## Borel, Armand

## Borel, Armand

# BOREL, ARMAND

(*b*. la Chaux-de-Fonds, Switzerland, 21 May 1923; *d*. Princeton, New Jersey, 11 August 2003)

*mathematics, Lie groups, algebraic groups, arithmetic groups, topology*.

Lie groups and linear algebraic groups played a central role in Borel’s work. His exploration of these was basic in important trends in pure mathematics in the second half of the twentieth century, for example in the development of the theory of modular forms.

**Biography.** Borel grew up in the French-speaking part of Switzerland. After finishing his secondary school education in Geneva, he entered the Eidgenössische Technische Hochschule (ETH) in Zürich in 1942, studying mathematics and physics. He graduated in 1947, having also fulfilled his obligatory military service. Distinguished mathematicians were teaching at the ETH, including Heinz Hopf, who introduced Borel to topology, and Eduard Stiefel, who taught him the arcana of the theory of Lie groups. After two years as an assistant at the ETH, Borel spent the academic year of 1949–1950 in Paris with a grant from the Centre National de Recherches Scientifiques. There he came in contact with French mathematicians of his generation (notably Jean-Pierre Serre). He studied the work of Jean Leray in topology, being one of the first to become familiar with that work. After fulfilling a temporary teaching position at the University of Geneva, he obtained his doctoral degree (Doctorat ès Sciences Mathématiques) in 1952 in Paris, under the aegis of Leray. Also in 1952 Borel married Gabrielle (Gaby) Aline Pittet. They had two daughters.

Borel spent the academic years of 1952–1954 at the Institute for Advanced Study in Princeton, New Jersey. The period was crucial for the broadening of his mathematical interests. The next academic year (1954–1955) he was a visiting lecturer at the University of Chicago, where André Weil stimulated his interests in algebraic geometry and number theory. In 1955 Borel returned to Switzerland to take up a professorship at the ETH in Zürich. He then accepted, in 1957, a professorship at the Institute for Advanced Study in Princeton. This remained his base, although over the years he had many visiting positions elsewhere. An institution frequently visited was the Tata Institute of Fundamental Research in Bombay (Mumbai), of which he was an honorary fellow. In 1983–1986 he also held a professorship at the ETH in Zürich.

Borel retired from the Institute for Advanced Study in 1993, but he remained vigorous and active until the very end of his life. In 1999–2001 he was engaged in a program on Lie groups at the University of Hong Kong. Shortly after his eightieth birthday he was struck by cancer. He died a few months later.

Almost all of Borel’s published papers can be found in the four volumes of his *Oeuvres* (Collected Papers). The fourth volume contains a list of all publications, including seventeen books. The numbering of that list is used below (numbers in square brackets).

The theory of Lie groups is central in Borel’s work. Many of its aspects are explored. The work is of great depth and reveals not only a profound insight but also a remarkable clarity in the presentation of difficult material.

Borel had a somewhat reserved personality and at first sight could seem unapproachable. But once one knew him a little better he turned out to be a very friendly person, with a good sense of humor. Many mathematicians were helped or stimulated by him. Frequently, mathematical contacts with him resulted in collaborations, in which he freely shared his ideas. His mathematical erudition was extraordinary. Outside mathematics, his interests and knowledge were also impressive. He traveled widely and he was very knowledgeable about the culture of the countries he and his wife had visited. A particular interest of Borel’s was Carnatic (South Indian) classical music, of which he had expert knowledge.

He received several honors, including an honorary doctorate of the University of Geneva and memberships in the French Académie des Sciences and of the National Academy of Sciences. In 1992 he received the Balzan Prize for mathematics.

**Lie Groups and Topology.** Lie groups embody the continuous symmetries in mathematics and physics. Their theory was started in the nineteenth century by the Norwegian mathematician Sophus Lie. At first the notion of a Lie group was “localized,” which led to the theory of Lie algebras. But in the beginning of the twentieth century, for example in the work of Élie Cartan and Hermann Weyl, “global” aspects came into play, involving the geometry of the Lie group viewed as a geometric entity. A little later, in the 1940s, Borel’s teacher Heinz Hopf made important contributions to the study of the topology of Lie groups. A fundamental problem in that study is the determination of the homology groups or better, of the cohomology ring, of compact Lie group. Borel in his thesis [23] uses a new approach to the problem.

