· 2020. 10. 29. · preface algebraic topology is one of the most important creations in...

Mahima Ranjan Adhikari

Basic AlgebraicTopology andits Applications

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Basic Algebraic Topology and its Applications


Mahima Ranjan Adhikari

Basic Algebraic Topologyand its Applications

123


Mahima Ranjan AdhikariInstitute for Mathematics, Bioinformatics,Information Technology and ComputerScience (IMBIC)

KolkataIndia

ISBN 978-81-322-2841-7 ISBN 978-81-322-2843-1 (eBook)DOI 10.1007/978-81-322-2843-1

Library of Congress Control Number: 2016944483

© Springer India 2016This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or partof the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmissionor information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilarmethodology now known or hereafter developed.The use of general descriptive names, registered names, trademarks, service marks, etc. in thispublication does not imply, even in the absence of a specific statement, that such names are exempt fromthe relevant protective laws and regulations and therefore free for general use.The publisher, the authors and the editors are safe to assume that the advice and information in thisbook are believed to be true and accurate at the date of publication. Neither the publisher nor theauthors or the editors give a warranty, express or implied, with respect to the material contained herein orfor any errors or omissions that may have been made.

Printed on acid-free paper

This Springer imprint is published by Springer NatureThe registered company is Springer (India) Pvt. Ltd.


Tomy parents Naba Kumar andSnehalata Adhikariwho created my interest in mathematicsat my very early childhood


Preface

Algebraic topology is one of the most important creations in mathematics whichuses algebraic tools to study topological spaces. The basic goal is to find algebraicinvariants that classify topological spaces up to homeomorphism (though usuallyclassify up to homotopy equivalence). The most important of these invariants arehomotopy groups, homology groups, and cohomology groups (rings). The mainpurpose of this book is to give an accessible presentation to the readers of the basicmaterials of algebraic topology through a study of homotopy, homology, andcohomology theories. Moreover, it covers a lot of topics for advanced students whoare interested in some applications of the materials they have been taught. Severalbasic concepts of algebraic topology, and many of their successful applications inother areas of mathematics and also beyond mathematics with surprising resultshave been given. The essence of this method is a transformation of the geometricproblem to an algebraic one which offers a better chance for solution by usingstandard algebraic methods.

The monumental work of Poincaré in “Analysis situs”, Paris, 1895, organizedthe subject for the first time. This work explained the difference between curvesdeformable to one another and curves bounding a larger space. The first one led tothe concepts of homotopy and fundamental group; the second one led to the conceptof homology. Poincaré is the first mathematician who systemically attacked theproblems of assigning algebraic invariants to topological spaces. His vision of thekey role of topology in all mathematical theories began to materialize from 1920.This subject is an interplay between topology and algebra and studies algebraicinvariants provided by homotopy, homology, and cohomology theories. Thetwentieth century witnessed its greatest development.

The literature on algebraic topology is very vast. Based on the author’s teachingexperience of 50 years, academic interaction with Prof. B. Eckmann and Prof.P.J. Hilton at E.T.H., Zurich, Switzerland, in 2003, and lectures at different insti-tutions in India, USA, France, Switzerland, Greece, UK, Italy, Sweden, Japan, andmany other countries, this book is designed to serve as a basic text of modernalgebraic topology at the undergraduate level. A basic course in algebraic topology

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presents a variety of phenomena typical of the subject. This book conveys the basiclanguage of modern algebraic topology through a study of homotopy, homology,and cohomology theories with some fruitful applications which display the greatbeauty of the subject. For this study, the book displays a variety of topologicalspaces: spheres, projective spaces, classical groups and their quotient spaces,function spaces, polyhedra, topological groups, Lie groups, CW-complexes,Eilenberg–MacLane spaces, infinite symmetric product spaces, and some otherspaces. As well as, the book studies a variety of maps, which are continuousfunctions between topological spaces.

Characteristics which are shared by homeomorphic spaces are called topologicalinvariants; on the other hand, characteristics which are shared by homotopyequivalent spaces are called homotopy invariants. The Euler characteristic is anintegral invariant, which distinguishes non-homeomorphic spaces. The search ofother invariants has established connections between topology and modern algebrain such a way that homeomorphic spaces have isomorphic algebraic structures.Historically, the concepts of fundamental groups, higher homotopy groups, andhomology and cohomology groups came from such a search. The natural emphasisis: to solve a geometrical problem of global nature, one first reduces it to ahomotopy-theoretic problem; this is then transformed to an algebraic problemwhich provides a better chance for solution. This technique has been the mostfruitful one in algebraic topology. The notions initially introduced in homology andhomotopy theories for applications to problems of topology have found fruitfulapplications in other parts of mathematics. Homological algebra and K-theory aretheir outstanding examples.

The materials discussed here have appeared elsewhere. Each chapter opens witha short introduction which summarizes the information that sets out its centraltheme and closes with a list of sources for the use of readers to expand theirknowledge. This does not mean that other sources are not good. Our contribution isthe selection of the materials and their presentation. Each chapter is split intoseveral sections (and subsections) depending on the nature of the materials whichconstitute the organizational units of the text. Each chapter provides exercises withfurther applications and extension of the theory. Some exercises carry hints whichshould not be taken as ideal ones. Many of them can be solved in a better way. Thetitle of the book suggests the scope and power of algebraic topology and its text isexpanded over 18 chapters and two appendices displayed below.

Chapter 1 assembles together some basic concepts of set theory, algebra,analysis, set topology, Euclidean spaces, manifolds with some standard notationsfor smooth reading of the book.

Chapter 2 is devoted to the study of basic elementary concepts of homotopy theorywith illustrative examples. A homotopy formalizes the intuitive idea of continuousdeformation of a continuous map between two topological spaces. It displays avariety of phenomena and related problems such as homotopy classification ofcontinuous maps up to homotopy equivalence introduced by Hurewicz (1904–1956)in 1935, contractible spaces, H-groups (Hopf groups) and H-cogroups through theirhomotopy properties. Finally, this chapter presents interesting immediate

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applications of homotopy in dealing with some extension problems, lifting problems,and proving “Fundamental theorem of algebra” by using homotopic concepts.

