Table Of ContentTEXTS AND READINGS
27
IN MATHEMATICS
Aigebraic Topology
A Primer
Texts and Readings in Mathematics
Advlsory Editor
C. S. Seshadri, Chennai Mathematical Inst., Chennai.
Managlng Editor
Rajendra Bhatia, Indian Statistical Inst., New Delhi.
Editors
V. S. Borkar, Tata Inst. of Fundamental Research, Mumbai.
Probai Chaudhuri, Indian Statistical Inst., Kolkata.
R. L. Karandikar, Indian Statistical Inst., New Delhi.
M. Ram Murty, Queen's University, Kingston.
V. S. Sunder, Inst. of Mathematical Sciences, Chennai.
M. Vanninathan, TIFR Centre, Bangalore.
T. N. Venkataramana, Tata Inst. of Fundamental Research,
Mumbai.
Aigebraic Topology
A Primer
Satya Deo
Rani Durgawati University
Jabalpur
HINDUSTAN
fl::[glJg0l o
U l!!J UBOOK AGENCY
Published by
Hindustan Book Agency (lndia)
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Copyright © 2003, Hindustan Book Agency (lndia)
Copyright © Corrected Reprint, 2006, Hindustan Book Agency (lndia)
No part of the material protected by this copyright notice may be
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copyright owner, who has also the sole fight to grant licences far
translation into other languages and publication thereof.
This edition for sale in India only. Unauthorized export is a violation
of Copyright Law and is subject to legal action.
Produced from camera ready copy supplied by the Author.
ISBN 978-81-85931-68-5 ISBN 978-93-86279-13-2 (eBook)
DOI 10.1007/978-93-86279-13-2
PREFACE
Aigebraic Topology is an important branch of topology having several con
nections with many areas of modern mathematics. Its growth and inftuence,
particularly since the early forties of the twentieth century, has been remark
ably high. Presently, it is being taught in many universities at the M.A.jM.Sc.
or beginning Ph.D. level as a compulsory or as an elective course. It is best
suited for those who have already had a beginning course in topology as weIl
as in algebra. There are several exceIlent books, starting with the first mono
graph "Foundations of Aigebraic Topology" by S. Eilenberg and N.E. Steenrod,
which can be prescribed as textbooks for a first course on algebraic topology
by making proper selections. However, there is no general agreement on what
should be the "first course" in this subject. Experience suggests that a compre
hensive coverage of the topology of simplicial complexes, simplicial homology
of polyhedra, fundamental groups, covering spaces and some of their classical
applications like invariance of dimension of Euclidean spaces, Brouwer's Fixed
Point Theorem, etc. are the essential minimum which must find a place in a
beginning course on algebraic topology. Having learnt these basic concepts and
their powerful techniques, one can then go on in any direction of the subject
at an-advanced level depending on one's interest and requirement.
This book is designed to serve as a textbook for a first course in algebraic
topology described above. In order to lay down the real foundation of algebraic
topology, we have also included abrief introduction to singular homology and
cohomology. Approach to the contents has been dictated by several advanced
courses taught by the author on different topics of algebraic topology at many
universities. It is necessary to mention that the subject appears a bit abstract
to begin with, but after a while it presents concrete topological as weIl as geo
metrical results of great insight and depth. The beginning student is expected
to have some patience before appreciating the depth of these results. Maxi
mum care has been taken to emphasize the subtle points of the concepts and
results in a lucid manner. Examples are given at every stage to illustrate and
bring out the underlying concepts and the results. It is hoped that the detailed
explanations will help the student to grasp the results correctly and to have
asound understanding of the subject with confidence. The prerequisites for
the study of this book are very little, and these have been briefty discussed in
Chapter 1 and the Appendix to make it self-contained. These are included to
vi PREFACE
maintain eontinuity between what the student has already learnt and what he
is going to learn. It may be either quiekly reviewed or even skipped.
