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An Obstruction Theory for Fibre-Preserving Maps Ryuji Maehara Iowa State University

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An Obstruction Theory for Fibre-Preserving Maps Ryuji Maehara Iowa State University Iowa State University Capstones, Theses and Retrospective Theses and Dissertations Dissertations 1972 An obstruction theory for fibre-preserving maps Ryuji Maehara Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/rtd Part of the Mathematics Commons Recommended Citation Maehara, Ryuji, "An obstruction theory for fibre-preserving maps " (1972). Retrospective Theses and Dissertations. 4754. https://lib.dr.iastate.edu/rtd/4754 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Retrospective Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. INFORMATION TO USERS This dissertation was produced from a microfilm copy of the original document. While the most advanced technological means to photograph and reproduce this document have been used, the quality is heavily dependent upon the quality of the original submitted. The following explanation of techniques is provided to help you understand markings or patterns which may appear on this reproduction. 1. The sign or "target" for pages apparently lacking from the document photographed is "Missing Page(s)". If it wras possible to obtain the missing page(s) or section, they are spliced into the film along with adjacent pages. This may have necessitated cutting thru an image and duplicating adjacent pages to insure you complete continuity. 2. When an image on the film is obliterated with a large round black mark, it is an indication that the photographer suspected that the copy may have moved during exposure and thus cause a blurred image. You will find a good image of the page in the adjacent frame. 3. When a map, drawing or chart, etc., was part of the material being photographed the photographer followed a definite method in "sectioning" the material. It is customary to begin photoing at the upper left hand corner of a large sheet and to continue photoing from left to right in equal sections with a small overlap. If necessary, sectioning is continued again — beginning below the first row and continuing on until complete. 4. The majority of users indicate that the textual content is of greatest value, however, a somewhat higher quality reproduction could be made from "photographs" if essential to the understanding of the dissertation. Silver prints of "photographs" may be ordered at additional charge by writing the Order Department, giving the catalog number, title, author and specific pages you wish reproduced. University Microfilms 300 North Zeeb Road Ann Arbor. Michigan 48106 A Xerox Education Company 73-9461 MAEHARA., 1935- M OBSTRUCTION THEORY FOR FIBRE-PRESERVING MAPS. Iowa State University, Ph.D., 1972 Matheniatics University Microfihns, A XEROX Company, Ann Arbor, Michigan THIS DISSERTATION HAS BEEN MICROFILMED EXACTLY AS RECEIVED. An obscrucuion theory for fibre-preserving maps by i^uji Maehara A Dissertation Submitted to the Graduate Faculty in Partial Fulfillment of The Requirements for the Degree of DOCTOR OF PHILOSOPHY Maj or: •Mathematics Approved: Signature was redacted for privacy. In Charge of Ma^or Work Signature was redacted for privacy. For the M jo Department Signature was redacted for privacy. For the Graduate College Iowa State University Ames, Iowa 1972 PLEASE NOTE: Some pages may have Indistinct print. Filmed as received. University Microfilms, A Xerox Education Company ii TABLE OF CONTENTS Page ACKNOWLEDGEMENTS iv I. INTRODUCTION AND PRELIMINARY NOTIONS 1 A. Introduction 1 1. Outline of the method 2 2. Applications 4 3. Notations 6 B. The Category CG of Compactly Generated Spaces 8 C. The Set L (E, E' ) and the Space E^*^ 13 II. OBSTRUCTIONS TO EXTENDING FIBRE-PRESERVING MAPS 25 A. The Topology of a Fibre Space over a CW-Complex 25 B. Characteristic Maps for Blocks 31 C. Extensions of Fibre-preserving Maps 37 D. 1-extensibility 43 III. COHOMOLOGICAL EXPRESSION OF THE OBSTRUCTIONS 48 A. A Modification of n^(X, X^, x^) 48 B. The Obstruction Cocycle and the Difference Cochain 52 C. An Existence Leirma and a Generalization of Eilenberg's Theorem 57 D. The Fibre-preserving Homotopy Problem 61 E. The Naturality Property and an Application 64 IV. APPLICATIONS 72 A. Preliminaries 72 B. Induced Principal Fibrations 77 iii Page C. Horaotopy Classes of Fibre-preserving Maps 86 D. A Consequence of Gottlieb's Theorem 101 V. BIBLIOGRAPHY 112 iv ACKNOWLEDGEMENTS I am indebted to my thesis advisor and to my committee chairman: Professor Danier H. Gottlieb of Purdue University, the