Chapter 3 continues the study of homotopy theory through the concept offundamental groups invented by H. Poincaré in 1895 which conveys the firsttransition from topology to algebra by assigning an algebraic structure to the set ofrelative homotopy classes of loops in a functorial way. The group structure of thesehomotopy classes of loops is proved in Sects. 3.1 and 3.2 in two different ways.This group earlier called Poincaré group is now known as fundamental group. It isthe first influential invariant of homotopy theory and is also the first of a series ofalgebraic invariants πn, called homotopy groups studied in Chap. 7. This chaptercomputes fundamental group of the circle by using the degree of a continuous mapf : S1 ! S1, and studies Brouwer fixed point theorem for dimension 2, fundamentaltheorem of algebra, vector field problems on D2 and knot groups by using the toolsof fundamental groups.

Chapter 4 continues the study of the fundamental groups and presents a thoroughdiscussion of covering spaces which are deeply connected with fundamental groups.Algebraic features of the fundamental groups are expressed by the geometriclanguage of covering spaces. This chapter is designed to utilize the power of thefundamental groups and also to establish an exact correspondence between thevarious connected covering spaces of a given topological space B and subgroups ofits fundamental group π1ðBÞ, like Galois theory, with its correspondence betweenfield extensions and subgroups of Galois groups, which is an amazing result. Thischapter also studies the concepts offibrations and cofibrations with their applicationsborn in geometry and topology.

Chapter 5 continues the study of homotopy theory through fiber bundles, vectorbundles, and K-theory. Covering spaces provide tools to study the fundamentalgroups. Fiber bundles provide likewise tools to study higher homotopy groups(which are generalization of fundamental groups and described in Chap. 7). Theimportance of fiber spaces was realized during 1935–1950 to solve several problemsrelating to homotopy and homology. The motivation of the study of fiber bundlesand vector bundles came from the distribution of signs of the derivatives of the planecurves at each point. This chapter also discusses homotopy classification of vectorbundles, Milnor’s construction of a universal fiber bundle for a topological group Gwith homotopy classification of principal G-bundles and presents the introductoryconcept of K-theory born in connecting the rich structure of vector bundles over aparacompact space B with the set of homotopy classes of maps from B into theGrassmann manifold of n-dimensional subspaces in infinite-dimensional space. Thistheory plays a vital role in applications of algebraic topology to analysis, algebraicgeometry, topology, ring theory, and number theory.

Chapter 6 builds up interesting topological spaces called polyhedra fromsimplexes followed by a study of their homotopy properties and develops some toolsfor computing the fundamental groups of a large class of topological spaces. Thegeometrical objects such as points, edges, triangles, and tetrahedra are examples oflow-dimensional simplexes. Simplicial complexes provide a convenient way to

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study manifolds. This chapter considers how simplexes may be fitted together toconstruct simplicial complexes which play an important role to construct interestingtopological spaces such as polyhedra for the study of algebraic topology. They formbuilding blocks of homology theory which begins in Chap. 10. The concept oftriangulation is utilized to solve extension problems and that of edge-path to showthat edge-group EðK; vÞ is isomorphic to the fundamental group π1ðjKj; vÞ for anysimplicial complex K. Finally, van Kampen theorem is proved by usinggraph-theoretic results. This chapter also proves simplicial approximation theoremgiven by L.E.J. Brouwer (1881–1967) and J.W. Alexander (1888–1971) around1920 by utilizing a certain good feature of simplicial complexes introduced byAlexander. This theorem plays a key role in the study of homotopy and homologytheories.

Chapter 7 continues to study homotopy theory displaying the construction of asequence of functors πn given by W. Hurewicz (1904–1956) in 1935 from topologyto algebra by extending the concept of fundamental group invented by H. Poincaré in1895. The basic idea of homotopy groups is to classify all continuous maps from Sn

to pointed topological space X up to homotopy equivalence. To study topologicalspaces X of low dimension, the fundamental group π1ðXÞ is very useful. But it needsrefined tools for the study of higher dimensional spaces. For example, fundamentalgroup cannot distinguish spheres Sn with n� 2. Such a limitation of low dimensioncan be removed by considering the natural higher dimensional analogues of π1ðXÞ.More precisely, this chapter defines the nth (absolute) homotopy group andgeneralizes it to a (relative) homotopy group of a triplet and studies algebraic,functorial and fibering properties with the exactness of homotopy sequence offibering, Hopf maps introduced by H. Hopf (1894–1971) in 1935 for the investi-gation of certain homotopy groups of Sn, action of π1 on πn, Freudenthal suspensiontheorem given by H. Freudenthal (1905–1990) in 1937 for the investigation of thehomotopy groups πmðSnÞ for 0\m\n, weak fibration which plays a key role in thestudy of higher homotopy groups, and the nth cohomotopy set πnðX;AÞ on which K.Borsuk (1905–1982) endowed in 1936 with an abelian group structure under certainconditions on ðX;AÞ. This chapter also discusses some interesting applications ofhigher homotopy groups.

Chapter 8 continues to study homotopy theory through a suitable special class oftopological spaces, called CW-complexes introduced by J.H.C. Whitehead (1904–1960) in 1949 to meet the need for further development of homotopy theory. Thisclass of spaces is broader and has some better categorical properties than simplicialcomplexes, but still retains a combinatorial nature that allows for computation (oftenwith a much smaller complex). The concept of CW-complexes is introduced as anatural generalization of the concept of polyhedra by relaxing all “linearity condi-tions” in simplicial complexes, instead cells are attached by arbitrary continuousmaps starting with a discrete set, whose each point is regarded as a 0-cell.A CW-complex is built up by successive adjunctions of cells of dimensions1; 2; 3; . . .: There is an analogy between what can be done topologically with a space,and what can be done algebraically with its chain groups. In the class ofCW-complexes this analogy attains its highest strength. The category of

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CW-complexes is a suitable category for a systematic study of algebraic topology.Algebraic topologists now feel that a study of CW-complex should be included inthe basic course of algebraic topology, and this study should move to the theoremthat every continuous map between CW-complexes is homotopic to a cellular map.This chapter studies the basic aspects of CW-complexes and relative CW-complexeswith their homotopy properties and proves Whitehead theorem, Freudenthalsuspension theorem (general form), and cellular approximation theorem with theirapplications.

