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## Lectures on an Introduction to Grothendieck's Theory of the Fundamental Group

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Would you also like to submit a review for this item? 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.

Review by: Jerome P Levine. Mathematical Reviews , MR 39 Some of these results have appeared in an unpublished work of the author: "On isolated singularities of hypersurfaces".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 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 and abelian group K 0 or K 1 respectively. The theory has been developed by many authors, but the work of Hyman Bass has been particularly noteworthy, and Bass's book "Algebraic K-theory" , is the most important source of information.

Review by: Jean-Pierre Jouanolou. Mathematical Reviews , MR 50 Supplemented notes of a seminar held in Princeton in , 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 , ] which, for obvious reasons, was limited to the study of groups of Grothendieck and Whitehead. 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.

## Lectures on An Introduction to Grothendieck's Theory of the Fundamental Group by J.P. Murre

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 , as well as lectures at Princeton University in Review by: Louis H Kauffman. Mathematical Reviews , MR 58 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. Review by: E H Spanier. In 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 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. It is a valuable and welcome addition to the literature.

## Gasbarri : Heights of vector bundles and the fundamental group scheme of a curve

Review by: Friedrich Hirzebruch. Mathematical Reviews , MR 55 This book emerged from lectures that Milnor held at Princeton University in 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 , the older of the two authors apologizes in the preface "for the delay in publication", the reviewer apologizes for "the delay in reviewing".

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. Review by: Serge L Tabachnikov.

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Mathematical Reviews , MR 95c 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", , answers a question from a course of differential geometry Milnor attended in , 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", , 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. 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. Mathematical Reviews , MR g 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 s 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.

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 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. Review by: John Hubbard. I am a Milnor fan. Although I first met Milnor in , 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. Review by: Dierk Schleicher. Mathematical Reviews , MR i 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.