This approach is based on the ideas developed by Jean Leray (during his internment as a prisoner of war, 1940–1945). His ideas led to the cohomology theory of sheaves. Also, he devised a technical tool called “spectral sequence,” which has become a standard ingredient in homological algebra. Leray lectured about these new and unusual matters in Paris around 1950. Borel attended the lectures, digested Leray’s ideas, contributed to them, and gave a course on Leray’s work at the ETH (which led to the lecture notes [18]).

To elucidate, let *G* be a compact connected Lie group. In his thesis Borel introduces a “universal space” E_{G} for *G*, an acyclic principal fiber space with fiber *G*(actually, finite dimensional approximations to E_{G} are used). The basis B_{G} of the fiber space is a *classifying space* for *G*. A central object in Borel’s thesis is the Leray spectral sequence associated to the map E_{G}*→* B _{G.}. A difficult algebraic result gives, in good situations, a connection between the graded cohomology algebras *H ^{*}*(

*G, F*) and

*H*(B

^{*}_{G},

*F*) over a field

*F*, say. The first one will be a graded exterior algebra with generators of odd degrees say 2

*d*

_{1}– 1, 2

*d*

_{2}– 1, … 2d

_{1}– 1, and the second one a graded polynomial algebra with generators of degrees 2

*d*

_{1}, 2

*d*

_{2}, …, 2d

_{1,}, with an explicit connection between the respective generators (involving the Leray spectral sequence). The result is quite powerful and Borel exploits it, for example to study the cohomology of homogeneous spaces

*G*/

*U*, where

*U*is a closed subgroup of

*G*. In subsequent publications Borel studied several aspects of the cohomology of

*G*. One of these is its torsion. In [53] the torsion primes

*p*, the prime numbers for which

*H*(

^{*}*G*, Z) has

*p*-torsion, are described in terms of the root system of

*G*. These primes turned up later in seemingly unrelated questions in the theory of algebraic groups.

In connection with the universal space of *G*, mention should be made of the “Borel construction,” which is somewhat hidden in the seminar notes [52]. Let *X* be a space with a *G*-action. The universal space E_{G} has a free *G*-action. The Borel construction associates to *X* the quotient ofrelative to the diagonal *G* -action. Then *H ^{*}*(

_{GX}

*X*,

*F*) is the

*equivariant cohomology*of

*X*over

*F*. It has become an important tool of algebraic topology.

During his stay at the Institute for Advanced Study in 1952–1954, Borel started a collaboration with Friedrich Hirzebruch on the algebraic topology of homogeneous spaces of Lie groups. The collaboration led to the three influential papers [43, 45, 47]. A spin-off of the work with Hirzebruch was the “Borel-Weil theorem” (obtained in 1953, but unpublished at the time, see [30]). It describes a realization of the irreducible representations of a compact Lie group by sections of certain line bundles, a description that has become methodically important. The joint paper [49] with John C. Moore is a contribution to algebraic topology, introducing the much used “Borel-Moore homology.”

**Linear Algebraic Groups.** Around 1960, topics related to the theory of linear algebraic groups became paramount in Borel’s work. Linear algebraic groups are algebraic versions of Lie groups. They had already occurred in the nineteenth century. A linear algebraic group is a subgroup *G* of a group GL_{n}(*k*) of invertible *n*× *n*-matrices, where *k* is an algebraically closed field, such that the elements of *G* are the solutions of a set of polynomial equations in the matrix coordinates (an orthogonal group is an example). If *k =* C, the field of complex numbers, *G* can be made into a Lie group.

