Promenade 12

Topology - Or how to survey the fogs

The innovative notion of the "scheme", we've seen, allows one to establish connections between the different geometries associated with each prime number (or "characteristics"). These geometries, however, are all of an essentially 'discontinuous' or 'discrete' nature, as opposed to the traditional geometry which is our legacy from previous centuries, 9 back to Euclid). The new concepts introduced by Zariski and by Serre have restored, to some extent, a 'continuous dimension' concept for these geometries, which was automatically picked up by the "schematic geometry" that had just been invented to unify them. However the "fabulous conjectures" of Weil were still a long way off. These "Zariski topologies" were, seen from this perspective, so crude that one might just as well have remained at the "discrete aggregate" stage.

It was clear that what was still lacking was some new principle that could connect these geometric objects ( or "varieties", or "schemes") to the usual "well behaved" (topological) "spaces": those, let us say, whose points are clearly distinguished one from the other, whearas in the "harum-scarum" spaces introduced by Zariski, the points have a sneaky tendency to cling to one another....

Most certainly it was through nothing less than through this "new principle" that the marriage of "number and magnitude",( or of "continuous and discontinuous" geometry) could give birth to the Weil conjectures.

The notion of space is certainly one of the oldest in mathematics. It is fundamental to our "geometric" perspective on the world, and has been so tacitly for over two millenia. Its only over the course of the 19th century that this concept has, bit-by-bit, freed itself from the tyranny of our immediate perceptions ( that is, one and the same as the "space" that surrounds us), and of its traditional theoretical treatment ( Euclidean), to attain to its present dynamism and autonomy. In our own times it has joined the ranks of those notions that are most freely and universally employed in mathematics, and is familiar, I would say, to every mathematician without exception. It has become a concept of multiple and varied aspects, of hundreds of thousands of faces depending on the kinds of structures one choses to impose on a space, from the most abundant and rich, ( such as the venerable 'euclidean' structures, or the 'affine' or 'projective' ones , or again the 'algebraic' structures of similarly designated 'varieties' which generalize and extend them), down to the most 'impoverished': those in which all 'quantitative' information has been removed without a trace, or in which only a qualitative essence of "proximity" or of "limit"(*), ( and, in its most elusive version, the intuition of form ( called 'topological spaces' )), remains.

(*)When I speak of the idea of a "limit" it is above all in terms of passage to a limit, rather than the idea that most non-mathematicians, of a "frontier"
The most "reductive" of all these notions over the course of half a century down to the present, has appropriated to itself the role of a kind of conceptual englobing substrate for all the others, that of the topological space. The study of these spaces constitutes one of the most fascinating and vital branches of geometry: Topology.

As elusive as it might appear initially, the "qualitatively pure" structure encapsulated in the notion of "space"( topological) in the absence of all quantitative givens, ( notably the metric distances between points) which enables us to relate it to habitual intuitions of "large" and "small", we have, all the same, over the last century, been able to confine these spaces in the locked flexible suitcases of a language which has been meticulously fabricated as the occasion arose. Still better, as the occasion arose, various 'weights and measures' have been devised to serve a general function, good or bad, of attaching "measures" ( called 'topoligical invariants'), to those sprawled-out spaces which appear to resist, like fleeting mists, any sort of metrizability. Most of these invariants, its true, certainly the most essential ones, are more subtle than simple notions like 'number' and 'magnitude' - often they are themselves fairly delicate mathematical structures bound ( by rather sophisticated constructions) to the space in question. One of the oldest and most crucial of these invariants, introduced in the last century ( by the Italian mathematician Betti) is formed from the various "groups" ( or 'spaces'), called the "Cohomology" associated with this space. (*)

(*) Properly speaking, the Betti invariants were homological invariants. Cohomology is a more or less equivalent or "dual" version that was introduced much later. This has gained pre-eminence over the intial "homological" aspect, doubtless as a consequence of the introduction, by Jean Leray, of the viewpoint of sheaves, which is discussed further on. From the technical point of view one can say that a good part of my work in geometry has been to identify and develop at some length, the cohomological theories which were needed for spaces and varieties of every sort, above all for the "algebraic varieties" and the schemes. Along the way I was also led to a reinterpretation of the traditional homological invariants in cohomological terms, and through doing so, to reveal them in an entirely new light.

