypothesis as above. X is a H-space iff it admits a

Postnikov decomposition for which each k-invariant, kn^(X) is primitive with, respect to the inductively constructed H-space structures on the { E1}.

( For application of this theorem to H-space structure on spheres see 126] and also the review for this paper in Steenrod's Review concerning theorem 2.2 of [26]. )

§6.3. The k-invariants for a Lifting

6.3.1. Let F be an (n-1)-connected space for some n >_ 2 and let

t h: F y Y=niin K(Z,k) q k=l a e ^T(F; Z ) 3 q q be a representation of "^(F) f°r some t. Let X be the fiber of h. We *k would like to be able to compute \> (X; Z ). Suppose the first non-zero q homotopy group of X occurs in dimension n . The main difficulties lie k in showing that (1) { [ f ]; f : S »• X } , is independent S a . acf (X; Z)

(2) { [ } _k U( [ over Z and falg tj^ } v is independent 3 q a aET(X; Z) * aeSNx, Z ) q over Z q 105

Using the Barratt-Puppe sequence, in the stable range, we have

the exact sequence: """ * ... > HJ(F) > HJ(X) Hj+1(Y) -^-»- Hj+1(F)

+± ( for j < 2n-2 ). A class x. , e tf? (Yi Z ) can be written in the form, - 3+1 q

x. , = ; a

3+1 a a ae 5t(F)

where is a primary cohomology operation, cl • •

: Hk(Y; Z .) > H3+1(Y; Z ) . a(q,k) q q This is the usual representation of H (K^Tr"*, k) ; IT) as primary cohomology

operations in the stable range ( see e.g. [44] ) . Hence if <$> . . is a(q,k)

non-zero then q" = q or either one or both of q, q" is zero. By the above

exact sequence,

J aet)t(F)

j ~ * is zero iff there exists a class, xT e H (X) such that

6.3.2. Definitions. We say an operation,

4>: Hk(X; Z) »• HJ+1(X; Z ) , q i k of type (k, Z; j+l, Z^) is primitive if there exists a map, g: S > S , k j+l such that is non-trivial in the complex, S e . Call an operation, g

: q q of type (k, Z^; j+l, Z^) primitive if

(1) the image of (J)" under the natural homomorphism, * * H (K(Z , k); Z) y H (K(Z, k); Z ), is primitive; or q q — (2) (j>^ = , the Bockstein operator associated with the sequence: 21 " i Examples. (1) Sq , for i - 0, 1, 2 or 3 and for k > 2 ; and are primitive.

2 1

(2) Sq Sq is also primitive.

See Thomas [58, Theorem 4] and Mahowald [30] for an indication of the proof of (1) and (2). j+l x H Z s re resente 6.3.3. THEOREM. Suppose j+1 £ ^' ^ ^ P d as

X = 5! i+1 MO D+1 ae (F) a a and q ^ 0 . Suppose also that for at least one a" e ^ (F) , / is t a j "*

x primitive. Then for any x e H (X) such that <5 x' = j+^ > there exists q a map, f: S >• X, such that f x' generates H (S ; Z ), i.e., x' is (Z ) spherical. q Proof. See [30]

The following proposition gives a special case of telling when k+2 some classes in H (X; Z^) are spherical.

6.3.4. PROPOSITION. Suppose a" e ^k(F; 7,^ for some k >_ 3 and k * k k if f'^: S > F is a map such that f'^(a') generates H (S ; Z ), then a a 2 assume 2[ f'^] = 0 where [ f."A is the homotopy class of , . Let x e a a a k+3 H (Y; Z^) be represented in the form

v 2 1 x = i 4> a such that tj> _ - Sq Sq . Then for any x'

ae dt(F) k+2 '* H (X; Z2) such that j x' = x , x' is spherical. The proof of this proposition uses Paechter's groups [46] and can be found in Mahowald [30].

