Fundamental Groups
and Covering Spaces
Elon Lages Lima
AK Peters 2003
ISBN 1-56881-131-4

Poincaré has been described as one of thelast great universalists, being the most prolific author of his time, surpassed only by
Euler in quantity: his legacy permeatesmany areas of contemporary mathematics.
Algebraic topology is one of these, and his
notion of the fundamental group of a spaceis one of many original and far-reachingideas. The prototype is the exponential
map x 7→ eix by which the circle is infinitely and evenly covered by the real line,
more familiar to undergraduates in the form
θ 7→ (cos θ, sin θ). Poincaré’s idea was to observe this as the circle “unwrapping” to produce the real line, and to abstract this idea
so that any reasonable (topological) space
X can be uniquely unwrapped to give a
simply-connected universal covering space
X̃ → X. This is achieved by observing
that a set of equivalence classes of paths
in a space, beginning and ending at thesame point, naturally has a composition andgroup structure. This group – the so-called‘fundamental group’ π1 (X) – acts naturallyon X̃, with X as its orbit space. Any subgroup H of π1 (X) also gives a quotient XHof X̃ with a canonical projection XH → X,with X̃ = XId corresponding to the trivial subgroup. Remarkably π1 (XH ) ∼= H,and so X̃ has trivial fundamental group (ieis ‘simply-connected’). Topological spaces
and group theory are inextricably united.

Poincaré had also been the first to observe the relationship between the complex analysis of Riemann surfaces, Fuchsiangroups, Klein’s ideas of transformations andgeometry, and discrete groups of hyperbolic
isometries. Accordingly, the new subject ofalgebraic topology abstracted the underpinnings of the work of Gauss, Abel, Galois,Cauchy, Riemann and many other mathematicians of the preceding century. It isinteresting to note that Poincaré’s initialideas preceded the concepts of a metric or
topological space developed by Fréchet andHausdorff more than a decade later, andeschewed the notions of sets developed byCantor. Just as one can construct the realline by taking a circle, cutting at a pointand gluing together infinitely many copies,
there is a ‘constructive’ approach to producing the universal cover X̃ from X itself. Oneof the simplest applications of the fundamental group is Brouwer’s fixed-point theorem, that every continuous map from thedisc to itself has a fixed point.
This and many other applications are
given in Lima’s book, together with a number of historical asides and numerous exercises. This is a book that all libraries
should have, not because it is encyclopedic, but because it introduces the student
to a variety of accessible applications andinsights too quickly skimmed over in standard texts. For an encyclopedic treatment,
famous for its succinct impenetrability fora novice, Edwin Spanier’s ‘Algebraic Topology’ is still classic. Elon Lima and Morris
Hirsch were students of Spanier together atChicago, both graduating in 1958. Interestingly Hirsch also has a proof of Brouwer’s
fixed point theorem credited to him, using

Book Reviews

differential topology (the title of his graduate level text). Lima’s book contains many
examples of the interplay between differential topology, covering spaces, fibre spaces,
matrix Lie groups, immersions of curves inthe plane and differential forms. It is theseexamples which make the book so useful fora student, since these are all ingredients ofcontemporary mathematics and mathematical physics. Fibre spaces are precursors offibre bundles, and path liftings to covering
spaces and fibre spaces are the precursors toliftings via connections in Riemannian geometry. The structure of the groups SO(n),SU (n), and Sp(n) is described in a way
understandable to students who may nothave been exposed to a course in transversality and differential topology, and the description of winding numbers of immersed
curves will prepare the reader for Smale’s
more sophisticated proof of the existenceof a smooth eversion of the sphere in R3 .
(Amusingly there is an account of Eilenberg’s initial disbelief of Smale’s result reminiscent of the story attributed to Feynman– those who had met Eilenberg would not
be surprised!)
A student often first encounters the fundamental group as a last topic in an introductory course in topology, or as partof the homotopy section of a course inalgebraic topology. For example, Greenberg and Harper’s classic ‘Algebraic Topology: a first course’ completes the discussion of π1 (X) and covering spaces in about
30 pages. Lima’s book spends a lot longer
going deeper into conceptual applications,is a very nice read, and is highly recommended for students. It also provides additional material a lecturer might wish topresent in a first course. A minor warning is that ‘surface’ should sometimes beconstrued as ‘hyper-surface’ or ‘manifold’.
A prior course in basic or metric topology
would be desirable.
Incidentally and appropriately, the given
proof of Brouwer’s theorem is of course notthe original, nor is it the one attributed to273