Chapter 9 continues to study homotopy theory through the different products inhomotopy groups such as the Whitehead product introduced by J.H.C. Whiteheadin 1941, mixed product introduced by McCarty in 1964, and Samelson product.Whitehead product provides a technique at least in some cases for constructingnonzero elements of the homotopy group πpþ q�1ðXÞ of a pointed topological spaceX. Moreover, this chapter finds a generalization of Whitehead product, establishescertain relation between Whitehead and Samelson products, and studies mixedproducts corresponding to a fiber space and a topological transformation groupacting on it.

Chapter 10 begins to study homology and cohomology theories. Homotopygroups are very difficult to compute. There is an alternative approach of constructionof a different topological invariant, the so-called homology group, which historicallycame earlier than homotopy groups. Homology (cohomology) theory is a covariant(contravariant) functor from the category of topological spaces to the category ofabelian groups. Homology (simplicial) invented by H. Poincaré in 1895 is one of themost fundamental influential inventions in mathematics. The basic idea of homologyis that it starts with a geometric object (a space) which is given by combinatorial data(a complex). Then the linear algebra and boundary relations determined by this dataare used to construct homology groups. The simplicial devices in simplicialhomology theory are gradually generalized to singular homology by using thealgebraic properties of the singular complex. There exist different homologytheories: simplicial, singular, cellular, and C

^

ech homology theories which arestudied in this chapter. The most important homology theory in algebraic topology isthe singular homology. Simplicial homology is the primitive version of singularhomology. Cohomology theory is also discussed. In some sense, homology theoryand cohomology theory are dual to each other. More precisely, this chapter beginswith a study of the concepts of chain complex, boundary, cycle introduced byW. Mayer (1887–1947) in 1929 from a purely algebraic viewpoint. This chapterpresents a construction of the homology groups of a simplicial complex in two steps:first by assigning to each simplicial complex a certain complex, called chain complexfollowed by assigning to the chain complex its homology group. This constructionassigns a group structure to cycles that are not boundaries with an extension to theconcept of relative simplicial homology groups and generalizes simplicial homologytheory to singular homology theory. These two theories are related by the basic resultthat the singular homology of a polyhedron is isomorphic to the simplicial homologyof any of its triangulated simplicial complexes. This chapter examines the relations

Preface xi


between absolute homology groups of simplicial chain complexes and the relativehomology groups of relative simplicial chain complexes by using the language ofexact sequences and shows that the relative homology groups HpðK; LÞ for any pairðK; LÞ of simplicial complexes fit into a long exact sequence. This chapter alsodiscusses homology groups HnðX;GÞwith an arbitrary coefficient group G (abelian),Mayer–Vietoris sequences in singular and simplicial homology theories, cupproduct, and gives the Künneth formula and Eilenberg–Zilber theorem which areused for computing homology or cohomolgy of product spaces, and Eulercharacteristic & Jordan curve theorem from the viewpoint of homology theory.

Chapter 11 studies a special class of CW-complexes having only one nonzerohomotopy groups, called Eilenberg–MacLane spaces which were introduced byS. Eilenberg (1915–1998) and S. MacLane (1909–2005) in 1945. Such spaces forma very important class of CW-complexes in algebraic topology. Their importance istwofold: they develop both homotopy and homology theories. They are closelylinked with the study of cohomology operations. This chapter presents Eilenberg–MacLane spaces with their construction and studies their homotopy properties. Theconstruction process of Eilenberg–MacLane spaces KðG; nÞ for all possible ðG; nÞdepends on a very natural class of spaces, called Moore spaces of type ðG; nÞ,denoted by MðG; nÞ: This chapter also studies Postnikov towers to meet the needfor construction of Eilenberg–MacLane spaces.

Chapter 12 presents an approach formulating axiomatizaton of ordinaryhomology and cohomology theories. These axioms, now called Eilenberg andSteenrod axioms were announced by S. Eilenberg (1915–1998) and N.E. Steenrod(1910–1971) in 1945, but first appeared in their celebrated book Foundations ofAlgebraic Topology in 1952. This approach came from the problem of comparingthe various definitions of homology and cohomology given in the previous years.Eilenberg and Steenrod initiated a new approach by taking a small number of theirproperties (not focussing on machinery used for construction of homology andcohomology groups) as axioms to characterize a theory of homology andcohomology. This axiomatic approach simplifies the proofs of many lengthy andcomplicated theorems and escapes the avoidable difficulty to motivate the studentswho are learning homology and cohomology theories for the first time as theirsystematic study. This axiomatic approach classifies and unifies different homologygroups on the category of compact triangulable spaces and inaugurates its dualtheory called cohomology theory. This approach is the most important contributionto algebraic topology since the invention of the homology groups by Poincaré in1895.

Chapter 13 continues the study of homology and cohomology theories bypresenting some of their interesting properties which directly follow from theEilenberg and Steenrod axioms for homology and cohomology theories such ashomotopy equivalence in these theories, relations between cofibrations andhomology theory, and finally computes the ordinary homology groups of Sn withcoefficients in an arbitrary abelian group G.