The theory of Lie groups uses analytic and local methods, unavailable for algebraic groups. Their theory requires tools from algebraic geometry, which had become available by the middle of the twentieth century. Borel took up the theory of linear algebraic groups during his stay in Chicago in 1954–1955. The foundations of the theory were laid in Borel’s paper [39]. A linear algebraic group *G* over the algebraically closed field *k* is an affine algebraic variety and as such is provided with a topology, the Zariski topology. There is an analogy with the theory of Lie groups.

Borel discovered an elementary but quite fundamental result, whose analogue in the theory of Lie groups is not true, the “orbit lemma.” It states that if *G* acts on an algebraic variety *X* (in the sense of algebraic geometry), then *G* has a Zariski-closed orbit in *X*. The lemma is an ingredient of the proof of the “Borel fixed point theorem.” This asserts that if the connected, solvable, linear algebraic group *G* acts on a complete variety *X*, then *G* fixes a point of *X*.

Borel showed the importance of the maximal closed, connected, solvable, subgroups *B* of *G* (soon after called *Borel subgroups*). Two Borel subgroups of *G* are conjugate in *G*. Moreover, a quotient *G/B* is a projective algebraic variety.

An (algebraic) *torus* is a linear algebraic group isomorphic to a group of diagonal matrices in some GL_{n(}(*k*). Another fundamental result of Borel’s paper (an analogue of a basic result about compact Lie groups) is the conjugacy of maximal tori of *G*. The theory of linear algebraic groups over algebraically closed fields was completed soon after the appearance of [39] by Claude Chevalley. He introduced in the theory of algebraic groups the combinatorial ingredients of the theory of Lie groups (root system, Weyl group). His work culminated in the classification of simple algebraic groups. The work was published in the seminar notes [Chev].

Borel next turned to the “relative” theory of linear algebraic groups, over a field *F*, which is not necessarily algebraically closed. His interest in the relative theory was certainly motivated by applications to arithmetical questions. Let *G* be a linear algebraic group over the algebraically closed extension *k* of *F*. Then *G* is *defined over F* if *G* and the group operations are defined over *F* in the sense of algebraic geometry. If *F* is a field of characteristic zero, this means that *G* is a subgroup of some GL_{n}(*k*) defined by polynomial equations with coefficients in *F*. If *G* is defined over *F* one has the subgroup *G*(*F*) of elements of *G* with coordinates in *F*.

An early publication on the relative theory is the joint paper [64] with Jean-Pierre Serre on Galois cohomology. Much of it has become standard material (see, e.g., Serre’s book [Ser]). Fundamental for the relative theory is the joint paper [66] on reductive groups with Jacques Tits. A reductive group is a linear algebraic group whose radical (the maximal closed, connected, normal, solvable subgroup) is a torus. The theory of reductive groups over *k* resembles the theory of compact Lie groups.

The object of study of [66] is a connected, reductive group *G*, which is defined over *F*. Important in the theory are the parabolic subgroups of *G*, that is, the closed subgroups of *G* containing a Borel group. They are introduced and studied in this work.

*G* need not have Borel groups which are defined over *F*. Instead, one considers parabolic subgroups that are defined over *F*, and in particular the minimal ones. It is shown that two minimal groups are conjugate by an element of *G*(*F*). This is the “relative” version of the conjugacy of Borel groups. *G* is said to be *isotropic* over *F* if there exist proper parabolic subgroups over *F*. If this is the case, Borel and Tits introduce a relative root system and a relative Weyl group. Of Borel’s later work pertaining to the general theory of algebraic groups, there is another joint paper [97] with J. Tits. This work studies the group-theoretical homomorphisms between groups of the type *G*(*F*), where *G* is a simple linear algebraic group defined over *F*. The paper brings the subject to a certain close. Borel’s last publication on the general theory of linear algebraic groups is the joint paper [158] with Frédéric Bien and János Kollár on rational connectedness of homogeneous spaces of algebraic groups. It is methodologically interesting, applying new insights of algebraic geometry to algebraic groups.

