## John W Milnor's books

John Milnor wrote a number of important books. Below we give extracts from reviews of eight of these. We also give and extract from Milnor's Preface for many of the books. His Collected Papers have been published by subject, rather than date. To date seven volumes have been published. These contain not only Milnor's published papers but also some unpublished one as well as a commentary by Milnor putting the papers into context. We present extracts from Milnor's Prefaces and extracts from some reviews of the seven volumes.

**1. Morse theory (1963).**

**1.1. From the Preface.**

This book gives a present-day account of Marston Morse's theory of the calculus of variations in the large. However, there have been important developments during the past few years which are not mentioned.

**1.2. Review by: N H Kuiper.**

*Bull. Amer. Math. Soc.*

**71**(1) (1965), 136-137.

The theory of Marston Morse deals with the topological analysis of a manifold or a function space together with a real function on this space. Calculus of variations in the large was originally the main purpose of the theory. A typical subject was the study of the geodesies connecting two given points in a complete Riemannian n manifold. ... Milnor's book is a lucid rapid introduction to the subject, with a highly geometrical flavour. It is well written, and points to many subjects of current research.

**1.3. Review by: Harold I Levine.**

*Mathematical Reviews*, MR0163331

**(29 #634)**.

This book is devoted to an exposition of Morse theory. Starting from scratch, it goes through the proofs of the periodicity theorems of Bott for the unitary and orthogonal groups. The path taken to these theorems is by no means a minimal geodesic but the detours along the way serve to display the power of the theory as it is developed.

**2. Lectures on the h-cobordism theorem (1965).**

**2.1. From the Introduction.**

These are notes for lectures of John Milnor that were given as a seminar on differential topology in October and November 1963 at Princeton University.

**2.2. Review by: Pierre E Conner.**

*Mathematical Reviews*, MR0190942

**(32 #8352)**.

These notes are devoted to an exposition of the differentiable h-cobordism theory entirely from the viewpoint of Morse theory. Although the basic facts of the study of critical points of functions as developed by Morse are clearly stated, it would be in order to suggest that the reader have a somewhat more expanded background in Morse theory. The object is the following theorem, due originally to Smale. Let W be a compact, smooth manifold with two boundary components V and V'. If V is simply connected, has dimension greater than 4 and if V and V' are both deformation retracts of W, then W is diffeomorphic to V x [0,1].

**3. Topology from the differentiable viewpoint (1965).**

**3.1. From the Preface.**

These lectures were delivered at the University of Virginia in December 1963. ... They present some topics from the beginnings of topology, centring about L E J Brouwer's definition, in 1912, of the degree of a mapping. The methods used, however, are those of differential topology, rather than the combinatorial methods of Brouwer. The concept of regular value and the theorem of Sard and Brown, which asserts that every smooth mapping has regular values, play a central role. To simplify the presentation, all manifolds are taken to be infinitely differentiable and to be explicitly embedded in Euclidean space. A small amount of point-set topology and of real variable theory is taken for granted.

**3.2. Review by: E H Brown.**

*Amer. Math. Monthly*

**74**(4) (1967), 461.

In this short volume several of the most important results and techniques of algebraic and differential topology are introduced through an exposition in the spirit of "advanced mathematics from an elementary point of view." The reader is assumed to be familiar with very elementary point set topology and a few of the basic theorems of real variable theory. The Sard-Brown Theorem is proved and the degree of a map is defined using the concept of a regular value of a smooth mapping. Hopf's Theorem is proved by showing that two maps of an n-sphere into itself are homotopic if and only if they have the same degree. The notion of cobordism is introduced and the relation between framed cobordism classes of manifolds and homotopy classes of maps of a sphere into a sphere is established.

**3.3. Review by: W Stephen Piper.**

*Mathematical Reviews*, MR0226651

**(37 #2239)**.

The lectures begin with definition and discussion of smooth manifolds and maps. After the proof of the theorem of Sard and Brown, and presentation of smooth homotopy, smooth isotopy, the Brouwer degree of mappings of oriented manifolds, vector fields, and the Euler number, the lectures conclude with framed cobordism, the Pontryagin construction, and Hopf's theorem.

**4. Singular points of complex hypersurfaces (1968).**

**4.1. Review by: Jerome P Levine.**

*Mathematical Reviews*, MR0239612

**(39 #969)**.

The purpose of this work is to study the local behavior of a complex hypersurface V in Euclidean space at a singularity z_0 ... Some of these results have appeared in an unpublished work of the author: ("On isolated singularities of hypersurfaces"). There is also some overlap with work of Pham, Brieskorn and Hirzebruch ...

