A Topological Aperitif

A Topological Aperitif

Topologyhasbeenreferredtoas“rubber-sheetgeometry”.Thenameisapt,for the subject is concerned with properties of an object that would be preserved, no matter how much it is stretched, squashed, or distorted, so long as it is not in any way torn apart or glued together. One’s ?rst reaction might be that such animprecise-soundingsubjectcouldhardlybepartofseriousmathematics,and wouldbeunlikelytohaveapplicationsbeyondtheamusementofsimpleparlour games. This reaction could hardly be further from the truth. Topology is one of the most important and broad-ranging disciplines of modern mathematics. It is a subject of great precision and of breadth of development. It has vastly many applications, some of great importance, ranging from particle physics to cosmology, and from hydrodynamics to algebra and number theory. It is also a subject of great beauty and depth. To appreciate something of this, it is not necessary to delve into the more obscure aspects of mathematical formalism. For topology is, at least initially, a very visual subject. Some of its concepts apply to spaces of large numbers of dimensions, and therefore do not easily submit to reasoning that depends upon direct pictorial representation. But even in such cases, important insights can be obtained from the visual - rusal of a simple geometrical con?guration. Although much modern topology depends upon ?nely tuned abstract algebraic machinery of great mathematical sophistication, the underlying ideas are often very simple and can be appre- ated by the examination of properties of elementary-looking drawings. 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 Apéritif 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 longexact 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. I have grown more and more sorry about this gap in my knowledge over the years (although some of the problem was solved in an ad hoc fashion: teaching the subject helps, even if it's done badly). But now there is another real and pressing need to correct the problem: techniques from modern algebraic geometry have become big players in my work. I like to learn new and unfamiliar material within a framework of analogies, if possible, and so it has been my way for a while now to try to understand some of the new and high-falutin' stuff concerning, say, derived categories, by means of juxtaposing these developments against the corresponding staples of algebraic topology. (Mapping cones come to mind right away, for example.) But the shortcomings in my education in topology, described self-indulgently above, have made this approach increasingly difficult, causing me to seek a more formal course of self-study. It would be nice to go a linear route, with as one objective the development of a sort-of latter-day topologist's intuition. I think the present book by Huggett and Jordan, distinguished by clear and wonderful