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Chapter 14 presents further interesting applications of the homotopy, homology,and cohomology theories. The notions initially introduced in these theories to solveproblems of topology that have fruitful applications, and proves many interestingtheorems such as Hopf’s classification theorem, hairy ball theorem, ham sandwichtheorem, Borsuk–Ulam theorem, Lusternik–Schnirelmann theorem, Lefschetz fixedpoint theorem, and Jordan curve theorem. It also proves some results related tograph theory, fixed point theory of continuous maps, vector fields, and applicationsto algebra. Moreover, this chapter indicates some applications of algebraic topologyin physics, chemistry, economics, biology, and medical science with specificreferences.

Chapter 15 conveys the concept of a spectrum originated by F.L. Lima (1929–)in 1958 and constructs its associated spectral homology and cohomology theories,and generalized homology and cohomology theories (which have been proved to bevery useful theories) to distinguish them from ordinary homology and cohomologytheories. Their properties and relations to homotopy theory are also discussed. Forexample, the ordinary homology group of certain topological spaces X can bethought of as an approximation to πnðXÞ. Moreover, this chapter constructs a newΩ-spectrum A, generalizing the Eilenberg–MacLane spectrum KðG; nÞ and alsoconstructs its associated cohomology theory h�ð;AÞ which generalizes the ordinarycohomology theory of Eilenberg and Steenrod. This chapter conveys K-theory as ageneralized cohomology theory and also studies the Brown representabilitytheorem, stable homotopy groups, the cohomology operations, and Poincaré dualitytheorem.

Chapter 16 studies a theory known as “obstruction theory” by utilizing the toolsof cohomology theory to encounter two basic problems in algebraic topology suchas extension and lifting problems. Obvious examples are the homotopy extensionand homotopy lifting problems. The homotopy classifications of continuous maps,together with the study of extension and lifting problems, play a central role inalgebraic topology. The term “obstruction theory” refers to a technique for defininga sequence of cohomology classes that are obstructions to finding solution to theextension, lifting or relative lifting problems. Obstruction theory leads to make anattempt to find a general solution. This theory originated in the classical work ofHopf, Eilenberg, Steenrod, and Postinikov in around 1940. Certain sets ofcohomology elements, called obstructions, are associated with both a single map inthe case of extension and with a pair of maps in the case of homotopies. These areinvariants depending only on the topological spaces and their continuous mappings.In polyhedra these are the characteristics for the existence or non-existence of thedesired extensions and homotopies. The underlying idea of associating cohomologyelements with continuous mappings was implicitly used by H. Whitney (1907–1989) and first explicitly formulated by N.E. Steenrod (1910–1971). This chapteruses cohomology theory to yield algebraic indicators for obstacles to extension andlifting problems of continuous maps and proves Eilenberg extension theorem. Itpresents some applications of obstruction theory to prove a homological version ofWhitehead theorem, stepwise extension of cross-section and obstruction forhomotopy between relative lifts.

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Chapter 17 presents some similarities and interesting relations among homotopy,homology, and cohomology. In earlier chapters, some relations between thesetheories have been discussed. This chapter continues to convey more relationsthrough Hurewicz homomorphism, Eilenberg–MacLane spaces, Dold–Thomtheorem, Brown’s representation theorem, Hopf invariant and Adams classicaltheorem on Hopf invariant. Historically, L.E.J. Brouwer first connected homologyand homotopy in 1912 by proving that two continuous maps of a two-dimensionalsphere into itself can be continuously deformed into each other if and only if theyhave the same degree (i.e., if and only if they are equivalent from the view point ofhomology theory). Hopf’s classification theorem generalizes Brouwer’s result to anarbitrary dimension. The homotopy groups resemble the homology groups inmany respects under suitable situations proved by Hurewicz in his celebrated“equivalence theorem”. There is also a lack of similarities between these twotheories essentially due to the absence in higher homotopy groups the excisionproperty for homology and also due to the absence in higher homotopy groups atheorem analogous to van Kampen theorem for fundamental group.

Chapter 18 focuses a brief history of algebraic topology highlighting theemergences of the ideas leading to new areas of study in algebraic topology andconveys the contributions of some mathematicians who introduced new concepts orproved theorems of fundamental importance or inaugurated new theories inalgebraic topology starting from the creation of fundamental group and homologygroup by H. Poincaré in 1895, which are the first basic and influential inventions inalgebraic topology. The literature on algebraic topology is very vast. Some conceptsstudied now in algebraic topology had been found in the work of B. Riemann(1826–1866), C. Felix Klein (1849–1925), and H. Poincaré (1854–1912). But thefoundation of algebraic (combinatorial topology) was laid in the decade beginning1895 by H. Poincaré through the publication of his famous series of memoirs“Analysis Situs” from 1895 onwards. J.W. Alexander (1888–1971) used the word“topological” in the titles of his research papers in the 1920s. This chapter alsoconveys more names with their contributions in algebraic topology. The earlydevelopment of homotopy theory was essentially due to H. Poincaré, L.E.F.Brouwer, H. Hopf, W. Hurewicz, H. Freudenthal, and many others. W. Hurewiczfirst established a connection between homology and homotopy groups for ðn� 1Þ-connceted spaces, when n� 2. H. Hopf pioneered a study of maps into spheresduring 1926–1935 and inaugurated the homotopy theory with the discovery of theHopf map followed by the research of W. Hurewicz, and Freudenthal. Since thenhomotopy theory has made a rapid progress and now plays an important role inmathematics. Homology, invented by Henry Poincaré during 1895–1901, is oneof the most fundamental influential inventions in mathematics. He started with ageometric object (a space) which is given by combinatorial data (a simplicialcomplex), then the linear algebra and boundary relations by these data are used toconstruct homology groups. There are other homology theories:

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(i) Homology groups for compact metric spaces introduced by L. Vietoris(1891–2002) in 1927;

(ii) Homology groups for compact Hausdorff spaces introduced by E.C^

ech(1893–1960) in 1932;

(iii) Singular homology groups are first defined by S. Lefschitz (1884–1972) in1933.