Hirsch. Those interested in historical anecdotes concerning Lima’s own experiences
might consider asking any Brazillian connected with IMPA; they may be surprised
at what hot gossip they learn!
Iain Aitchison
Department of Mathematics, University of Melbourne, VIC 3010
E-mail: I.Aitchison@ms.unimelb.edu.au

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Galileo Galilei:
When the World Stood StillAtle Næss, trans. James Anderson
Springer Berlin 2005
ISBN 3-540-21961-7
They worked faster in those days, but aneye for a grant came in handy as much thenas now. In June 1609 Galileo, then Professor of Mathematics at Padua but always
hopeful of a move back home to Florence,heard an interesting rumour from one of his
dubious intellectual contacts in Venice. It
was said that someone in Holland had put
lenses in a tube and was able to see things
larger at a distance. Galileo did some geometry as to what shaped lenses might do thatand over several months of experiment (andlearning to polish glass) he achieved a resultthat magnified 9 times with good clarity. InAugust he exhibited it to the senators inVenice. The military implications were obvious and they doubled his salary. By thewinter he had a version that magnified 20times, but with rumours all over Europe it
was going to be hard to keep ahead of thecompetition. On 7 January 1610 he turned
the telescope on Jupiter. It had three unknown stars next to it, in itself not surprising as many new stars were evident. The
next night, the tiny stars seemed to have got
ahead of Jupiter instead of being left a littlebehind. 9 January was overcast. On 10 January, the stars were still with Jupiter but in274

Book Reviews

a different position. Galileo realised he had
discovered moons, the first celestial discovery in recorded history (except for transient
events like comets). A fast enough printing job would ensure his priority, and he
had the naming rights. As a result of aprevious networking success, he had someyears before tutored the young Grand Duke
Cosimo de Medici of Florence, before hisaccession. On 13 February he wrote asking for gracious ducal permission to namethe new moons the “cosmic” stars, suggesting Cosimo, but adding that “Medicean”
might also be a possibility. The Grand
Duke’s marketing department replied graciously preferring the latter option, andwith the necessary last-minute typesettingchanges the book announcing the discovery of the Medicean stars appeared on 12March. The several pages of dedication toCosimo include the sentence “Scarcely havethe immortal graces of your soul begun toshine forth on earth than bright stars offer themselves in the heavens which, liketongues, will speak of and celebrate yourmost excellent virtues for all time.” There
was then an urgent need to fend off a claim
by a rival professor of mathematics that it
was all made up, but an urgent request toKepler and a rushed pamphlet from him(though he did not have a telescope powerful enough to see the new stars), sorted outthat contretemps. Galileo was appointed
grand ducal mathematician and philosopher
at Florence on 10 July, needless to say atan acceptably huge salary. Galileo’s people
skills did not work so well on the Inquisition. They could work fast as well – toofast. When Galileo visited Rome in 1616
talking up Copernicus’ heliocentric theory,
the Inquisition, in a process lasting six days
and without the benefit of any astronomicaladvice, declared it heretical. It is a classic
case of the saying, “everything is run by idiots”. Bizarrely, it seems to have occurred
to no-one except Galileo and an obscure
German visitor who had nothing to do withthe case that the Church might get egg on

