Topological Aperitif 2e

The appearance of this beautiful, modest-sized (166pp) book in this time and place is, for me personally, an instance of splendid serendipity. My reason for regarding this publication as a case of perfect timing is that my own work in number theory has begun to take a frighteningly topological turn over the last few years (in the French sense, so to speak), and some old lacunae have to be taken care of. Now! And A Topological Apritif looks to be a great place to start. The book would serve well for self-study and would play an important role in an introductory course as a supplementary level text to one of the established players, as I will discuss at greater length presently. It would actually also be perfect for a seminar in which various topics could then be pursued in greater depth using various supplementary sources. I should like to preface my review proper with a few personal reminiscences for which I beg the reader's indulgence. More years ago than I care to admit, I enrolled in undergraduate topology, at first full of enthusiasm and good intentions. But soon I proceeded to cut most of the classes because I was goofy and irresponsible and, if not in love, then heavily in like. I had become profoundly romantically infatuated with a pretty flautist whose music-department schedule allowed me to spend the hour ostensibly reserved for topology sitting on a bench near a fountain, gazing into her eyes rather than at a blackboard covered with what were to me very weird pictures. The romantic liaison did not last and I didn't learn topology (although I passed the class). In graduate school my elective courses outside of number theory ran toward Fourier analysis, trigonometric series, and commutative Banach algebras, even though no one would accuse me of being an analyst. As regards topology, it was by now strangely and sneakily fascinating to me, having glimpsed all those delectable commutative diagrams and long exact cohomology sequences in general department colloquia and on sundry blackboards in open offices. I even had the great pleasure of attending a seminar with Karoubi, whom I was ruefully unprepared to appreciate properly (though he was impossible not to like and admire). But all in all: no real learning of topology took place for me.