6.3.5. Let E = (E, p, BG, P) be the universal bundle with F

(n-1)-connected and let there be an MPT for E in the stable range

(6.1.7(2)):

Ml

*• BG

x1 = 1 , k) where n n n . K (Z

1 & q q k=l ae 3 (X ; Z ) x K+X x—1 3c i. Let k e H (E ; Z ) for a E ^ =>k (,X i ; Z ) be the k-invariant. An a q q application of 2.1.3 to the fibration h^ gives the following diagram

i-l where X, is a homotopy equivalence, i.e., u. is equivalent to h, i-l i-l i-l

Ai

Consider the induced fibration, (E' , p. E i-l•, , M. , , X ), in the fiber square, i-1 Look at the diagram: i-i

an Since V1-± ^ h are homotopically the same map we then have p.o(h.oX. ) - u • This then together with the fact that X. is a i I i-i i-1 i-i homotopy equivalence implies by the universal property of the fiber

square that (E , p. |E , M. , , X1) has a cross-section, p. Hence yi"1 1 ^i-1 1-1 E - M. . x X , and p is chosen in such a way that u. = u.op . We = 1-1 ii explain below. Assume we have shown that = M ( C BG ). Then we see that all other M.'s can be taken to be M. The new k-invariant• i i+l * i * — * k e H (E ) will be a class in Ker( p op, ) coming from the fibering cl 1 ! ~i • ., ..i+l h. : E •+ E with X as fiber. l

Look at the fiber square:

C E v-1 - E .y y. V-l V-l v-1 M C where v = i or i+l V-l i ~i o. The fibration, h.: X > X , induces a map, E ' > E , and

"V • A a. p . , :" M y E induces p . : E C E 1"1 "i-l 1 "i Vl

6.3.6. THEOREM. Suppose X1+^" is (j-l) regular. Assume that a e

oi-1 i+1 ^* 13 (X ; Z ) is such that 6 a ^ 0 where 6 is the transgression in q * ~* *

i • , .+1 • iP (X )• ^^(X1) —H3^). * —* x+1 ' — ~i Then i u.(k ) = 6 a , where i: X E ' 1 a . Vl The proof is just simply putting things together.

6.3.7. Since E £ M x X1 , by the Kunneth formula, Vi

X H*(E ; Z ) = H*(M; Z ) 0 H*(X } Z ) . q q q Let p. , : M >• E be the cross-section. We may assume in dimension x-1 y.^ 3 Ker(p ) 9£ I HJ (M) & H (X ) . v=l j * So a e H (E^ ; Z^) D Ker(p^ ^) is a linear combination of m_. ^9 x^

_ t> i where m e H (M; Z ) and x, e H (X ; Z ) . a q £> q

Definition. Let e = (e e , •••) where a£(v,k) e t5 (X1) and al a2 t e . . e Hk+"*" (E1 "S Z ) ( & = 1, 2, • * •) . We shall define an action of a e H3 (E Z ) r\ Ker(p ) for j < 2n on e . Vl q t Now, X1 = n . II n , . K (Z , k). Let a . ,\-be the A Q A(Q K) q k ae k(X1; Z ) '

' ... - ^£ image of the fundamental class of K (Z , k) in X . a q 110

Since n < Dim a < 2n for all a , as in 6.3.1 we can write — a a

x = T • • a e Hy (X1; Z ) y ae* (X1) 3 a q t "i ~ii '*—1 v ,D+i > 1J. Define (m. ® x )© e = (-1) y. ,m, ) . 4> e e HJ (E ; Z ) , 3-y y r-1 3-y ^^(^j * * where Sty = and S is the suspension, y. is clearly an isomorphism cl cl 1""X

since j-y < n . Extend by linearity to arbitrary a e H (Ew" ; Z^)AKer(p ) Vi' q With this new action we have the following lemma.

6.3.8. LEMMA. For fixed i let k = { k ; a e "5,(X ) }. Then * *_i * —

1 * i-1 * i-1 H (E1_1) Vi H (M) —> H (E , M) -> H (E )

* j—y * Just simply note that Q ( m. ® x ) = C-l) . m. "-6 x" , after identifica- j-y y D-y y • *-i * *-i *-i tion with the preimage of m. . Therefore p. 6 (m. 0x ) =-y. _m. *p. x . -1 * *_]_ * i D-y i D-y y 1 D-y i y Since j p. 6 4 a = d>k ,.. we see that the lemma is true for a =. m. ®x i a a a a .-- J-V y i * and hence for any a e H CE ; Z ) r\ Ker (p . ) . - yi-l q 1

6.3.9. 6.3.8 applies to give the following: * * i-l * THEOREM. Suppose h. : H (E ) > H (E) is an epimorphism in

i * dimension j. Then for each o e H (E ; Z ) AKerfp ) there exists a Vl q v E H (E ; Z ) O Ker(h.) such that y.v = a iff a ® k = 0 . q 1 1

The proof is easy; just apply the previous result and checking on the diagram above.

i+1 * 6.3.10. THEOREM. Suppose X is (j-2) regular, h^ ^ is an

_A epimorphism and y. is a monomorphism. Let k = (k , k , k ) be some X X s , vector in H3 (E1; Z ) x H3(E1; Z ) ( s-fold product ) and q ^ 0 be q q fixed such that the following conditions are satisfied.