All these homology theories lived in isolation. Algebraic topologists in around1940 started comparing various definitions of homology and cohomology given inthe previous years. Eilenberg and Steenrod initiated a new approach in 1945 by takinga small number of their properties (not focusing on machinery used for constructionof homology and cohomology groups) as axioms to characterize a theory ofhomology and cohomology. This approach is the most important contribution toalgebraic topology since the invention of the homology groups by Poincaré and iscalled the axiomatic approach given by a set of seven axioms announced byS. Eilenberg and N. Steenrod in 1945 and published in their book in 1952. Thisapproach classifies and unifies different homology groups on the category of compacttriangulated spaces and inaugurated its dual theory for cohomology theories. Thischapter also conveys the contributions of more mathematicians, S. MacLane,J.H. Whitehead, Serre, Brown, Milnor, and Grothendieck, to name a few.

Apppendix A studies classical topological groups and Lie groups that occupya vast territory in topology and geometry. Lie groups are special topologicalgroups and also manifolds carrying a differential structure. For example,GLðn;RÞ;GLðn;CÞ;GLðn;HÞ; SLðn;RÞ; SLðn;CÞ;Oðn;RÞ;Uðn;CÞ; SLðn;HÞ aresome important classical Lie Groups. Historically, S. Lie (1842–1899) investigatedgroup of transformations. He developed his theory of transformation groups tosolve his integration problems. Such groups are now called Lie groups after hisname. The Fifth Problem of Hilbert announced at the ICM 1900, Paris, is linked toSophus Lie theory of transformation groups which asserts that Lie groups act asgroups of transformations on manifolds.

Appendix B discusses category theory through the study of categories, functors,and natural transformations with an eye to study algebraic topology which consistsof the construction and use of functors from some category of topological spacesinto an algebraic category. This theory plays an important role for the study ofhomotopy, homology, and cohomology theories, which constitute the basic text ofthis book in addition to adjoint functor, representable functor, abelianizationfunctor, Brown functor, and infinite symmetric product functor. All constructions inalgebraic topology are in general functorial. Fundamental groups, higher homotopygroups, and homology and cohomology groups are not only invariants of theunderlying topological space, in the sense that two topological spaces which arehomeomorphic have the isomorphic associated groups (or modules) but theirassociated morphisms also correspond to a continuous mapping of topologicalspaces an induced group (or module) homomorphism on the associated groups(modules), and these homomorphisms can be used to show non-existence (or, much

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more deeply, existence) of mappings. So the readers of algebraic topology cannotescape learning the concepts of categories, functors and natural transformations.

The author acknowledges the Science and Engineering Research Board (SERB)under the Department of Science & Technology of the Government of India forsanctioning of the financial support towards writing this book under the Utilizationof Scientific Expertise of Retired Scientists (USERS) scheme (G.O. No HR/UR/17/2010 dated December 27, 2011) and also to West Bengal University ofTechnology (now renamed as Moulana Abul Kalam Azad University ofTechnology) for providing infrastructure towards implementing the scheme. Theauthor expresses his sincere thanks to Springer for publishing this book. The authoris very thankful to the reviewers of the manuscript and also to Dr. Avishek Adhikariof the University of Calcutta for their scholarly suggestions for improvement of thebook. Thanks are also due to K. Sardar for his cooperation in dealing with thetyping of one version after another of the manuscript and also to many otherindividuals who have helped in proofreading the book. Author’s thanks are due tothe institute IMBIC, Kolkata, for providing the author with the library and otherfacilities towards the manuscript development work of this book. Finally, the authoracknowledges, with heartfelt thanks, the patience and sacrifice of long-sufferingfamily of the author, specially author’s wife Minati, son Avishek, daughter-in-lawShibopriya, and grandson little Avipriyo.

Kolkata, India Mahima Ranjan AdhikariMarch 2016

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Contents

1 Prerequisite Concepts and Notations . . . . . . . . . . . . . . . . . . . . . . . 11.1 Set Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Groups and Fundamental Homomorphism Theorem . . . . . . . . 41.3 Group Representations, Free Groups, and Relations . . . . . . . . 7

1.3.1 Linear Representation of a Group . . . . . . . . . . . . . . 71.3.2 Free Groups and Relations. . . . . . . . . . . . . . . . . . . 81.3.3 Betti Number and Structure Theorem for Finite

Abelian Group . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.4 Exact Sequence of Groups. . . . . . . . . . . . . . . . . . . . . . . . . . 111.5 Free Product and Tensor Product of Groups . . . . . . . . . . . . . 13

1.5.1 Free Product of Groups . . . . . . . . . . . . . . . . . . . . . 131.5.2 Tensor Product of Groups . . . . . . . . . . . . . . . . . . . 14

1.6 Torsion Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.7 Actions of Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.8 Modules and Vector Spaces. . . . . . . . . . . . . . . . . . . . . . . . . 16

1.8.1 Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.8.2 Direct Sum of Modules . . . . . . . . . . . . . . . . . . . . . 171.8.3 Tensor Product of Modules . . . . . . . . . . . . . . . . . . 171.8.4 Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.9 Euclidean Spaces and Some Standard Notations . . . . . . . . . . . 221.10 Set Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.10.1 Topological Spaces: Introductory Concepts . . . . . . . 231.10.2 Homeomorphic Spaces . . . . . . . . . . . . . . . . . . . . . 251.10.3 Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 271.10.4 Connectedness and Locally Connectedness . . . . . . . 281.10.5 Compactness and Paracompactness . . . . . . . . . . . . . 291.10.6 Weak Topology . . . . . . . . . . . . . . . . . . . . . . . . . . 30

1.11 Partition of Unity and Lebesgue Lemma . . . . . . . . . . . . . . . . 301.11.1 Lebesgue Lemma and Lebesgue Number . . . . . . . . . 31

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1.12 Separation Axioms, Urysohn Lemma, and TietzeExtension Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

1.13 Identification Maps, Quotient Spaces, and GeometricalConstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

1.14 Function Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371.15 Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381.16 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401.17 Additional Reading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2 Homotopy Theory: Elementary Basic Concepts . . . . . . . . . . . . . . . 452.1 Homotopy: Introductory Concepts and Examples . . . . . . . . . . 47

