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Andre Weil and Algebraic Topology borel.qxp 2/12/99 10:41 AM Page 422 André Weil and Algebraic Topology Armand Borel ndré Weil is associated more with number and belongs to a conjugate of T. If T 0 is another theory or algebraic geometry than with alge- torus and t0 generates a dense subgroup of T 0, then braic topology. But the latter was very much any torus containing t0 will also contain T 0, whence on his mind during a substantial part of his the conjugacy statement. career. This led him first to contributions to This was the first new proof of that theorem, Aalgebraic topology proper, in a differential geometric completely different from the original one, which setting, and then also to the use in abstract algebraic relied on a study of singular elements (cf. H. Weyl, geometry and several complex variables of ideas bor- Collected Papers II, 629–633). It was rediscovered rowed from it. independently, about five years later, by H. Hopf According to [W3], I, p. 562, his first contacts with al- and H. Samelson (Comm. Math. Helv. 13 (1940–41), gebraic topology took place in Berlin, 1927, in long con- 240–251). versations with, and lectures from, Heinz Hopf. The first For about ten years, from 1942 on, topology was publication of H. Hopf on the Lefschetz fixed point for- present in several works of Weil, often pursued si- mula appeared the following year, so it is rather likely multaneously, which I first list briefly: that Weil heard about it at the time. At any rate, his first a) In algebraic geometry: foundations, intro- paper involving algebraic topology is indeed an applica- duction of fibre bundles, formulation of the Weil tion of that formula to the proof of a fundamental the- conjectures. orem on compact connected Lie groups (which Weil at- b) New proof of the de Rham theorems. To- tributes to E. Cartan, but is in fact due to H. Weyl): Let gether with Leray’s work, this was the launching G be a compact connected Lie group. Then the maximal pad for H. Cartan’s work in sheaf theory. tori (i.e. maximal connected abelian subgroups) of G are c) Characteristic classes for differentiable bun- conjugate by inner automorphisms and contain all ele- dles: Allendoerfer-Weil generalization of the Gauss- ments of G ([1935c] in [W3], I, 109–111). Bonnet theorem, theory of connections, the Chern- The proof is a repeated application of the Lefschetz Weil homomorphism, the Weil algebra. fixed point formula to translations by group elements on d) Joint work with Cartan, Koszul, and Cheval- the homogeneous space G/T, where T is a maximal ley on cohomology of homogeneous spaces. torus. Note that the isotropy groups on G/T are the con- e) A letter to H. Cartan (August 1, 1950) on com- jugates of T, so that an element belongs to a conjugate plex manifolds, advocating the use of analytic fibre of T if and only if it fixes some point in G/T. Weil first bundles in the formulation of problems such as points out that T is of finite index in its normalizer those of Cousin. N(T ). If t ∈ T generates a dense subgroup of T, then its There is a last item I would like to add, dating fixed points are the same as those of T, and a local com- from 1961–62: putation shows their indexes to be simultaneously equal f) Local rigidity of discrete cocompact subgroups to 1 or to −1. The Lefschetz number of t is then =06 . But of semisimple Lie groups. since t is connected to the identity, this number is equal On the face of it, it does not belong to algebraic to the Euler-Poincaré characteristic χ(G/T) of G/T, which topology, but can be fitted under my general title is therefore =06 . As a consequence, any element g ∈ G when stated as a theorem on group cohomology. has a non-zero Lefschetz number, hence a fixed point, This formulation was originally an afterthought, but turned out to be important to suggest further Armand Borel is professor emeritus of mathematics at the Insti- developments. tute for Advanced Study. His e-mail address is [email protected]. This article is based on a lecture at the Institute for Advanced Algebraic Geometry Study on January 8, 1999, as part of a conference on the Work The algebraic geometry, as developed mainly by the of André Weil and Its Influence, January 8–9, 1999. Italian School, did not offer a secure framework for The article is also appearing in the Gazette des Mathématiciens. the proof of the Riemann hypothesis for curves and 422 NOTICES OF THE AMS VOLUME 46, NUMBER 4 borel.qxp 2/12/99 10:41 AM Page 423 other researches of Weil in algebraic