**Arithmetic Groups.** Let GL_{n}(C) ∍ G be a linear algebraic group that is defined over the field of rational numbers Q. The group *G*(R) of real-valued points of *G* is a Lie group. *G*(Z) denotes the subgroup of integral-valued points of *G*(R). A subgroup Γ of *G* (R) is *arithmetic* if the intersection Γ (Z) has finite index in both Γ and *G* (Z).

Fundamental for all arithmetical applications is the joint paper [58] with Harish-Chandra. It brought a conceptual approach to arithmetical questions in group theory that so far had been handled by ad hoc methods, and it lays the foundations of the modern theory of automorphic forms. Let *G* be defined over Q and assume *G* to be semisimple (i.e., with trivial radical). Let Γ be an arithmetic subgroup of *G*(R). Borel and Harish-Chandra construct an open subset *U* of *G* (R) with *G*(R)*=* Γ*.U*, having good properties (a “fundamental subset”). These imply that Γ is finitely generated and that the quotient space *G*(R)*/* Γ has finite volume (relative to a natural measure). It is compact if and only if *G* is not isotropic over Q.

The study of the (infinite-dimensional) representation of the Lie group *G*(R) in the Hilbert space *L* (*G*(R) */* Γ) leads into the theory of automorphic forms. In the subsequent paper [60] Borel gave an “adelic” version, also basic for the theory of automorphic forms, of his work with Harish-Chandra. Maintaining the previous notations, let *K* be a maximal compact subgroup of *G*(R). The quotient *X = K* \*G* (R) is the symmetric space associated to the Lie group *G*(R). An arithmetic group Γ acts on it and the quotient space *X/* Γ is an image of *G*(R)*/* Γ.

Borel and Walter L. Baily, Jr., in [69] studied the case that *X* is hermitian symmetric, that is, has an invariant complex structure. Then *V = X/*Γ is a normal analytic variety. In this work a “compactification” is constructed, that is, a compact analytic variety *V ^{*}*, containing

*V*as an open subvariety. (Actually,

*V*is a projective algebraic variety.) The boundary

^{*}*V*is a union of pieces, each related to classes of parabolic subgroups of

^{*}-V*G*that are defined over Q. Here the relative theory of algebraic groups is crucial. Embeddings of

*V*in complex projective spaces are constructed using automorphic forms.

^{*}For arbitrary semisimple *G* a compactification of the space *V* was constructed by Borel and Serre [98]. Now the compactification *V ^{*}* is a real manifold with corners.

Much of Borel’s later work deals with the cohomology of arithmetic groups. Let Γ be as before and let *E* be a Γ-module. To study the Eilenberg-MacLane cohomology groups *H ^{*}*(Γ

*, E*) the space

*V*is brought into the picture. One shows that

*H*(Γ

^{*}*, E*) is isomorphic to a relative Lie algebra cohomology group, with coefficients in a vector space related to functions on

*V*. In Borel’s talk [101] at the 1974 International Congress in Vancouver he gave a brief review.

The case *E =* R is discussed in [100], using the BorelSerre compactification. A beautiful result, important in algebraic *K* -theory, is the determination of the stable cohomology of groups like *H ^{*}*(SL

_{n}(

**o**), R), where

**o**is the ring of integers of an algebraic number field. Borel’s article [108] gives further arithmetical applications.

In Borel’s book [172] with Nolan R. Wallach the connection with relative Lie algebra cohomology is thoroughly explored. Later Borel gave various refinements of his previous work, for example in [125]. One of the refinements is the study of the *L*^{2}-cohomology of *V*, basic results about which are established in the joint paper [126] with W. Casselman. Of a somewhat different nature is the work with Gopal Prasad on finiteness results for arithmetic subgroups of semisimple groups (see [139]).

**Other Publications.** Borel’s publications include seventeen books. Several are based on lectures given by him, about various subjects related to his interests, for instance, the widely used textbook *Linear Algebraic Groups* [142]. At the Institute for Advanced Study, he organized seminars on current topics of research, and he was involved with the publication of notes of these seminars, such as [52] (on “Transformation Groups,” 1960), [Ind] (on the “Index Theorem,” 1965), and [Sem] (on “Algebraic Groups and Related Finite Groups,” 1970). Similar publications, outgrowths of seminars held in Switzerland, are [IC] (on “Intersection Cohomology,” 1984) and [DM] (on “Algebraic *D*-modules,” 1987).