* —* (1) i y_^ k^ are linearly independent.

(2) If i u. k = x = T a , then for at least one a, i v . o »_v a a ae *3 (F) cj) is primitive.

(3) Let 01(q)^ be the y-component of the Steenrod algebra mod q,

<31(q) . Then the vector space of H3 1 (X^*^) generated by

a=l has co-dimension s in H3 1 (X1*^) .

(4) y,. k^ e Kert P ) •

Then there exists a class t^3 (X1^; Z ) such that k is the collection, q { ki+1 ; a E *5j-V+1, Z ) }. a q Proof. See Mahowald [30, Theorem 3.4.2].

6.3.11. Remark. Thomas [58] gave a computation of some of the spherical sets of the fibration, 112

V -*> BSO(n+2), for n > 2 , using n+2,2 -> Bso(n) Paechter's groups [46]. When n = 4s+l , the computation there is much * simpler. He actually computed Ker(p JnKerd:^) to find Ker(h ). Since

2TT ( V ) = 0, an application of 6.3.4 gives the required sets,

is the modified transgression in 2.3.5(5).

§6.4. Obstruction Theory for Orientable

Fiber Bundles

We now consider the question of how a MPT can be used to give an obstruction theory for liftings of orientable fiber bundles. Let

£ = (&, p, BG, F) be the universal bundle and £ = (E, p, B, F) be a

G-bundle with F as fiber and ^ : B > BG as classifying map.

Suppose B is a CW-complex of cohomology dimension t. Suppose further that F is (m-1)-connected. Consider the MPT for £ of dimension t in the stable range: i\, Oil- 1>i The fiber of h.. E *• E is t-connected. if E is the last space in x ^i" —i" the MPT. Then any map, ^: B >• E , from B to E can be lifted to a map, '* B y E.

6.4.1. £ has a cross-section iff • BG can be lifted to „ ^ a>l ^2 a map, : B > E, which is equivalent to lifting of ^ to E , E -i" and so on upto E . The following is well known:

THEOREM, b^^'- B • E1""1 can be lifted to a map ^ : B »• E1 iff (. k1) = 0 for all a e ^5 ' (X1) i-i a t.

For the first lifting we have the following criterion:

* * * 6.4.2. THEOREM. Suppose 6 : H (F) > H (E, F) is the coboundary -1 homomorphism. Then : B > BG can be lifted to a map <|>: B > E iff 6 (a) = 0 for all a e O (F). 1 * ]_ * * * Proof. Just use the fact that p k = 6 i a whence S a = 0 a iff k^ = 0 ( relative Serre Theorem ). Here i is the inclusion of the a fiber, F, in E.

We have the following diagram:

•+ E

BG 114

By 6.1.5 ^ ls so chosen that ^ = ^i°p w^ere"P 15 a cross-section. Then 1 "1 — — E • = B x X . . Thus if p is chosen. so that V^op = P°yQ i then in the _i* * —* stable range y^ is an isomorphism between Ker(p ) and Ker(p ).

Suppose B is a CW-complex. Then * * 2 * —* * 2 P *Ik a = " hl 11 \ +1 ka

[30, §5.3.3], where U is the inverse map ( defined only in the stable range ), H: Ker(p ) >• Ker(p ), to y |Ker(p ). We thus get 1 4c 4c 4c H (E ) SsL Ker(p ) 0 Im(p ). —* * 2 * * fly (k ) e Ker(p ) implies there exists a class z e H (B) 1 1 a * _* * 2 * 2 * * **2 such that p. (z) = If y_ cj>. (k ) - , (k ) . Thus z = p p (z) = - p (k ) 1 llala l i a *2 * * 2 *** = - cb" (k ) . So since h cj> (k ) =0 and h p = p , we thus establish X cl X X cl XX 4c 4c O 4c 4c 4c 4c ""4c 4c O -p A- (O = p (z) = h. p. (z) = h 1f y. 4> (k ) . la ll ilia

The proof of the next theorem uses the above formula

6.4.3. Suppose we have a lifting all the way up to E of the classifying map 4>^.: B *• BG.

*i-2

B y BG

Then define for each a e *£ _^(X1 x), i

i * k (cb,.) = { z e H (B) ; there exists a lifting, cj>r . / a £ i-l of cj>R with ij>' ,K = z } 1T2 i-l a 115

k Let z = ( z ., z • • •, z ) c H (B; Z) be indexed by each a e al a2, a*

^ (X1 x) , i.e., z = { z , Consider the action we described in t a(v,k) i *

6.3.7.. Analogously we define an action of a E H (E ; Z ) n Ker (p )

on z ( j < 2n ). First define action of m. 0 x on z by * *-l v

1 1 a D-y y r-2 i-2 D-y ^o^ " ) a

E j * and extend by linearity to arbitrary a H (E^ ; Z^)n Ker(p^_2) .