its face by being proved wrong; it was thensupposed that scientific questions, like legal
and theological ones, were subject to neverending debate. Galileo was instructed not
to teach or defend the theory. He did the exact opposite, publishing the polemical Dialogue Concerning the Two Chief World Systems, which, though not explicitly favouring
Copernicanism, made the alternatives lookridiculous. Galileo, then aged 70 and ill,
was dragged before the Inquisition, foundguilty of disobeying its previous orders, andordered to recant his theory and live under house arrest. The end of the story issurely inevitable: Galileo lived out his few
remaining years in illness, a broken man unable to accomplish anything further . . . well,actually, no. Though he was ill and eventually blind, Galileo’s last years were amazingly productive. The Archbishop of Siena
installed him in comfort and urged him towrite something less controversial. In Siena
and later at home in Florence, he produced
his Dialogue Concerning Two New Sciences,
the summary of his researches on motion
and mechanics that is his true masterwork.
Copernicanism was gaining ground anyway
and might have been better off without hishelp, but his work on the quantification ofmotion is the beginning of the true Scientific(or perhaps better Mathematical) Revolution, the work on which Newton and all later
applied mathematicians built. The search
for laws of proportionality like Boyle’s, Pascal’s, Hooke’s and Newton’s laws, perhaps
the central discoveries of the Scientific Revolution, was begun in earnest by Galileo.
Atle Næss knows how to write a readable
book. He works fast, too, and wastes notime on padding. He weaves the history, thepersonalities and the technicalities together
expertly, with a sure eye for an interesting detail and an ability to cut through themyths, propaganda and speculation thatsurround the Galileo case. A mathematical audience might prefer him to have given
more of the mathematics, such as Galileo’s

Book Reviews

remarkable “bare hands” proof that it is impossible for anything to move from rest suchthat its speed is proportional to the distancecovered. But the aim of the book is a short
introduction to introduce the man and whetthe appetite for his ideas. It is a resounding
success.

James Franklin
School of Mathematics, University of New South
Wales, Sydney 2052
E-mail: j.franklin@unsw.edu.au

Lagrange-Type Functions
in ConstrainedNon-Convex Optimization

A. Rubinov and X.Q. Yang
Kluwer Academic Publishers 2003
ISBN 1-4020-7627-4

Lagrange and penalty functions provide a
powerful approach for study of constrainedoptimization problems. The applicationsof classical Lagrange and penalty functions are, however, restricted with some(although very important) special classesof constrained optimization problems, andthere is a clear need to introduce andstudy more general types of such functions,which would have a wider range of applications. The book gives a systematic andunified presentation of many important results that have been obtained in this area
during last several years. The authors introduce and examine non-linear Lagrange-typefunctions, which can be represented as a
convolution of the objective and constraints.
They also consider generalized augmented
Lagrangians with a very broad class of275

augmenting functions and they pay a special attention to level-bounded and peakat-zero augmenting functions. Both nonlinear Lagrange-type functions and generalized Lagrangians are studied in the framework of abstract convexity (the area of expertise of the first author) and with useof tools of modern non-smooth analysis(the area of expertise of the second author) for examining the first and second
order optimality conditions. Among thenon-linear and generalized augmented Lagrangians that are considered there are generalized penalty functions and, in particular, the so-called lower order penalty functions that are useful in the study of error
bounds and mathematical programs withequilibrium constraints.
The book develops a unified approach toduality and penalization and to convergence
analysis of the first and second order optimality conditions. The authors examine
the properties of weak duality, zero duality gap, as well as those of existence of exact Lagrange multipliers, penalty parameters and saddle points, without usual assumptions about convexity and constraint
qualification. It is known that the weak duality gives an estimation of a global minimum and that the zero duality gap property
allows one to reduce a constrained problem to a series of unconstrained ones. Thus
new dual models investigated in the bookmay provide a better estimation of the lower
bound of the optimal value of the originalproblem. It is also known that the duality
theory provides tools for analysis of optimalalgorithms. It is yet to be seen how the newmodels studied in the book can be used tohelp developing this line of research.
A number of impressive new results on
the existence of an exact penalty parameter have been obtained in the book, someof these being of interest even in a classical penalty functions setting. In particular, an analytic expression for the least exact penalty parameter for a large class ofpenalty functions has been obtained.