**5. Introduction to algebraic K-theory (1971).**

**5.1. From the Preface.**

The name "algebraic K-theory" describes a branch of algebra which centres about two functors K

_{0}and K

_{1}, which assign to each associative ring

_{0}

_{1}

**5.2. Review by: Jean-Pierre Jouanolou.**

*Mathematical Reviews*, MR0349811

**(50 #2304)**.

Supplemented notes of a seminar held in Princeton in 1966, this book is an excellent introduction to the algebraic K-theory. The focus is on the definition, proposed by the author, and universally accepted, of the group K

_{2}of a ring, which was the main motivation of the seminar. From this point of view, it complements the classic book of H Bass [

*Algebraic K-theory*, 1968] which, for obvious reasons, was limited to the study of groups of Grothendieck and Whitehead.

**6. (with Dale Husemoller) Symmetric bilinear forms (1973).**

**6.1. From the Preface.**

The theory of quadratic forms and the intimately related theory of symmetric bilinear forms have a long and rich history, highlighted by the work of Legendre, Gauss, Minkowski, and Hasse. ... Our exposition will concentrate on the relatively recent developments which begin with and are inspired by Witt's 1937 paper "Theorie der quadratischen Formen in beliebigen Körpern." We will be particularly interested in the work of A Pfister and M Knebusch. However, some older material will be described, particularly in Chapter II. The presentation is based on lectures by Milnor at the Institute for Advanced Study, and at Haverford College under the Phillips Lecture Program, during the Fall of 1970, as well as lectures at Princeton University in 1966.

**6.2. Review by: Louis H Kauffman.**

*Mathematical Reviews*, MR0506372

**(58 #22129)**.

Perhaps the most enticing of all the varieties of mathematics are those that begin in extreme simplicity only to branch out almost at once, touching diverse areas and yet at the same time retaining integrity and deep structure. Just such a topic is the subject of this wonderful book.

**7. (with James D Stasheff) Characteristic classes (1974).**

**7.1. Review by: E H Spanier.**

*Bull. Amer. Math. Soc.*

**81**(5) (1975), 862-866.

In 1957 there appeared notes by Stasheff of lectures on characteristic classes by Milnor at Princeton University. These notes are a clear concise presentation of the basic properties of vector bundles and their associated characteristic classes. Since their appearance they have become a standard text regularly used by graduate students and others interested in learning the subject. The present, long-anticipated book is based on those notes. It follows the order of the notes but is considerably expanded with more detail and discussion. In addition, exercises have been added to almost each section, there are many useful references to the textbooks on algebraic topology that are available now, and there is an epilogue summarizing main developments in the subject since 1957. All of these strengthen the book and make it even more valuable as a text for a course as well as a book that can be read by students on their own. The material covered should be required for doctoral students in algebraic or differential topology and strongly recommended for those in differential geometry. ... there is a lot of mathematics included in the book. It is a valuable and welcome addition to the literature.

**7.2. Review by: Friedrich Hirzebruch.**

*Mathematical Reviews*, MR0440554

**(55 #13428)**.

This book emerged from lectures that Milnor held at Princeton University in 1957. Notes written by Milnor and Stasheff drafting the lectures then circulated at many universities and was for many students a foundation of their training in algebraic topology. Many students of the lecturers at the University of Bonn have benefited from these notes for almost 20 years. The lectures are finally published as a book in 1974, the older of the two authors apologizes in the preface "for the delay in publication", the reviewer apologizes for "the delay in reviewing".

**8. Collected papers. Vol. 1. Geometry (1994).**

**8.1. From the Preface.**

This is the beginning of an effort to organize my published and unpublished papers. It seemed most useful to arrange these papers by subject, rather than date. This first volume consists of papers, spanning more than forty years, which have a strong geometric flavour. It included four papers which have not previously been published.