6.4.4. 1 THEOREM. Notation is as above, k (r) is a coset of the a c, — k group, a" © z where z . , . ranges over all of H (B; Z ) and a' = a a(v,K) V a

V* , k1 e H*(E ) for a e ^(X1-1) . i-l a y. „ t i-2

Proof. See [30, 5.4.3].

6.4.5. Remarks.

(1) Mahowald has sucessfully used this theory in determining the regular spherical sets for V ( orientable ) bundles. Notice he n+m,m used cohomology operations without twisting. In general one would expect * the set Ker(p )fl Ker(t^) to have a twisting action. Mahowald [30, §6] gave a computation of the k-invariants for V bundles. For the non- n+m,m orientable case see McClendon [37; 38; 39], Thomas [59] and Maunder [33].

(2) For applications to questions of embedding of real projective spaces in Euclidean spaces see Mahowald [30, §7] and Schwarzenberger and

Epstein [15].

(3) For an application to the number of ( linearly ) independent vector fields on manifold, for a least lower bound of such a number see [59] 116

(4) If M is a spin manifold of dimension n , and n = 3 mod 8 , 2n-4 n > 3 , then M immerses in R [ 59, Theorem 1.6].

§6.5. An Illustration

6.5.1. Remarks.

(1) Let £ be a stable vector bundle over a CW-complex, X .

Following Atiyah [5], % is said to be of geometric dimension n , if n

is the least positive integer k such that the following lifting problem

is solvable:

^»BO(k)

X > BO where BO is the classifying space for the infinite orthogonal group and

£ is the classifying map of £ .

One can use the method outlined in the last sections to find a bound for the geometric dimension of £ . Following Thomas' line, some

results concerning the geometric dimension of stable vector bundles over lens space have been obtained together with an immersion theorem

for lens space in Euclidean spaces I D. Sjerve, 50].

(2) For other applications, see D. Sjerve 151], to geometric dimension of stable vector bundles over orbit manifolds, to mod p connected K-theory, see J.C. Alexander 14]. 117

6.5.2. Consider the question of lifting a map, E: X >• BSO(n+l), to BSO(n), i.e., the problem' of. obtaining ."a commutative diagram:

BSO(n)

P

X ->- BSO (n+1)

Observe that BSO (n+1) is 1-connected so that the Serre Theorem applies to give the exact sequence:

- Hn(Sn) -»• Hn+1(BSO(n+l)) -*• Hn+1(BSO(n)) -> 0.

Thus Ker(P ) = Im(t) in dimension n+1 is generated by the Euler-Poincare

class, Xn+]_ •

We shall construct a Postnikov system for the fibration and compute some of its k-invariants. We know of course that the map, E, lifts to BSO(n) if and only if all the classes kX(£) vanish successively,

x where 'k.1 (E) is defined as in 6.4.3 when k1 (E) vanishes. We shall attempt to calculate the k1 for i <_ n+5 . So we have the following diagram: . . ^ ,

n S -> BSO(n)

Sl

K(Z, n) -*• E

'n+1 BSO(n+1) -»• K(Z, n+1) i.e., the first stage of a Postnikov system in which we have T(S.) = ± x • 1 n+1 118

Ker(p*) = Ker(p*) as x -, ~ " •, ( mod 2 ) ( (n+l)-th Stiefel-Whitney 1 " n+l n+l class ). By lemma 2.4.3(1) we see that the following is a fiber square:

uo'(lxq )=v K(Z, n) x BSO(n) > E

Pl BSO(n) y BSO(n+l)

11 1 where y: K(Z, n) x E >• E is the action of K(Z, n) on E . The Serre

Theorem applies (2.3.5(5)) to give an exact sequence:

T 0 y Ht(E1) •+ Hfc(K(Z, n) x BSO(n)) >• BSO(n+l) for t < 2n+l .