2.1.1 Concept of Homotopy . . . . . . . . . . . . . . . . . . . . . . 472.1.2 Functorial Representation. . . . . . . . . . . . . . . . . . . . 57

2.2 Homotopy Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582.3 Homotopy Classes of Maps . . . . . . . . . . . . . . . . . . . . . . . . . 622.4 H-Groups and H-Cogroups . . . . . . . . . . . . . . . . . . . . . . . . . 64

2.4.1 H-Groups and Loop Spaces . . . . . . . . . . . . . . . . . . 652.4.2 H-Cogroups and Suspension Spaces . . . . . . . . . . . . 75

2.5 Adjoint Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 792.6 Contractible Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

2.6.1 Introductory Concepts . . . . . . . . . . . . . . . . . . . . . . 832.6.2 Infinite-Dimensional Sphere and Comb Space . . . . . 85

2.7 Retraction and Deformation . . . . . . . . . . . . . . . . . . . . . . . . . 882.8 NDR and DR Pairs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 952.9 Homotopy Properties of Infinite Symmetric

Product Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 962.10 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

2.10.1 Extension Problems . . . . . . . . . . . . . . . . . . . . . . . 972.10.2 Fundamental Theorem of Algebra . . . . . . . . . . . . . . 99

2.11 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1002.12 Additional Reading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

3 The Fundamental Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1073.1 Fundamental Groups: Introductory Concepts . . . . . . . . . . . . . 108

3.1.1 Basic Motivation . . . . . . . . . . . . . . . . . . . . . . . . . 1083.1.2 Introductory Concepts . . . . . . . . . . . . . . . . . . . . . . 1093.1.3 Functorial Property of π1 . . . . . . . . . . . . . . . . . . . . 1173.1.4 Some Other Properties of π1 . . . . . . . . . . . . . . . . . 119

3.2 Alternative Definition of Fundamental Groups . . . . . . . . . . . . 1243.3 Degree Function and the Fundamental Group

of the Circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1263.4 The Fundamental Group of the Punctured Plane. . . . . . . . . . . 1303.5 Fundamental Groups of the Torus . . . . . . . . . . . . . . . . . . . . 131

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3.6 Vector Fields and Fixed Points . . . . . . . . . . . . . . . . . . . . . . 1323.7 Knot and Knot Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1333.8 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

3.8.1 Fundamental Theorem of Algebra . . . . . . . . . . . . . . 1363.8.2 An Alternative Proof of Brouwer Fixed Point

Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1373.8.3 Borsuk–Ulam Theorem . . . . . . . . . . . . . . . . . . . . . 1393.8.4 Cauchy’s Integral Theorem of Complex

Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1403.9 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1413.10 Additional Reading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

4 Covering Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1474.1 Covering Spaces: Introductory Concepts and Examples . . . . . . 148

4.1.1 Introductory Concepts . . . . . . . . . . . . . . . . . . . . . . 1484.1.2 Some Interesting Properties of Covering Spaces . . . . 1514.1.3 Covering Spaces of RPn . . . . . . . . . . . . . . . . . . . . 152

4.2 Computing Fundamental Groups of Figure-Eightand Double Torus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

4.3 Path Lifting and Homotopy Lifting Properties . . . . . . . . . . . . 1554.4 Lifting Problems of Arbitrary Continuous Maps . . . . . . . . . . . 1584.5 Covering Homomorphisms: Their Classifications

and Galois Correspondence . . . . . . . . . . . . . . . . . . . . . . . . . 1614.5.1 Covering Homomorphisms and Deck

Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . 1614.5.2 Classification of Covering Spaces

by Using Group Theory. . . . . . . . . . . . . . . . . . . . . 1634.5.3 Classification of Covering Spaces and Galois

Correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . 1674.6 Universal Covering Spaces and Computing π1(RP

n) . . . . . . . . 1704.6.1 Universal Covering Spaces . . . . . . . . . . . . . . . . . . 1704.6.2 Computing π1(RP

n) . . . . . . . . . . . . . . . . . . . . . . . 1724.7 Fibrations and Cofibrations . . . . . . . . . . . . . . . . . . . . . . . . . 174

4.7.1 Homotopy Lifting Problems . . . . . . . . . . . . . . . . . . 1744.7.2 Fibration: Introductory Concepts. . . . . . . . . . . . . . . 1764.7.3 Cofibration: Introductory Concepts . . . . . . . . . . . . . 179

4.8 Hurewicz Theorem for Fibration and Characterizationof Fibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

4.9 Homotopy Liftings and Monodromy Theorem . . . . . . . . . . . . 1844.9.1 Path Liftings and Homotopy Liftings . . . . . . . . . . . 1854.9.2 Monodromy Theorem . . . . . . . . . . . . . . . . . . . . . . 185

4.10 Applications and Computations . . . . . . . . . . . . . . . . . . . . . . 1864.10.1 Actions of Fundamental Groups . . . . . . . . . . . . . . . 1864.10.2 Fundamental Groups of Orbit Spaces . . . . . . . . . . . 187

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4.10.3 Fundamental Group of the Real ProjectiveSpace RPn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

4.10.4 The Fundamental Group of Klein’s Bottle . . . . . . . . 1894.10.5 The Fundamental Groups of Lens Spaces . . . . . . . . 1904.10.6 Computing Fundamental Group of Figure-Eight

by Graph-theoretic Method . . . . . . . . . . . . . . . . . . 1914.10.7 Application of Galois Correspondence. . . . . . . . . . . 191


5 Fiber Bundles, Vector Bundles and K-Theory . . . . . . . . . . . . . . . . 1975.1 Bundles, Cross Sections, and Examples. . . . . . . . . . . . . . . . . 198

5.1.1 Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1995.1.2 Cross Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . 2005.1.3 Morphisms of Bundles . . . . . . . . . . . . . . . . . . . . . 2015.1.4 Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