**8.2. Review by: Serge L Tabachnikov.**

*Mathematical Reviews*, MR1277810

**(95c:01043)**.

This volume contains geometrical papers of one of the best modern geometers and topologists, John Milnor. This book covers a wide variety of topics, and includes several previously unpublished works. It is delightful reading for any mathematician with an interest in geometry and topology, and, I daresay, for any person with an interest in mathematics (a number of papers in the collection, intended for a general mathematical audience, have been published in the American Mathematical Monthly). Each paper is accompanied by the author's comments on further development of the subject. The volume contains twenty-one papers, and is partitioned into three parts: Differential geometry and curvature, Algebraic geometry and topology, and Euclidean and non-Euclidean geometry. Although some of the papers have been written quite a while ago ("On total curvature of knots", 1950, answers a question from a course of differential geometry Milnor attended in 1948-1949), they appear more modern than many of today's publications. Milnor's excellent, clear, and laconic style makes the book a real treat ("Eigenvalues of the Laplace operator on certain manifolds", 1964, is less than a page long; the paper contains the first example of isospectral but not isometric compact Riemannian manifolds). This volume is highly recommended to a broad mathematical audience, and, in particular, to young mathematicians who will certainly benefit from their acquaintance with Milnor's mode of thinking and writing.

**9. John Milnor: collected papers. Vol. 2. The fundamental group (1995).**

**9.1. From the Preface.**

The field of topology has grown at a prodigious rate during my mathematical lifetime, the collective effort of a large and wonderful collection of mathematicians. It has been a pleasure, while preparing this collection and looking back at papers written twenty or thirty or forty years ago, to see hoe the field has changed, branching out in unforeseen directions with surprising successes in many cases, although many extremely difficult problems remain. In preparing introductions to the four sections of this volume, I have tried to describe some aspects of this growth and change.

**9.2. Review by: Serge L Tabachnikov.**

*Mathematical Reviews*, MR2307956

**(2008g:01014)**.

The second volume of the Collected Papers of J Milnor contains 16 papers united by their common theme, the fundamental group. The papers are divided into four parts: knot theory, free actions on spheres, torsion and three-dimensional manifolds. Each part has an introduction in which each paper is briefly described and relevant further developments are mentioned; each introduction has its own list of references. The volume is dedicated to Ralph Fox. The papers in the collection range from short gems ("Some curious involutions of spheres", joint with M Hirsch) to substantial surveys ("Whitehead torsion", 70 pp.). Many topics of the papers included in this collection have witnessed a spectacular progress in recent years. For example, it is very interesting to compare the papers in Part 1 with work in knot theory after its explosion in the 1980s after the introduction of the Jones polynomial and related developments. Another notable example is a short joint paper with W Thurston, "Characteristic numbers of 3-manifolds", that represents the early stage of the program of applying geometrical methods to topological problems; this paper is seen differently today, after the recent breakthrough in the proof of Thurston's Geometrization Conjecture.

**10. Dynamics in one complex variable (1999).**

**10.1. From the Preface.**

This volume studies the dynamics of iterated holomorphic mappings from a Riemann surface to itself, concentrating on the classical case of rational maps of the Riemann sphere. They are based on introductory lectures given at Stony Brook during the Fall Term of 1989-90. These lectures are intended to introduce the reader to some key ideas in the field , and to form a basis for further study. The reader is assumed to be familiar with the rudiments of complex variable theory and of two-dimensional differential geometry, as well as some basic topics from topology.

**10.2. Review by: John Hubbard.**

*Bull. Amer. Math. Soc. (N.S.)*

**38**(4) (2001), 495-498.

I am a Milnor fan. Although I first met Milnor in 1982, as a sophomore (now 37 years ago) I read Topology from a differentiable viewpoint , still in my view the best mathematics book ever written. It had a lot to do with why I became a mathematician, and what sort of mathematician. Other Milnor books followed: Morse theory, Lectures on the H-cobordism theorem, Singular points of complex hypersurfaces, each a turning point in my view of mathematics. My undergraduate advisor, Raoul Bott, himself a great expositor, always said: to learn to write mathematics well, read Milnor and try to emulate his style. There aren't many who have Milnor's extraordinary ability to take a complicated subject and present it in such a luminous way that everything seems natural, easy and perfectly clear. All this to say that I am by no means a detached and unprejudiced observer: I came to the book under review expecting to love it, and this expectation was fulfilled. ... Without explicitly quoting whole proofs, it is hard to convey exactly what makes Milnor's book so pleasant to read. I have used it in several graduate classes, and the students have consistently reacted favourably, although they find the book more difficult to read than Beardon's, and generally find the problems extremely challenging.