Borel was a co-organizer of two Summer Research Institutes of the American Mathematical Society, on “Algebraic Groups and Discontinuous Subgroups” (Boulder, 1966) and on “Automorphic Forms, Representations, and *L* -functions” (Corvallis, 1977). An aim of the conferences was to present and explain new developments. The conference proceedings ([Proc 9] and [Proc 23]), containing several contributions by Borel himself, have been very influential.

Borel was an outstanding expositor. He gave several talks about the work of others, for example in talks at the Séminaire Bourbaki in Paris. An important expository paper is the report [44] written with J.-P. Serre on Grothendieck’s Riemann-Roch theorem. It remains the most accessible description of Grothendieck’s work. Borel was a member of the Bourbaki group of mathematicians (see [165]). Originally, their goal was to write textbooks for large parts of mathematics. Some of the published books have become standard references. The group was most influential around the middle of the twentieth century. Borel was involved with the preparation of several Bourbaki books.

In the last ten years of his life he made several contributions to the history of twentieth-century mathematics, for example in reminiscences about other mathematicians. Borel’s book *Essays on the Theory of Lie Groups and Algebraic Groups* (American Mathematical Society, 2001) is unique: a historical study of mathematical research in the twentieth century written by one of the leading researchers.

## BIBLIOGRAPHY

*The Borel archive is kept in Geneva, in the Bibliothèque Publique et Universitaire de Genève*.

*Borel’s four-volume* Oeuvres *(Collected Papers) lists almost all of his published papers. The fourth volume contains a list of all publications, including seventeen books. The numbering of that list is used below*.

### WORKS BY BOREL

*Oeuvres: Collected Papers*. Berlin: Springer-Verlag, vol. I, II, III, 1983; vol. IV, 2001.

18.*Cohomologie des espaces localement compacts d’après J. Leray*. *Lecture Notes in Mathematics* vol. 2. Berlin: Springer-Verlag, 1961.

23. “Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts.” *Annals of Mathematics* 57 (1951): 115–207.

30. “Représentations linéaires et espaces homogènes kähleriens des groupes simplement connexes compacts.” (1954, unpublished at the time).

39. “Groupes linéaires algébriques.” *Annals of Mathematics* 64 (1956): 20–82. 43, 45, 47. With Friedrich Hirzebruch. “Characteristic Classes and Homogeneous Spaces, I.” *American Journal of Mathematics* 80 (1958): 458–538; II, ibid. 81 (1959): 315–382; III, ibid. 82 (1960): 491–504.

44. With Jean-Pierre Serre. “Le théorème de Riemann-Roch, d’après Grothendieck.” *Bulletin Société Mathématique de France* 86 (1958): 97–136.

49. With John C. Moore. “Homology Theory for Locally Compact Spaces.” *Michigan Mathematical Journal* 7 (1960): 137–159.

52. “Seminar on Transformation Groups.” *Annals of Mathematical Studies* 46. Princeton, NJ: Princeton University Press, 1961.

53. “Sous-groupes commutatifs et torsion des groupes de Lie compacts connexes.” *Tôhoku Mathematical Journal* 13 (1961): 216–240.

58. With Harish-Chandra. “Arithmetic Subgroups of Algebraic Groups.” *Annals of Mathematics* 75 (1962): 485–535.

60. “Some Finiteness Properties of Adele Groups over Number Fields.” *Publications Mathématiques de l’Institut des Hautes Études Scientifiques* 16 (1963): 5–30.

64. With Jean-Pierre Serre. “Théorèmes de finitude en cohomologie galoisienne.” *Commentarii Mathematici Helvetici* 39 (1964): 111–164.

66. With Jacques Tits. “Groupes réductifs.” *Publications Mathématiques de l’Institut des Hautes Études Scientifiques* 27 (1965): 55–150.