geometry. He had to develop new foundations, with as one of its main goals a theory of intersections of subvarieties. It had also to be over any field. This implied a mas- sive recourse to algebra, but Weil still wanted to keep a geometric language and picture. Until then, only projective, affine, or quasi-projective vari- eties had been considered, i.e. subvarieties of some standard spaces. He wanted a notion of “abstract variety” which would be the analogue of a mani- fold (albeit with singularities). His first version [W1] is a bit awkward, as acknowledged in the foreword to the second edition, because no topol- ogy is introduced. From ([1949c], [W3], I, 411–413) on, however, he uses the language of the Zariski topology (introduced in 1944 by O. Zariski), and I shall do so right away. Fix a “universal field” K, i.e. an algebraically closed field of infinite transcen- Photograph courtesy of Sylvie Weil. dence degree over its prime field. Let V be an al- Weil (left) with Armand Borel in Chicago about 1955. gebraic subset of Kn, i.e. an affine variety. In the Zariski topology, the closed subsets of V are the abelian varieties, and it supplied essentially the only algebraic subsets. The open sets are, of course, their framework for algebraic geometry over any field complements and are quite big. If V is irreducible, until Grothendieck’s theory of schemes (from about any two nonempty ones intersect in a dense open 1960 on). one, so that the topology is decidedly not Haus- Algebraic topology also underlies the formula- dorff (unless V is a point), which may explain tion of the conjectures in ([1949b], cf. [W3], I, some reluctance to use it initially. To define an (ir- 399–410), soon to be called the Weil conjectures, reducible) abstract variety V, start from a finite col- which suggest looking for a cohomology theory for lection (Vi,fji), (i,j ∈ I), where Vi is an irreducible complete smooth varieties in which a Lefschetz affine algebraic set, fji a birational correspon- fixed point formula would be valid. This vision, dence from Vi to Vj satisfying certain conditions, which turned out to be prophetic, was unique at −1 so that, in particular: fii is the identity, fij = fji , the time. there exist open subsets Dji ⊂ Vi such that fji is In ([1949c], cf. [W3], I, 411–12), Weil introduces a biregular mapping of Dji onto Dij , and in algebraic geometry fibre bundles with an alge- fji = fjk ◦ fki. Two points Pi ∈ Dji and Pj ∈ Dij braic group, say G, as structural group. Given a va- are equivalent if fji(Pi)=Pj . The “abstract vari- riety B and a finite open cover {Vi}(i ∈ I) of B, as- ety” V is by definition the quotient of the disjoint sume one is given regular maps sij : Vi ∩ Vj → G union V˜ of the Vi by that equivalence relation. (i,j ∈ I; sii is the constant map to the identity), with Note that V is obtained by gluing together dis- the usual transitivity conditions. Let F be a vari- joint affine sets. For lack of suitable concepts, it ety on which G operates. Then a fibre bundle E was not possible to start from a topological space on B, with typical fibre F, is obtained by gluing the and require that it be endowed locally with a given products Vi × F by means of the sij, as usual. Weil structure, as is done for manifolds (as was done also considers the case of principal bundles (F = G, later by Serre using the notion of ringed space acted upon itself by right translations). In partic- ∗ [S2]). As a result, the Vi and fji are part of the struc- ular, if G = C is the multiplicative group of non- ture, which is rather unwieldy and requires a some- zero complex numbers, the isomorphism classes what discouraging amount of algebra to be worked of such bundles correspond to linear equivalence with. Nevertheless, Weil develops the theory of classes of divisors. It also allowed Weil to interpret such varieties and of the intersection of cycles. For in a more conceptual way earlier work on algebraic the latter, the analogy with the complex case and curves (see [W3], I, 531, 541, 570 for comments). the intersection product in the homology of man- A detailed exposition is given in [W2], where the ifolds (on which he had lectured earlier at the classification of such bundles is studied in some Hadamard Seminar ([W3], I, 563)) is always present. simple cases. In particular, a key property is the analogue of In view of the big size of the neighborhoods on Hopf’s inverse homomorphism (see [W1], Intro- which such a bundle is trivial, it was not a priori duction, xi–xii).
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