**10.3. Review by: Dierk Schleicher.**

*Mathematical Reviews*, MR1721240

**(2002i:37057)**.

*Dynamics in one complex variable*is an introduction to the iteration theory of rational maps, written by a master of the subject and a master of exposition. The subject itself is currently experiencing a second phase of great activity. It contains as special cases the well-known Newton-Raphson method to find zeroes of polynomials or rational maps, as well as the theory of iterated polynomials with its universality phenomena, explaining the ubiquitous "Mandelbrot set". The first active phase of complex dynamics took place during the first two decades of the 20th century after Paul Montel had introduced normal families, which gave rise to a burst of activity by Fatou, Julia and others. The second active phase started around 1980 when Sullivan introduced quasiconformal maps into the field, and when computer experiments became available which revealed much of the structure (and beauty) of the subject. Milnor's

*Dynamics in one complex variable*started as lecture notes in 1990. They have had great impact, by giving direction to the field, and by helping to educate many of the younger researchers (the writer of this report is no exception).

**11. Collected papers of John Milnor. III. Differential topology (2007).**

**11.1. From the Preface.**

This is the third volume of my collected papers to appear. Most of the material which follows was written during a ten year period in the late 1950s and early 1960s. It was my good luck to begin working on the topology of manifolds, and especially on the topology of smooth manifolds, at a time when the field was ripe for development. The study of cohomology and homotopy theory and the study of fibre bundles were in full flower, and I was particularly fortunate to learn about these subjects from Norman Steenrod. Three important developments during the 1950s played a key role. The first was Serre's thesis, which opened up the study of homotopy groups. The second was Thom's work on cobordism, which provided powerful new tools towardsunderstanding smooth manifolds, and in particular led to a proof of Hirzebruch's signature formula. The third was Bott's work on the homotopy groups of classical groups, which again played an essential role. I hope that the present collection of papers will help to keep this circle of ideas alive for the next generation of students.

**11.2. Review by: Serge L Tabachnikov.**

*Mathematical Reviews*, MR2307957

**(2008h:01022)**.

The third volume of

*Collected papers of John Milnor*is divided into four parts: Exotic Spheres, Expository Lectures, Relations with Algebraic Topology and Cobordism. Each part has an introduction in which each paper is briefly described and relevant further developments, with references to the literature, are mentioned. The volume contains a very warm four-page dedication to G de Rham and H Whitney (and "to their beloved mountains") accompanied by photographs. There are other lesser known photographs in the book: R Bott, R Thom, F Hirzebruch and J Milnor himself. The papers in the volume mostly represent the "golden years" of differential topology (late 1950s-early 1960s), to which Milnor was one of the principal contributors; the book certainly belongs on every working or budding topologist's bookshelf. Furthermore, a graduate student studying topology will benefit from reading Part 2, which consists of three previously unpublished survey lectures. Part 1 includes the celebrated papers on exotic smooth structures on spheres, including the 28 different smooth seven-dimensional spheres. Part 4 contains R Thom's Bourbaki Seminar talk on Milnor's work in cobordism theory. The volume concludes with a gift to the reader, a three-page note: "A concluding amusement: symmetry breaking", featuring, with comments, a closed immersed plane curve that bounds two essentially different immersed disks ("a doodle generated when my attention wandered during a lecture in the early 60s").

**12. Collected papers of John Milnor. IV. Homotopy, homology and manifolds (2009).**

**12.1. From the Preface.**

This fourth volume of my collected papers gathers together works on topology that did not appear in the three earlier volumes. Most of these were written during the late 1950's or the 1960's, but some recent expository papers are also included. I was fortunate to be in Princeton at a time when Norman Steenrod and John Moore were leaders in the development of algebraic topology. Seminars and teaching led to questions about the limits of application of the familiar functors of topology, about general constructions and their properties, and about the fine structure of invariants and families of invariants. Many of these papers deal with such questions. Further developments of the algebraic topology of manifolds, particularly those that are unrelated to cobordism, are also collected here. As in volume three, the previously published papers have been copied without change. Unpublished papers have been typeset and edited for readability and correctness. In one case, a lengthy rewriting was undertaken to achieve the original goal of the lectures. It is hoped that this collection will enrich the record of the development of algebraic topology and the education of the next generation of topologists.