There are numerous other "topological invariants" which have been introduced by the topologists to deal with this or that property or this or that topological space. Next after the "dimension" of a space and the (co)homological invariants, come the "homotopy groups". In 1957 I introduced yet another one, the group (known as "Grothendieck) K(X), which has known a sensational success and whose importance ( both in topology and arithmetic) is constantly being re-affirmed. A whole slew of new invariants, more sophisticated than the ones presently known and in use, yet which I believe to be fundamental, have been predicted by my "moderated topology" program ( one can find a very summary sketch of this in the "Outline for a Programme" which appears in Volume 4 of the Mathematical Reflections) . This programme bases itself on the notion of a "moderated theory" or "moderated space", which constitutes, a bit like the topos, a second "metamorphosis of the concept of space". It is at the same time more self-evident and less prodound than the latter. I predict that its immediate applications to topology "properly speaking" will be decidedly more incisive, that in fact it will turn upside down the "profession" of topological geometer, through a far-reaching transformation of the conceptual context appropriate to it. ( As was the case with Algebraic Geometry with the introduction of the point-of-view of the scheme.) Furthermore, I've already sent copies of my "Outline" to several of my old friends and some illustrious topologists, yet it seems to me that that haven't been inclined to take any interest in it....

It was the Betti numbers that figure ("between the lines" naturally) in the Weil conjectures, which are their fundamental "reason for being" and which ( at least for me, having been "let in on the secret" by Serre's explications) give them meaning. Yet the possibility of associating these invariants with the "abstract" algebraic varieties that enter into these conjectures, in such a manner as to response to the very precise desiderata demanded by the requirements of this particular cause - that was something only to be hoped for. I doubt very much that, outside of Serre and myself, there isn't anyone else ( including André Weil!)who really believes in it.(*)
(*) It is somewhat paradoxical that Weil should have an obstinate, even visceral block against the formalism of cohomology, particularly since it had been in large part his "famous" conjectures that inspired the development, starting in 1955, of the great cohomological theories of algebraic geometry, (launched by J.P. Serre with his foundational article "FAC", already alluded to in a footnote.)

Its my opinion that this "block" is part of a general aversion in Weil against all the global formalisms, (whether large or small), or any sort of theoretical construction. He hasn't anything of the true "builder" about him, and it was entirely contrary to his personal style that he saw himself constrained to develop, starting with the 30's, the fundamentals of "abstract" algebraic geometry, which to him ( by his own dispositions), have proved to be a veritable "Procrustean bed" for those who use them.

I hope he doesn't hold it against me that I chose to go beyond him, investing my energy in the construction of enormous dwelling places, which have allowed the dreams of a Kronecker, and even of himself, to be cast into a language and tools that are at the same time effective and sophisticated. At no time did he ever comment to me about the work that he saw me doing, or which had already been done. Nor have I received any response from him about Récoltes et Semailles, which i sent to him over three months ago, with a warm hand-written personal dedication to him.

Soon afterwards our understanding of these cohomological invariants was profoundly enriched and renovated by the work of Jean Leray ( carried out as a prisoner of war in Germany in the early part of the 40's). The essential novelty in his ideas was that of the (Abelian) sheaf over a space, to which Leray associated a corresponding collection of cohomology groups ( called "sheaf coefficients"). It is as if the good old standard "cohomological metric" which had been used up to them to "measure" a space, had suddenly multiplied into an unimaginablely large number of new "metrics" of every shape, size and form imaginable, each intimately adapted to the space in question, each supplying us with very precise information which it alone can provide. This was the dominant concept involved in the profound transformation of our approach to spaces of every sort, and unquestionably one of the most important mathematical ideas of the 20th century.

Thanks above all to the ulterior work of Jean-Pierre Serre, Leray's ideas have produced in the half century since their formulation, a major redirection of the whole theory of topological spaces, ( notably those invariants designated as "homotopic", which are intimately allied with cohomology), and a further redirection, no less significant, of so-called "abstract" algebraic geometry ( starting with the FAC article of Serre in 1955). My early work in geometry, from 1955 onwards, was conceived of as a continuation of the work of Serre, and for that reason also a continuation of the work of Leray.

Promenade 13: Toposes - or The Double Bed

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