* Our aim now is to compute Ker(T^)^ Ker(s^) . Let l®b e * a H (K(Z, n)xBSO(n)) . Then T (l®b) = 0 for all b e H^(BSO(n)) and q <_ 2n , * since T^(1®1) = 0 and is an H (BSO(n+l))-morphism. Since q^ = vos^ where s^ is a cross-section of K(Z, n)xBSO(n), we see that, by definition * * of v , to find Ker(q1) it suffices to compute Ker(t1) /iKer(s1). In the

1 y fibration, K(Z,n) y E BSO(n+l) , we have T(I ) = ± x ., where n n+l i e H°(K(Z, n); Z) and T is the transgression in the fibration. Let n . i be the fundamental calss of sn. It is well known that T(I) = OJ e n+l n+2 — H (BSO(n+l)). Thus T (I 01) = tii . Hence in mod 2 coefficients .1 n n+l

T^tUl) = SqS^l) = Sq\+1 = Vl.Ui * using Wu's formula. Since is a H (BSO(n+l))-morphism, we have ®p T_ (i a) = io 'a In . n+l * ' *v for any a e H (BSO(n+l)) where i is the mod 2 reduction of i n n 6.5.3. Computation of Ker(q^). Here i denotes the mod 2 reduction of the fundamental class of K(Z, n) . . '.

Dimension n: T_ (x0l) = to ,. . 1 n+1 * Ignore this dimension as KerCs^) = 0 . * Dimension n+1: Ignore this dimension as Ker(s^) = 0 , since

Hn+1(K(Z, n)xBSO(n)) Hn+1(BSO(n)) . 2 2 Dimension n+2: T., Sq = Sq to . = to , _ • to_ and 1 ^ ^ n+1 n+1 2 U M T (10U ) = n+l' 2 1 2 • 2

Thus Sq (i®l) + \®to2 e Ker(r^) . 3 Dimension n+3: T.Sq (\<81) = to ., *to_ and 1 ^ n+1 3 T. (i®to_) = to '10 . 1 3 n+1 3 3

Thus Sq (iSl) + i©to3 e Kerd^) . 4

Dimension n+4: T,Sq (i®l) = to , *to. ; 1 ^ n+1 4 T l0 = U l< V Vl* 4 ;

2 0)^.0.2 ; TlSq (T0(O2) = («n+1'»2)'«2 - l U > = U Tl< « 2 Vl' 2 * 4 2 2 Thus Sq \®1 + i®to4 , Sq i®to2 + \®u)2 e Ker(x1) . 5 Dimension n+5: T,Sq (i81) = to , *to ; 1 n+1 o

3 •rlSq (i^2) = Vl'Vs U •V^da^) °Vl,U3 2 *0 - a. . ; ^•(1 )5) n+1 Ws

(l9w U > = 'Tl 2 3 Vi'Va *'

Tl(;®to3to2) = • 5 3 + x and So Sq \®t05 , Sq (i0to2) + &m^bi^,. 2

Sq (i®to3) + t®"3to2 e KerCx^) . 120

As a summary we list a table of Ker(t^) upto dimension n+5 , ignoring those not in Ker(s ) .

* Dimension Ker(T^) n Ker(s^)

<. n+l 0

2 + x®to n+2 A = Sq i®l 2

1 3 101 ©w n+3 Sq A = Sq + l 3

2 4 n+4 Sq A, B = Sq 101 +

3 12 3 2 n+5 Sq A = Sq Sq A = Sq (xBt*^) + Sq 100^;

2 1 5 Sq Sq A = Sq 101 + l©(w_ + to '(0 b 2 3 2 ;©co ; + Sq 3 1„ 5 Sq B = Sq 101 + 10U),.

6.5.4. Apply the previous discussion to the fibration,

Vn+2 2 ^ BSO(n) >- BSO (n+2) for n _ 2 .

Using Paechter's groups we get when n = 4s+l ( and so n = 1 mod 4 ) and s >_ 1 the following table:

* Dimension Ker(S;L) o Ker(TI)

<_ 4s+2 0 2 4s+3 A = Sq 101 + I0u2 1 12 1 4s+4 Sq A = Sq Sq 101 + Sq 100^ ; 2 1 1 B = Sq Sq 101 + Sq 10U)2

4s+5 Sq1B = Sq2Sq2l01 + Sq1!©!^

2

= Sq A + A*0)2 * 1 * Hence there exist k^,. e H (E ) such that v k^ A and v k, 12 'ii^ 83 and so Sq k4 + Sq k + k 0 .

