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 {<j) } commute with the change of coefficient operator ? n 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 <j>®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 .