Anything by Adams or Milnor

Een eeuw wiskunde in boeken, Nieuw Archief voor Wiskunde (5) 1 (2000), p. 308.

When I first went up to Cambridge as a third year undergraduate, Frank Adams did not teach any courses at that level, and very sadly he died before I returned to Cambridge a couple of years later as a Ph.D. student. So I never had the intellectual pleasure of attending a lecture course by Adams, one of the greatest algebraic topologists there ever was. But fortunately he had the rare gift to write books and scientific papers that give the impression as if he was lecturing personally to you during a walk through the countryside, without making any concessions regarding mathematical rigour.

All his writings are enlivened by witty remarks that he also used as a lecturer to keep the attention of his audience. Some of these remarks have become classics, like his description of a spectral sequence -- one of the machineries of algebraic topology to compute invariants by a complicated iterative algebraic procedure -- as ``an Elizabethan drama, full of action, in which the business of each character is to kill at least one other character, so that at the end of the play one has the stage strewn with corpses and only one actor left alive (namely the one to speak the last few lines).'' In case the machinery of algebraic topology ever seemed too daunting, he would comfort the reader with the words ``let us be glad we don't work in algebraic geometry.''

One of his books that I found particularly useful as a student was his Algebraic Topology -- A Student's Guide. It begins with a thirty page survey of the material which a student of algebraic topology faces, and a guide to the most convenient sources for these topics. The bulk of the book consists of a collection of classic expositions taken from lecture notes, proceedings and other sources which are not easily accessible. The introductory survey ends with the immensely sane advice to the student that before he ``writes anything himself he should soak himself in papers which are well written. For this purpose I would recommend practically anything written by J.-P. Serre or J.W. Milnor.'' -- or J.F. Adams, I might add.

With my research turning towards differential topology, I thoroughly soaked myself in writings of John Milnor. Many of these only exist as mimeographed Princeton notes that would certainly warrant compiling a companion volume to the book by Adams. One of Milnor's little gems is his Topology from the Differentiable Viewpoint. Milnor manages to cover on sixty pages some of the most fundamental ideas of differentiable topology, such as the Pontrjagin construction and framed cobordisms, and yet keep the level of presentation accessible to second year undergraduates, without ever resorting to quoting theorems without proof. Simply miraculous.

J. Frank Adams, Algebraic Topology -- A Student's Guide, Cambridge University Press, 1972;
John W. Milnor, Topology from the Differentiable Viewpoint, The University Press of Virginia, 1965 (reprinted by Princeton University Press, 1997).

H. Geiges