“Seminar on the Index Theorem.” *Annals of MathematicalStudies* no. 57. Princeton, NJ: Princeton University Press, 1965. [Ind.]

With George D. Mostow, eds. “Algebraic Groups and Discontinuous Subgroups.” *Proceedings of Symposia in Pure Mathematics* no. 9, American Mathematical Society (1966). [Proc. 9]

69. With Walter L. Baily, Jr. “Compactification of Arithmetic Quotients of Bounded Symmetric Domains.” *Annals of Mathematics* 84 (1966): 442–528. “Seminar on Algebraic Groups and Related Finite Groups.” *Lecture Notes in Mathematics* no. 131. Berlin: Springer-Verlag, 1970. [Sem.]

97. With Jacques Tits. “Homomorphismes ‘abstraits’ de groupes algébriques simples.” *Annals of Mathematics* 97 (1973): 499–571.

98. With Jean-Pierre Serre. “Corners and Arithmetic Groups.” *Commentarii Mathematici Helvetici* 48 (1973): 436–491.

100. “Stable Real Cohomology of Arithmetic Groups.” *AnnalesScientifiques Ecole Normale Supérieure* 7 (1974): 235–272.

101. “Cohomology of Arithmetic Groups.” *ProceedingsInternational Congress of Mathematicians, Vancouver, 1974*, I (1975): 435–442.

108. “Cohomologie de SL_{n} et valeurs de fonctions zeta aux points entiers.” *Annali Scientifici Scuola Normale Superiore Pisa* 4 (1977): 613–636; Correction, ibid. 7 (1980): 373.

With William Casselman, eds. “Automorphic Forms, Representations and *L*-functions.”*Proceedings of Symposia in Pure Mathematics* no. 23 (2 vols.), American Mathematical Society (1979). [Proc. 23]

125. “Regularization Theorems in Lie Algebra Cohomology.” *Duke Mathematical Journal* 50 (1983): 605–623.

126. With William Casselman. “*L* -cohomology of Locally Symmetric Manifolds of Finite Volume.” *Duke Mathematical Journal* 50 (1983): 625–647.

*Intersection Cohomology*, Progress in Mathematics, no. 50. Boston: Birkhäuser, 1984. [IC]

*Algebraic D-modules*. Perspectives in Mathematics. Boston: Academic Press, 1987. [DM]

139. With Gopal Prasad. “Finiteness Theorems for Discrete Subgroups of Bounded Covolume in Semi-simple Groups.” *Publications Mathématiques de l’Institut des Hautes Études Scientifiques* 69 (1989): 119–171; Addendum, ibid. 71 (1990): 173–177.

142.*Linear Algebraic Groups*. 2nd ed. Graduate texts in Mathematics, no. 126. New York: Springer-Verlag, 1991.

158. With Frédéric Bien and János Kollár. “Rationally Connected Homogeneous Spaces.” *Inventiones Mathematicae*124 (1996): 103–127.

“Twenty-five years with Nicolas Bourbaki, 1949–1973.” *Notices of the American Mathematical Society* 45 (1998): 373–380.

172. With Nolan R. Wallach. “Continuous Cohomology, Discrete Subgroups and Representations of Reductive Groups.” *Annals of Mathematical Studies* 94. Princeton, NJ: Princeton University Press, 1980; 2nd ed., American Mathematical Society, 1999.

### OTHER SOURCES

Arthur, James, et al. “Armand Borel (1923–2003).” *Notices of theAmerican Mathematical Society* 51 (2004): 498–524. [AMS]

Chevalley, Claude. “Classification des groupes algébriques semi-simples” (The Classification of Semi-simple Algebraic Groups). *Collected Works of Claude Chevalley*, vol. 3. Edited by Pierre Cartier. New York: Springer-Verlag, 2005.

Serre, Jean-Pierre. “Cohomologie galoisienne.” *Lecture Notes inMathematics* no. 5, 5th ed. Berlin: Springer-Verlag, 1994. English translation, “Galois Cohomology.” Berlin: Springer-Verlag, 1997. [Ser.]

*T. A. Springer*