Fiber Bundles and Homotopy

337 Pages·2021·76.417 MB·English

by  Dai Tamaki

FIBER BUNDLES
AND HOMOTOPY
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FIBER BUNDLES
HOMOTOPY
AND
Dai Tamaki
Shinshu University, Japan
NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI • TOKYO
1122330088__99778899881111223377999977__TTPP..iinndddd 22 3300//44//2211 1100::3333 AAMM
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FIBER BUNDLES AND HOMOTOPY
Copyright © 2021 by World Scientific Publishing Co. Pte. Ltd.
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Preface
In this book, we first study basic properties of fiber bundles and then,
by abstracting their properties, we introduce fundamental concepts and
methodologies in homotopy theory.
Homotopytheoryisarelativelynewresearchareainthelonghistoryof
mathematics. Theideaofcontinuousdeformationwasmadeprecisearound
1920sor1930sintheattemptstomaketheideaofPoincar´erigorous. After
theWorldWarII,homotopytheoryhasbeendevelopedrapidlyandbecame
a matured field utilizing abstract and complex concepts.
Thenotionofhomotopy,i.e.continuousdeformation, naturallyappears
inmanyfields,especiallyingeometry. Oneoffirstapplicationsofhomotopy
theory is the study of manifolds, and related geometric objects. Homotopy
theory, then, grew up as an independent field of “theory of continuous
deformations”. The modern homotopy theory does not look like a kind of
geometry.
The aim of this book is to give an introduction to this highly abstract
modern homotopy theory by showing how fundamental concepts in homo-
topy theory arise from geometric problems. We choose fiber bundles as a
main example.
This book consists of six chapters and an appendix:
• Chapter 1: How to Bundle Fibers.
The aim of this chapter is to give an intuition of fiber bundles by
simple examples before a precise definition is introduced.
• Chapter 2: Covering Spaces as a Toy Model.
Basic properties of covering spaces are reviewed as a toy model of
relationsbetweenfiberbundlesandhomotopystudiedinChapter4.
v
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vi Fiber Bundles and Homotopy
• Chapter 3: Basic Properties of Fiber Bundles.
We begin with one of the most primitive forms of fiber bundles by
usingonlylocaltrivializationsandthenintroducestructuregroups
by using coordinate transformations, including principal bundles.
• Chapter 4: Classification of Fiber Bundles.
The aim of this chapter, and the first half of this book, is to show
that,fora“goodgroup”Ganda“goodspace”X,thesetP (X)of
G
isomorphismclassesofprincipalG-bundlesoverX canbeidentified
withthehomotopyset[X,BG]fromX totheclassifyingspaceBG
of G.
Afterpreparingnecessarytoolssuchasthecoveringhomotopythe-
orem,CWcomplexes,andhomotopygroups,theclassificationthe-
oremisprovedunderthehypothesisthatauniversalbundleexists.
And then two kinds of constructions of universal bundles are ex-
plained.
At the end of this chapter, we review properties of covering spaces
from the viewpoint of fiber bundles.
• Chapter 5: Fibrations.
We introduce the notion of fibration, more precisely, Hurewicz fi-
brationandSerrefibration,asageneralizationoffiberbundle,and
thenstudytheirbasicproperties. Wealsointroduceafurthergen-
eralization, called quasifibration.
• Chapter 6: Postscript.
The meaning and role of homotopy theory is explained from the
author’s personal viewpoint.
• Appendix A.
Theappendixconsistsofthreesections,whosetopicsarethemean-
ing of compact-open topology, vector bundles, and simplicial tech-
niques, respectively.
Fiberbundlesthemselveshavebeenoneofmostfundamentalstructures
used in many kinds of geometry, especially in differential geometry. In this
book,however,wewillnotdiscusssuchapplications,mainlybecauseofthe
author’s limited knowledge. The reader is encouraged to take a look at
other books for geometric applications of fiber bundles.
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Acknowledgments
IbegantolecturefiberbundlesinApril,1993. Itwasjusttwomonthsafter
I finished my Ph.D. at the University of Rochester under the guidance of
ProfessorFredCohen. ItwasmyfirstexperiencetoteachJapanesestudents
in Japanese. I have taught calculus and linear algebra in the united states,
but have never taught such advanced topics as fiber bundles. To make
things worse, I found most students are juniors on the contrary to my plan
to teach seniors and graduate students,
I am very grateful to the students who attended and listened to me in
the class. I was really encouraged to see them listening to me and taking
notes.
At the end of my 1993 class I distributed the first version of this note.
In 1996, I had the second chance to teach fiber bundles. I distributed a
revised version at the end of the class again. I was happy that several
students found mistakes and typos in the notes. Juno Mukai was generous
tooffermehisgranttoprintthisnotein2001. Iusedtheprintedversionas
atextbookforseniorsreadingcoursein2001,duringwhichlotsofmistakes
and typos were found. Their work made this note accurate enough to be
publicized. I owe a lot to the members, Aoki, Ohtou, Shimada, Tsuruta,
and Shiina.
Irealizedtheimportanceofhomotopytheoreticpropertiesoffiberbun-
dles when I was a graduate students at the University of Rochester. There
was a good atmosphere of discussing various topics in topology among
graduate students. My viewpoint on mathematics was formed during the
exchanges with the algebraic topologists at the University of Rochester
vii
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viii Fiber Bundles and Homotopy
including graduate students. I would like to thank them and professor
Akira Kono at Kyoto University who helped me to go to Rochester.
Ibecamefamiliarwithfibrationsandquasifibration,togetherwithclas-
sifying spaces, during my struggle to understand the paper [Rothenberg
and Steenrod (1965)] and May’s book [May (1972)] when I was a student
at Kyoto University with lots of help from Professor Kouyemon Iriye, Pro-
fessor Goro Nishida, and Professor Hiroshi Toda. I also learned practical
usesoffibrationsfromProfessorFredCohenattheUniversityofRochester.
I am deeply grateful to them.
Professor Takao Sato carefully read through a Japanese version and
pointed out lots of typos and mistakes. I am also grateful to him.
April9,2021 11:59 ws-book9x6 FiberBundlesandHomotopy 12308-main pageix
Contents
Preface v
Acknowledgments vii
List of Figures xi
1. How to Bundle Fibers 1
1.1 Simple Examples . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Tangent Bundles . . . . . . . . . . . . . . . . . . . . . . . 4
2. Covering Spaces as a Toy Model 7
2.1 Covering Spaces . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Paths and Their Lifts. . . . . . . . . . . . . . . . . . . . . 10
2.3 The Fundamental Group . . . . . . . . . . . . . . . . . . . 18
2.4 Universal Covering . . . . . . . . . . . . . . . . . . . . . . 27
3. Basic Properties of Fiber Bundles 33
3.1 Defining Fiber Bundles . . . . . . . . . . . . . . . . . . . . 33
3.2 Fiber Bundles with Structure Groups . . . . . . . . . . . . 38
3.3 Topological Groups . . . . . . . . . . . . . . . . . . . . . . 44
3.4 Compact-Open Topology . . . . . . . . . . . . . . . . . . . 50
3.5 Fiber Bundles and Group Action . . . . . . . . . . . . . . 57
3.6 Quotient Spaces by Group Actions . . . . . . . . . . . . . 67
3.7 Principal Bundles . . . . . . . . . . . . . . . . . . . . . . . 86
4. Classification of Fiber Bundles 99
4.1 Maps between Fiber Bundles . . . . . . . . . . . . . . . . 99
ix