We introduce and study the fundamental groups and its properties in Chap
ter 2. Starting with the eoneept of pointed spaees we show that the fundamental
groups are topological invariants of a path-eonneeted spaee. After computing
the fundamental group of the eircle, we show how it ean be used to eompute
fundamental groups of other spaees by geometrie methods. In Chapter 3 we ex
plain the topology of simplicial complexes, introduee the notion of baryeentrie
sub division and then prove the simplicial approximation theorem. In Chapter
4 we introduce the first classical homology theory, viz., the simplicial homology
of a simplicial complex and then proeeed to define the simplieial homology of
a compact polyhedron. We provide detailed discussion of how a continuous
map between compact polyhedra induees a homomorphism in their simplicial
homology, and then prove the topological invarianee of these groups. A few
classical applications of the techniques of homology groups include the proof
of the fact that the Euclidean spaces of different dimensions are not home
omorphie, Brouwer's Fixed Point Theorem, Borsuk-Ulam Theorem, and the
Lefschetz Fixed Point Theorem ete. In Chapter 5, we study the theory of
covering projections and its relation with fundamental groups. Important re
sults on lifting of a map, classification of eovering projections and the universal
coverings have been discussed at length.
The final topic dealing with the singular homology and eohomology has
been diseussed in Chapter 6. We have presented the important properties of
singular homology including proofs of the homotopy axiom and the excision
axiom. All of this is done in a way so that singular homology gets established
as a "homology theory" on the eategory of all topologieal pairs in the sense of
Eilenberg-Steenrod. Then we go on to formally explain the definitions of an
abstract homology as weil as an abstract eohomology theory. Apart from the
singular homology, we show that simplieial homology is also a homology theory
on the category of all compact polyhedral pairs. Related topics such as homol
ogy and cohomology with coefficients, the Universal Coeffieient Theorem, the
Künneth Formula, the Mayer-Vietoris Sequenee and the cohomology algebra of
aspace etc., have been discussed and adequately illustrated byexamples. The
student can test his understanding of the subject by working out the exercises
given in every chapter. All of the basic material covered in this book should
make the subject fascinating for beginners and should enable them to pursue
advanced courses in any branch of algebraic topology.
Except for a few remarks, we have refrained from giving the detailed history
of the subject which is, of course, always very enlightening and interesting.
However, we feel that it can be appreciated adequately when the student has
learnt the subject at a more advanced level than what is presented here. The
material presented he re is contained in most of the books on algebraic topology.
My contribution is basically in organizing and presenting it afresh. Definitions,
vii
propositions, corollaries etc. are numbered by 3 digits, i.e., Theorem 4.5.1
means Theorem 1 of Section 5 of Chapter 4. A suggested guideline for teaching
a two-semester (or one academic year) course from this book is as follows: in the
first semester one can cover Fundamental Group (Sections 2.1 to 2.6), Simplicial
Complexes (Sections 3.1 to 3.4), Simplicial Homology (Sections 4.1 to 4.8) and
Covering Projection (Sections 5.1 to 5.6). The chapter on Covering Projection
can be taught immediately after the chapter on Fundamental Group. Proofs
of some of the technical theorems given in Chapters 4 and 5 can be omitted.
The remaining section of Chapter 4 on Simplicial Homology, viz. Section 4.9
and the whole of Chapter 6 on Singular Homology can be easily covered in
the second semester. It is hoped that the material prescribed for the first
semester is quite adequate for acquainting the student with the basic concepts
of algebraic topology. The material for the second semester, on the other hand,
will a110w hirn to take a deeper plunge in any of the .1dvanced topics of modern
algebraic topology.
The author would like to express his thanks to many of his friends and
colleagues, specially to G.A. Swarup, A.R.· Shastri, Ravi Kulkarni and K.
Varadarajan, all of whom have read the manuscript and made important sug
gestions für improvement of the book. The author appreciates the facili
ties made available to hirn by the Department of Mathematics, University of
Arkansas, where the first draft of the whole book was completed. Rajendra
Bhatia, the Managing Editor of the series "Texts and Readings in Mathemat
ics" of the Hindustan Book Agency is thanked for his keen interest in the
editorial work. I thank my wife Prema Tripathi for her insistence that the job
of completing the book should get preference over administrative duties. Fi
nally, J.K. Maitra, one of my students and a colleague, deserves special thallks
for his constant assistarlce in typesetting the book on Latex.