* Dimension KerCq^

< 4s+2 0

4s+3 V

1 4s+4 Sq k3 , k4

4s+5 Sq k4 , Sq k3

These agree well with Mahowald's computations.

So E lifts iff all the k-invariants, k^(tj), vanish. The above shows that the main difficulties lie in computing them.

§6.6. Some Examples of The Use of Postnikov Towers

6.6.1. Suppose X is a finite dimensional CW-complex. If ^ is the fundamental class of SP, then, given any v e HP(X; Z) , there exists an integer N such that there exists a lifting of N*v to SP, provided

.N'V •* K(Z, p)

(1) p is odd, or

(2) p is even and v = 0 122

Proof. Case 1. Consider the standard Postnikov Tower for 8P: SP

-> K(Z, p) :

k+l

Vhk.T

p+1

Sq

Fp = K(Z, p) -> K(Z2, p+2) .

Since (S^ ) is always finite for k+l > p and p odd [24], we see that

for some integer n' , if n»v is liftable to F » •(n«v) is liftable to . Thus for p odd and for sufficiently large M, if N*v is liftable to F ' , then N-v can be lifted all the way and hence it lifts to SP. M 2 Case 2. p is even and v = 0 . If p is even then TT (SP) for m > p > 2 is always finite except m — for m = 2'p-l . As before there exists an integer N such that N «v lifts * 0 0 to F„ . Now TT _ (SP) is isomorphic to a direct sum of an infinite 2p-2 2p-l cyclic group and a finite abelian group [24] ..••As before if we can somehow alter N «v so that it lifts past F^ _ , then it lifts all the way. o 2p-2

Consider the Postnikov Tower for SP. Let 6P e HP(SP; Z) be the p p fundamental class. Plainly G = 0 . Suppose Vwv = 0 ; we see that

N vwN v = 0 too . We can even take k„ „: F„ „ *" K(ir (S*'), 2p) 0 o 2p-2 2p-2 to be induced by the cup-squaring operation. So this means the integral 123

F 2p-l

h2p-2 2p-2 SP , . F ^ K( (SPj ^ 2p) /" 2p-2 2p-l

2 / 0 Sq

X y K(Z, p) —: y K(Z , p+2)

part of k 2o(No»v) is homotopic to Zero. The rest of the proof proceeds as before to obtain a lifting of N*v for some positive integer N.

6.6.2. STEENROD CLASSIFICATION THEOREM [55].

Let X be a CW-complex and 6^ e Hn(sn; Z) be a generator where * n n n > 2 . Then the map induced by 6^ , 8^: [X, S ] y H (X; Z), maps onto { u E Hn(X; Z) ; Sq2u = 0 } if Dim(X) <_ n+2 ; and 6^ 1 (u) is in one-one correspondence with the quotient abelian group,

n+1 2 n_1 H (X; Z2)/Sq H (X; Z) , if Dim(X) <_ n+1 .

Proof. Looking at a Postnikov tower for Sn, we see that if

Dim(X) <_ n+2 the only obstruction to lifting a class u E Hn(X; Z) to

n 2 n+2 S is Sq (u) e H (X; Z2) .

The second statement can be proved using classical

obstruction theory for K(Z2, n+2) principal fibrations ( i.e. a

fibration of type K(Z2, n+2) ), noting that the higher difference obstructions all vanish if Dim(X) <_ n+1'.."'( See Spanier [53, Chapter 8]

ft for details.- " For a generalisation to H-spaces see B. Drachman 124

"A generalization of the Steenrod classification theorem to H-spaces",

Trans. Amer. Math. Soc. 153 '(1971), pp.53-88 .)

6.6.3. Remarks. (1) Thomas [60] using his method proved the following:

THEOREM. If M is a closed, connected, smooth manifold of dimension

4s+3 and s >_ 0 and if u, (M) = to (M) = 0 , then span M [59; 60] >_ 2 .

Generalising the technique by using cohomology operations he is able to show that if furthermore 6 to(M) = 0 or k is odd, then 4k span M > 3 .

(2) For a Stiefel manifold fibering ( cf. 6.5.4 )

McClendon is able to generalize the technique used by Thomas [59] to construct a Postnikov system for such fibering ( not necessarily orientable ) and obtain a lifting theorem [38, Theorem 6.1].

(3) There is a Postnikov system defined for spectra [4]. This fact does not seem to have been exploited until only recently. BIBLIOGRAPHY'

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