thesis advisor, suggested the problem and pro­ vided invaluable advice during the period of research on the material presented in this dissertation and during the preparation of the manuscript. Professor James L. Cornette of Iowa State University, the committee chairman, gave me encouragement, good counsel and patient help throughout my studies. It was due to his kind suggestion and arrangement that I could have an opportunity to study algebraic topology under the guidance of Professor Gottlieb. I am more grateful to them than I can express here. I would also like to thank Dr. Richard J. Tondra for the discussions through which some problems became clear to me. Finally, I would like to acknowledge my indebtedness to the Department of Mathematics at the University of the Syukyus, of which I am a faculty member, for granting me a study leave for two and one-half years. Fall 1972 Ames, Iowa i^uji Maehara 1 I. INTRODUCTION AND PRELIMINARY NOTIONS A. Introduction Let p: 5 -»• B and p ': E ' ->• B ' be fibrations. A continuous map f : S ->• 3' is called a fibre-preserving map if for each b € 3 there is a b' € B' such that f(?-1 (b) ) (= p'-1 ^(b*). Suppose that there is given a fibre- preserving map f: p ^(A) -+ E' where A is a subset of B. Then the main problems which will be investigated in this thesis can be stated as follows. (1) Is there a fibre-preserving extension g: E ^ E' of f? (2) If g^,g^: E -> E* are two such extensions, does there exist a homotopy {g^}: E ^ E' (0 _< t £ 1) between them such that, for each t, g^ is a fibre-preserving extension of f? The basic assumption we impose is that (B, A) is a con­ nected relative CM-complex. When p and p' are 'fibre bundles', similar problems have been studied by Dugundji, Heller, Conner and Floyd (see [11],[19],[7]). They require f and its extensions to be 'equivariant' with respect to the actions of the group of the fibre bundle. Heller [19] has shown that the equivariant extension problem can be reduced to the cross section problem in an appropriate fibre bundle. Such a reduction, however, is not likely to be applicable to problems (1) and (2). 2 If E = B, E'=B' and P/P' are the identity maps, then (1) and (2) become the classical problems on 'free' extensions B B' of the map f: A B' to which the obstruction theory due to Eilenberg is applicable. Our main aim is to establish a generalization of Eilenberg's theory. 1. Outline of the method We consider, briefly, problem (1). For simplicity of the description, let us assume (B, A) to be a pair of simplicial complexes. Let B ^ denote the relative n-skele- ton of (B, A) and let = p ^(B ^). Assuming that f has a fibre-preserving extension g; E^ E', we will look for a fibre-preserving extension to g': E^_^^ E'; if g has no such extension, we will reconsider g in the hope of finding another extension of g| E^_^ which admits a fibre-preserving extension over For each (n+1)-simplex a of B - A, the partial fibra- tion p (CT) ^ a is equivalent to the projection o % F o where F = p ^Xb^) (b^ € B), and so we obtain the following diagram: a X F —I—> p"l(o) U U Ô X F —I >p~^(â) —— > E' where <f) denotes a fibre homotopy equivalence, a is the boundary of a, and g^ = g| p ^(a). It will be shown that 3 has a fibre-preserving extension g^: p ^(a) E' if and only if the map g^ ô x f has a fibre-preserving extension to a x F (Lemma 2.10). On the other hand, the obstruction to the latter property can be expressed by the homotopy class, say c^^^(g)(a), of the adjoint map à E' where E' is an appropriately defined subspace of the mapping space E'F (Corollary 1.11). Roughly speak­ ing, the assignment c^^^(g): a#—»c^^^(g)(a) € Tr^(E'^^^) becomes an (n+1)-cochain of (B, A) with coefficients in (F) •Tjj(E ' ) and so we obtain a cohomology class [C*+I(g)] € H^'^^(B,A; n^^E'fF))). This class measures the obstruction for the fibre-preserving extensibility of g| over (Theorem 3.9). In the arguments outlined above, however, we will soon encounter several difficulties. The first difficulty arises (F) from the fact that the continuity of a map a —» E * does not necessarily inç>ly the continuity of its adjoint a X F E'. Second, in order to construct a global continu­ ous map g': -»• E' from maps g^; p~^(a) —» E' we need the 'weak topology' of To overcome these difficulties we shall restrict the category in which we work to Steenrod's category ÇG of conpactly generated spaces (see Section B). Then, without any other assumption, we can formulate the (n+1)-extensibility of a fibre-preserving map g: ->• E' by 4 the triviality of a certain homomorphism of homotopy groups (Theorem 2.14).
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