5.2 Fiber Bundles: Introductory Concepts . . . . . . . . . . . . . . . . . . 2065.3 Hopf and Hurewicz Fiberings . . . . . . . . . . . . . . . . . . . . . . . 210

5.3.1 Hopf Fibering of Spheres. . . . . . . . . . . . . . . . . . . . 2105.3.2 Hurewicz Fibering . . . . . . . . . . . . . . . . . . . . . . . . 212

5.4 G-Bundles and Principal G-Bundles . . . . . . . . . . . . . . . . . . . 2135.5 Homotopy Properties of Numerable Principal G-Bundles. . . . . 2185.6 Classifying Spaces: The Milnor Construction . . . . . . . . . . . . . 2205.7 Vector Bundles: Introductory Concepts . . . . . . . . . . . . . . . . . 2235.8 Charts and Transition Functions of Bundles. . . . . . . . . . . . . . 2295.9 Homotopy Classification of Vector Bundles. . . . . . . . . . . . . . 2335.10 K-Theory: Introductory Concepts . . . . . . . . . . . . . . . . . . . . . 2365.11 Principal G-Bundles for Lie Groups G . . . . . . . . . . . . . . . . . 2405.12 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2415.13 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2415.14 Additional Reading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

6 Geometry of Simplicial Complexes and FundamentalGroups of Polyhedra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2496.1 Geometry of Finite Simplicial Complexes . . . . . . . . . . . . . . . 2506.2 Triangulations and Polyhedra . . . . . . . . . . . . . . . . . . . . . . . . 2536.3 Simplicial Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2576.4 Barycentric Subdivisions . . . . . . . . . . . . . . . . . . . . . . . . . . . 2586.5 Simplicial Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 2616.6 Computing Fundamental Groups of Polyhedra . . . . . . . . . . . . 2646.7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265

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6.7.1 Application to Extension Problem. . . . . . . . . . . . . . 2656.7.2 Application to Graph Theory . . . . . . . . . . . . . . . . . 2666.7.3 van Kampen Theorem . . . . . . . . . . . . . . . . . . . . . . 267


7 Higher Homotopy Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2737.1 Absolute Homotopy Groups: Introductory Concept . . . . . . . . . 2747.2 Absolute Homotopy Groups Defined by Hurewicz . . . . . . . . . 2777.3 Functorial Properties of Absolute Homotopy Groups. . . . . . . . 2787.4 The Relative Homotopy Groups: Introductory Concepts . . . . . 2817.5 The Boundary Operator and Induced Transformation . . . . . . . 282

7.5.1 Boundary Operator . . . . . . . . . . . . . . . . . . . . . . . . 2827.5.2 Induced Transformations . . . . . . . . . . . . . . . . . . . . 283

7.6 Functorial Property of the Relative Homotopy Groups . . . . . . 2847.7 Homotopy Sequence and Its Exactness . . . . . . . . . . . . . . . . . 285

7.7.1 Homotopy Sequence and Its Exactness . . . . . . . . . . 2857.7.2 Some Consequences of the Exactness

of the Homotopy Sequence . . . . . . . . . . . . . . . . . . 2877.8 Homotopy Sequence of Fibering and Hopf Fibering . . . . . . . . 289

7.8.1 Homotopy Sequence of Fibering. . . . . . . . . . . . . . . 2897.8.2 Hopf Fiberings of Spheres . . . . . . . . . . . . . . . . . . . 2907.8.3 Problems of Computing πmðSnÞ . . . . . . . . . . . . . . . 290

7.9 More on Hopf Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2917.10 Freudenthal Suspension Theorem and Table of πiðSnÞ

for 1� I; n� 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2927.10.1 Freudenthal Suspension Theorem . . . . . . . . . . . . . . 2937.10.2 Table of πiðSnÞ for 1� I; n� 8 . . . . . . . . . . . . . . . . 294

7.11 Action of π1 on πn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2947.12 The nth Cohomotopy Sets and Groups . . . . . . . . . . . . . . . . . 2957.13 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2977.14 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3007.15 Additional Reading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303

8 CW-Complexes and Homotopy . . . . . . . . . . . . . . . . . . . . . . . . . . . 3058.1 Cell-Complexes and CW-Complexes: Introductory

Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3068.1.1 Cell-Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . 3078.1.2 CW-Complexes. . . . . . . . . . . . . . . . . . . . . . . . . . . 3088.1.3 Examples of Spaces Which Are Neither

CW-Complexes Nor Homotopy Equivalentto a CW-Complex . . . . . . . . . . . . . . . . . . . . . . . . . 312

8.2 Cellular Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312

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8.3 Subcomplexes of CW-Complexes . . . . . . . . . . . . . . . . . . . . . 3138.4 Relative CW-Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . 3148.5 Homotopy Properties of CW-Complexes, Whitehead

Theorem and Cellular Approximation Theorem . . . . . . . . . . . 3158.5.1 Homotopy Properties of CW-Complexes . . . . . . . . . 3158.5.2 Whitehead Theorem . . . . . . . . . . . . . . . . . . . . . . . 3178.5.3 Cellular Approximation Theorem . . . . . . . . . . . . . . 318

8.6 More on Homotopy Properties of CW-Complexes . . . . . . . . . 3198.7 Blakers–Massey Theorem and a Generalization of

Freudenthal Suspension Theorem . . . . . . . . . . . . . . . . . . . . . 3208.8 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3218.9 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3238.10 Additional Reading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326

9 Products in Homotopy Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 3299.1 Whitehead Product Between Homotopy Groups

of CW-Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3299.2 Whitehead Products Between Homotopy Groups

of H-Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3339.3 A Generalization of Whitehead Product. . . . . . . . . . . . . . . . . 3359.4 Mixed Products in Homotopy Groups . . . . . . . . . . . . . . . . . . 336

9.4.1 Mixed Product in the Homotopy Categoryof Pointed Topological Spaces . . . . . . . . . . . . . . . . 336

9.4.2 Mixed Product Associated with Fibrations . . . . . . . . 3379.5 Samelson Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338