276Book Reviews

Along with a theoretical study, the bookcontains some recommendations concerning numerical solution of some optimization problems. In particular the authors
have found that a special class of penalty
functions is useful in solving some nonconvex problems including problems of concave minimization problems.
The outline of the book is as follows.
Chapter 1 contains motivation for studying Lagrange-type functions and provides
an outline of main topics studied in thebook.
Chapter 2 is devoted to abstract convexity and theory of IPH (increasing positively
homogeneous) functions and, also, it provides a description of the technique, which
is used for the examination of the zero duality gap property and penalty-type functions
for problems with a single constraint.
Chapter 3 introduces a general scheme ofLagrange-type functions. This is based on a
separation of certain sets in the image-spaceof the problem. Also, optimality conditions
are discussed in this chapter.
In Chapter 4 penalty-type functions forproblems with a single constraint are studied using an IPH convolution function. The
estimation of the least exact penalty parameter for various problems is given.
In Chapter 5 the authors examine thezero duality gap property and exactness forvarious augmented Lagrangian functions.
Chapter 6 is devoted to the analysisof the first- and the second-order optimality conditions for non-linear penalty-typefunctions and augmented Lagrangian functions in relation to the corresponding conditions of the original constrained optimization problem.

Vladimir Gaitsgory
School of Mathematics and Statistics, University ofSouth Australia, Mawson Lakes, SA 5095
E-mail: Vladimir.Gaitsgory@unisa.edu.au

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Coding Theory
A first course
San Ling and Chaoping Xing
Cambridge University Press 2004
ISBN-13: 9780521529235
For some years I have taught a 12 lecture introductory course in coding theory to level3 mathematics, computer science and engineering undergraduates, using as major
reference the excellent text “A first course
in coding theory”, by Raymond Hill (OUP,1986). Since I am revising and extending thesyllabus for this course I was interested toreview this book to see if it would be suitable as a more up-to-date reference. The authors have used the book as a text for a 2 semester course to advanced undergraduatesat the National University of Singapore.
The assumed prerequisite is some basic
linear algebra, although a healthy dose of“mathematical maturity” would be needed
to cope with some of the more advanced
topics. The first two chapters introduce
channel coding, the binary symmetric channel, redundancy, error detection and correction and nearest neighbour decoding. Chapter 3 gives a fairly complete and detailed
treatment of finite fields. This treatment
would be quite difficult for students withno previous course in modern algebra, although here and elsewhere there are manyroutine and less routine exercises. Chapter 4 treats the standard material on linear codes, encoding and syndrome decoding. The next chapter gives the main bounds
in coding theory and also introduces theHamming and extended binary Hammingcodes, Golay codes (via the generator matrix) and some non-linear codes, including
the Hadamard codes. Chapter 6, on the construction of linear codes, gives methods offorming new codes from old and introduces
the Reed-Muller codes. Chapter 7 gives afull account of cyclic codes from an algebraic point of view and gives a relatively
simple decoding algorithm via the divisionalgorithm for polynomials. The burst-error

Book Reviews

correcting capabilities of cyclic codes arealso treated. In chapter 8 we meet special
cyclic codes, including BCH codes and theirdecoding algorithm, Reed-Solomon codes
and Quadratic Residue codes. Finally, in
chapter 9, we meet the generalized ReedSolomon codes and their decoding algorithm, alternant codes and Goppa codes.
As can be seen from this description,
most of the well-known classes of codes aretreated. There are many examples and there
is a wealth of well-chosen exercises, someof which provide significant extensions tothe theory. Some topics that are not treatedor are treated very briefly, in comparison
to Hill, are perfect codes, codes and Latinsquares, decimal codes and the weight enumerator.

277

Two of the strengths of Hill, which areless present in Ling and Xing, are firstly
the way in which more complicated consructions, e.g., BCH codes, are introduced
via simple special cases; secondly the simple
and interesting examples of the practical useof codes, ranging from the transmission ofphotographs from deep-space to the use ofISBN numbers. Nevertheless, Ling and Xingis a valuable additional reference which covers a number of topics not present in Hill.
The treatment of cyclic codes is especiallyuseful.
Bob Clarke
School of Mathematical Sciences, University of Adelaide Adelaide, SA 5005
E-mail: robert.clarke@adelaide.edu.au