Satya Deo
July 1, 2003 Jabalpur
Added Preface
In this second printing several errors, typographical and otherwise, pointed
out by friends and colleagues, have been corrected. Exercises in various chap
ters have been increased. A small section on Tor and Ext functors has been
added in the Appendix for completeness. Thanks are due to V. V. Awasthi for
typesetting this second printing on Latex.
Satya Deo
Harish-Chandra Research Institute,
July 24, 2006 Allahabad.
Contents
PREFACE v
1 Basic Topology: a review 1
1.1 Basic Topology . . . . . . . . . . . . . . . . . 1
1.1.1 Euclidean Spaces and their Subspaces 1
1.1.2 Product Spaces ............ . 4
1.1.3 Continuous Maps and Homeomorphisms . 5
1.1.4 Quotient Spaces ............. . 10
1.1.5 Connected and Path-Connected Spaces 18
1.1.6 Compact Spaces and Locally Compact Spaces . 24
1.1.7 Compact Surfaces .. 28
1.2 What is Aigebraic Topology? 31
2 The Fundamental Group 36
2.1 Introduction ................ . 36
2.2 Homotopy ................. . 39
2.3 Contractible Spaces and Homotopy Type 42
2.4 Fundamental Group and its Properties 51
2.5 Simply Connected Spaces ..... 69
2.6 The Fundamental Group of Circle 74
3 Finite Simplicial Complexes 85
3.1 Simplicial Complexes . . . . 85
3.2 Polyhedra and Triangulations 95
3.3 Simplicial Approximation . . 103
3.4 Barycentric Subdivision - Simplicial
Approximation Theorem ...... . 106
4 Simplicial Homology 116
4.1 Introduction.... 116
4.2 Orientation of Simplicial Complexes ... 117
4.3 Simplicial Chain Complex and Homology 120
4.4 Some Examples . . . . . . . . . . . . . . 127
4.5 Properties of Integral Homology Groups . 137
x CONTENTS
4.6 Induced Homomorphisms . . . . . . . 151
4.7 Some Applications . . . . . . . . . . . 155
4.8 Degree of a Map and its Applications . 159
4.9 Invariance of Homology Groups ... . 167
4.9.1 Sub division Chain Map ... . 167
4.9.2 Homomorphism Induced by Continuous Map 172
4.9.3 Homotopy Invariance ..... 172
4.9.4 Lefschetz Fixed Point Theorem 174
4.9.5 The Borsuk - Ulam Theorem 178
5 Covering Projections 185
5.1 Introduction ................. . 185
5.2 Properties of Covering Projections .... . 189
5.3 Applications of Homotopy Lifting Theorem 194
5.4 Lifting of an Arbitrary Map ...... . 198
5.5 Covering Homomorphisms ....... . 200
5.6 Universal Covering Space - Applications 205
6 Singular Homology 212
6.1 Introduction............ 212
6.2 Singular Chain Complex ..... 213
6.3 One-Dimensional Homology and
Fundamental Group . . . . . . . . . . . . 218
6.4. Homotopy Axiom for Singular Homology . 226
6.5 Relative Homology and the Axioms. 229
6.6 The Excision Theorem . . . . . . . . 233
6.7 Homology and Cohomology Theories 239
6.8 Singular Homology with Coefficients 245
6.9 Mayer-Vietoris Sequence ...... . 253
6.10 Singular Cohomology and Cohomology Algebra 258
Appendix 271
A.1 Basic Algebra - a Review ....... . 271
A.1.1 Groups and Homomorphisms .. 271
A.1.2 Direct Product and Direct Sum . 273
A.1.3 The Structure of a Finite Abelian Group 274
A.1.4 Free Groups and Free Products 275
A.1.5 Modules and their Direct Sum 280
A.2 Categories and Functors . . . . . 282
A.2.1 The Hom(M, N) Functor .. . 286
A.2.2 Exact Sequences ....... . 286
A.2.3 The Tensor Product of Modules
and Homomorphisms. . . . . . . . . . . . 290
A.2.4 Chain Complexes and Homology ..... 300
A.2.5 Tensor Product of two Chain Complexes . 303