9.5.1 The Samelson Product. . . . . . . . . . . . . . . . . . . . . . 3389.5.2 The Iterated Samleson Product . . . . . . . . . . . . . . . . 339

9.6 Some Relations Between Whitehead and SamelsonProducts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340

9.7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3419.7.1 Adams Theorem Using Whitehead Product . . . . . . . 3419.7.2 Homotopical Nilpotence of the Seven Sphere S7 . . . . 342


10 Homology and Cohomology Theories . . . . . . . . . . . . . . . . . . . . . . 34710.1 Chain Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34910.2 Simplicial Homology Theory . . . . . . . . . . . . . . . . . . . . . . . . 352

10.2.1 Construction of Homology Groups of a SimplicialComplex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353

10.2.2 Induced Homomorphism and FunctorialProperties of Simplicial Homology . . . . . . . . . . . . . 360

10.2.3 Computing Homology Groups of Polyhedra . . . . . . . 361

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10.3 Relative Simplicial Homology Groups . . . . . . . . . . . . . . . . . 36210.4 Exactness of Simplicial Homology Sequences . . . . . . . . . . . . 36410.5 Simplicial Cohomology Theory: Introductory Concepts . . . . . . 36510.6 Simplicial Cohomology Ring . . . . . . . . . . . . . . . . . . . . . . . . 36710.7 Singular Homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368

10.7.1 Singular Homology Groups . . . . . . . . . . . . . . . . . . 36910.7.2 Reduced Singular Homology Groups. . . . . . . . . . . . 37210.7.3 Relative Singular Homology Groups . . . . . . . . . . . . 373

10.8 Eilenberg–Zilber Theorem and Künneth Formula . . . . . . . . . . 37410.8.1 Eilenberg–Zilber Theorem . . . . . . . . . . . . . . . . . . . 37410.8.2 Künneth Formula . . . . . . . . . . . . . . . . . . . . . . . . . 374

10.9 Singular Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37510.10 Relative Cohomology Groups . . . . . . . . . . . . . . . . . . . . . . . 37610.11 Hurewicz Homomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . 37710.12 Mayer–Vietoris Sequences. . . . . . . . . . . . . . . . . . . . . . . . . . 379

10.12.1 Mayer–Vietoris Sequences in SingularHomology Theory . . . . . . . . . . . . . . . . . . . . . . . . 379

10.12.2 Mayer–Vietoris Sequences in SimplicialHomology Theory . . . . . . . . . . . . . . . . . . . . . . . . 380

10.13 Computing Homology Groups . . . . . . . . . . . . . . . . . . . . . . . 38110.13.1 Homology Groups of a One-Point Space . . . . . . . . . 38110.13.2 Homology Groups of CW-complexes . . . . . . . . . . . 382

10.14 Cellular Homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38310.15 Čech Homology and Cohomology Groups. . . . . . . . . . . . . . . 38410.16 Universal Coefficient Theorem for Homology and

Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38510.16.1 Homology with Arbitrary Coefficient Group. . . . . . . 38510.16.2 Universal Cohomology Theorem

for Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . 38710.17 Betti Number and Euler Characteristic . . . . . . . . . . . . . . . . . 388

10.17.1 Euler Characteristics of Polyhedra. . . . . . . . . . . . . . 38810.17.2 Euler Characteristic of Finite Graphs. . . . . . . . . . . . 39010.17.3 Euler Characteristic of Graded Vector Spaces. . . . . . 39010.17.4 Euler–Poincaré Theorem for Finite

CW-complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . 39110.18 Cup and Cap Products in Cohomology Theory . . . . . . . . . . . 392

10.18.1 Cup Product. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39310.18.2 Cap Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395

10.19 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39610.19.1 Jordan Curve Theorem . . . . . . . . . . . . . . . . . . . . . 39610.19.2 Homology Groups of

_

α2aSnα . . . . . . . . . . . . . . . . . .

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10.20 Invariance of Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . 39810.21 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39910.22 Additional Reading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406

11 Eilenberg–MacLane Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40711.1 Eilenberg–MacLane Spaces: Introductory Concept . . . . . . . . . 40711.2 Construction of Eilenberg–MacLane Spaces KðG; nÞ. . . . . . . . 409

11.2.1 A Construction of KðG; 1Þ . . . . . . . . . . . . . . . . . . . 40911.2.2 A Construction of K(G, n) for n[ 1. . . . . . . . . . . . 40911.2.3 Moore Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 41011.2.4 Killing Homotopy Groups . . . . . . . . . . . . . . . . . . . 41011.2.5 Postnikov Tower: Its Existence and Construction . . . 41111.2.6 Existence Theorem . . . . . . . . . . . . . . . . . . . . . . . . 413

11.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41311.4 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41511.5 Additional Reading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417

12 Eilenberg–Steenrod Axioms for Homology and CohomologyTheories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41912.1 Eilenberg–Steenrod Axioms for Homology Theory . . . . . . . . . 42012.2 The Uniqueness Theorem for Homology Theory . . . . . . . . . . 42212.3 Eilenberg–Steenrod Axioms for Cohomology Theory . . . . . . . 42412.4 The Reduced 0-dimensional Homology

and Cohomology Groups. . . . . . . . . . . . . . . . . . . . . . . . . . . 42712.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427

12.5.1 Invariance of Homology Groups. . . . . . . . . . . . . . . 42812.5.2 Invariance of Cohomology Groups . . . . . . . . . . . . . 42812.5.3 Mayer–Vietoris Theorem . . . . . . . . . . . . . . . . . . . . 428


13 Consequences of the Eilenberg–Steenrod Axioms. . . . . . . . . . . . . . 43313.1 Immediate Consequences. . . . . . . . . . . . . . . . . . . . . . . . . . . 43313.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440

13.2.1 Cofibration and Homology. . . . . . . . . . . . . . . . . . . 44013.2.2 Computing Ordinary Homology Groups of Sn . . . . . 441


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