**12.2. Review by: Serge L Tabachnikov.**

*Mathematical Reviews*, MR2590677

**(2011g:01020)**.

The fourth volume of

*Collected papers of J. Milnor*has the same format as the previous one: it is divided into four parts: Homotopy theory; Cohomology and homology; Manifolds; and Expository papers. Each part has an introduction in which each paper is briefly described and relevant further developments, with references to the literature, are mentioned. The volume is dedicated to Norman Steenrod. There are two photographs in the book: of Milnor himself (circa 1965) and of Steenrod. The papers in the volume were mostly written during the late 1950s or the 1960s; two papers were not previously published. Similarly to the previous volumes, one finds here classical results that belong to textbooks. To mention one example, Part 1 opens with the papers "Construction of universal bundles, I, II" (1956) in which the celebrated construction of the universal G-bundle as the infinite join of the group G with itself is given. Part 4 contains an excellent, and previously unpublished, survey of foliations, based on lectures given by Milnor at MIT in 1969. There are also two recent surveys on the Poincaré conjecture (published in 2003 and 2006), obviously a topic of a great current interest. The last article in Part 4, "Fifty years ago: topology of manifolds in the 50's and 60's" (2006) includes questions from the audience and concludes with Milnor's limerick involving Papakyriakopoulos. ... In my opinion, the full collection belongs in every mathematical library and on the bookshelf of every working topologist.

**13. Collected papers of John Milnor. V. Algebra (2010).**

**13.1. From the Preface.**

During the late 1950's progress in differentiable topology seemed to rely more and more on classical results in algebra and number theory. Thus the study of differentiable structures on spheres made essential use of work by T Clausen and K G C von Staudt (circa 1840) on Bernoulli numbers. Work on the classification of even-dimensional manifolds made essential use of deep results about quadratic forms. The field of algebraic K-theory, then coming into being, introduce groups. K

_{0}and K

_{1}which were intimately connected with topological problems. ... Thus attempts to solve topological problems led directly to serious questions in algebra, which of course had an addictive fascination of their own. Fortunately, I already had some exposure to ideas of algebraic number theory, through contact with John Tate and Serge Lang, and especially with Emil Artin, whose beautiful and highly polished lectures were an inspiration.

**13.2. Review by: Balasubramanian Sury.**

*Mathematical Reviews*, MR2841244

**(2012h:01022)**.

It is wonderful to see these papers which appear as a part of volume 5 of the collected works of John Milnor. Particularly, there are some unpublished papers which were preliminary versions of some of the author's path-breaking papers. There are also some articles which appeared in conference proceedings and which may not have received as much exposure as one might desire. Milnor's work in the subjects appearing in this volume has given birth to whole programs and set the agenda of research in these subjects for the last five decades. ... In conclusion, this volume is a treasure-house.

**14. Collected papers of John Milnor. VI. Dynamical systems (1953-2000) (2012).**

**14.1. Review by: Peter Haissinsky.**

*Mathematical Reviews*, MR3059583.

This book collects some of the contributions of J W Milnor to dynamical systems, starting from his 1953 paper on the Poincaré-Bendixson theory of vector fields on the 2-sphere. The articles are organized into two main parts, depending on whether the dynamical systems are real or complex. Both parts are introduced by a detailed overview of their contents. Two main themes can be put forth: the study of the entropy of a dynamical system and the analysis of parameter spaces of dynamical systems, i.e., their classification. Many of the collected papers have been - and should remain - very influential. This is of course the case with respect to Milnor's work with W P Thurston on unimodal maps, with S Friedland on polynomial automorphisms of the complex plane and his approach to the classification of rational maps of the Riemann sphere with two free critical points, just to quote a few. One can also find some maybe less popularized but nonetheless quite interesting other contributions, like his example of a measurable subset of the square of full measure which intersects each leaf of a foliation by analytic curves at most once. Like the other volumes, it constitutes a rich selection of very well-written mathematics which deserves to be read, and re-read.

**15. Collected papers of John Milnor. VII. Dynamical systems (1984-2012) (2014).**

**15.1. From the Preface.**

The field of holomorphic dynamics was initiated by Pierre Fatou and Gaston Julia a hundred years ago, and brought to new life in the early 1980s by the work of Adrien Douady, John Hubbard, Dennis Sullivan, Bill Thurston, and many others. I was very happy to become involved in this exciting area of research, and to become a member of the large and friendly family of mathematicians who have been exploring it. Most of the papers in this collection were published in the twelve years between 2000 and 2012. The only exceptions are a manuscript from 1984 titled "Notes on surjective cellular automaton-maps," and a manuscript from 2000 titled "Tsujii's monotonicity proof for real quadratic maps," published for the first time in this collection. Four of these papers have been produced from the electronic files used in the publications in which they originally appeared. However, the remaining nine needed to be reTEXed. In these cases, there has been some editing for clarity and consistency of notation, and the references have been updated; but without significant mathematical change.