Leopold Vietoris (1891–2002)

Heinrich Reitberger

n April 9, 2002, shortly before his 111th the University of Vienna. Before that, he had passed birthday, Leopold Vietoris died in a the exam for high-school teachers. During the fol- sanitarium at Innsbruck after a brief ill- lowing period of teaching he received a postcard ness. The mathematical community from Escherich congratulating him on his thesis and Ohas lost a well-known researcher. offering him an assistant position at the Techni- Vietoris was the recipient of several high awards. cal University in Graz. Two years later Vietoris received his Habilitation in Vienna on the recom- Biography mendation of H. Hahn. L. Vietoris was born in Radkersburg (Styria) on In 1925 Vietoris started working in combinato- June 4, 1891. After his graduation from the rial topology. He spent three semesters as a Rock- “Benediktinergymnasium” in Melk, he studied math- efeller fellow with L. E. J. Brouwer in Amsterdam, ematics and descriptive geometry at the University where P. Alexandrov and K. Menger (whom he knew and the Technical University in Vienna. In his sixth as a student from Vienna) were staying at the same semester he heard a lecture on topology by time. It was during this stay that he began writing W. Gross in 1912 based on the axioms of accumu- the papers on algebraic topology for which he is lation points by F. Riesz—extended by Gross. At the best known (Mayer-Vietoris sequence and so forth). same time W. Rothe at the Technical University In 1927 he followed a call to Innsbruck as associ- raised the question of the notion of manifold. ate professor, in 1928 he went back to the Tech- Vietoris planned to create a geometrical notion of nical University in Vienna as full professor, and in manifold with topological means. He was drafted 1930 he finally settled in Innsbruck. in 1914 but continued working on his own on this In the autumn of 1928 Leopold Vietoris married problem. In September 1914 he was wounded, and Klara Riccabona. She later died after giving birth after his recovery he was sent to the southern to their sixth daughter. In 1936 he married Maria front. In 1916, while an army mountain guide, he Riccabona, Klara’s sister, who thenceforth was a obtained his first results, which he expanded dur- mother to her nieces and a devoted spouse. She died ing a three-month stay in Vienna (spring semester shortly before her husband. 1918), where he read for the first time Hausdorff’s 1 classic (published in 1914). On November 4, 1918, Foundations of General Topology he became an Italian prisoner of war. Due to de- To avoid Hausdorff’s countability axioms, Vietoris cent treatment, he was able to complete his thesis, added to the neighborhood axioms his separation which after his release he submitted to G. v. axiom of “regularity”; defined filter base (“Kranz” Escherich and W. Wirtinger in December 1919 at = wreath), directed set (“orientierte Menge”), nets, and the related convergence concept; and introduced the modern notion of compactness (under Heinrich Reitberger is professor of mathematics at the In- stitut für Mathematik der Universität Innsbruck, Inns- the name “lückenlos” = without gaps). bruck, Austria. His email address is heinrich. [email protected]. 1See also [Rei97], [Rei02].

1232 NOTICES OF THE AMS VOLUME 49, NUMBER 10 Discussing the notion of directed set in 1937, G. Birkhoff [Bir37] wrote the condition “(D3) given Ph.D. Students of Leopold Vietoris ∈ ∈ ∈ α A, β A, there exists γ A satisfying γ>α Kurt Hellmich, and γ>β” and remarked, “It is primarily condition Funktionen, deren Werte Mengen sind (1939) (D3) which was due to Moore and Smith, and which Martha Petschacher, distinguishes directed sets from other ‘partially Tafeln hypergeometrischer Funktionen (1946) ordered sets’”. Indeed, (D3) occurred as the “com- Hiltrud Jochum, position property” in a paper of E. H. Moore and Die Cayleyschen Formeln in der Kreisgeometrie und die 2 H. L. Smith [MS22] , but priority belongs to Vi- Brennpunkte in der Gaußschen Ebene (1952) etoris, who introduced this concept as “oriented Johann Leicht, set” in [Vie21, p. 184]. Interestingly enough, Birk- Zur intuitionistischen Algebra und Zahlentheorie (1952) hoff quoted this paper, concerning the separation Walter Dürk, axioms on p. 174, but seems not to have read any 2 Der Strukturkomplex t1t2 = b (1953) further! Helmut Grömer, Whereas Moore and Smith considered only gen- Über den Begriff der Wahrscheinlichkeit (1954) eralized sequences with numerical values, Vietoris Eva Ambach, studied right from the start “sets of second order”, Der Größenfehler einer durch das Adamssche i.e., systems of sets under Zermelo’s axioms, in- Interpolationsverfahren gewonnenen Näherungslösung dexed by a directed set, and gave the definition: A einer Differentialgleichung (1957) set of second order is called a “Kranz” [wreath], if Gerhard Riege, the intersection of any two elements contains again Das Axiom von Pasch in konvexen Räumen (1957) an element = ∅. With respect to inclusion, a “Kranz” Fortunat Pescolderung, forms a directed system of sets, and vice versa, the Über eine Fehlerabschätzung zur numerischen Integration remainders R(B):={x ∈ M : b

NOVEMBER 2002 NOTICES OF THE AMS 1233 Tietze in 1923. As This is again an -chain. The boundary of any Urysohn remarked, the -chain is defined by linear extension. The -chains term “regularity” goes with zero boundary are called -cycles. An -chain back to Alexandrov.4 xn is called η-homologous to zero in X, written n n n+1 n+1 The attempts to create x ∼η 0, if x = ∂y for an η-chain y in X. a convenient notion of n n n Vietoris called a sequence z =(z1 ,... ,zk ,...) of manifold led Vietoris to n → →∞ k -cycles zk in X fundamental if k 0 (for k ) look for a spatial struc- and for all >0 there exists N() such that for all ture on the power set of n ∼ n l,m > N(), we have zl zm , that is, a topological space. In n − n ∼ zl zm 0 in X. The fundamental sequences n “Regions of second order” form a group Zn(X,G). A fundamental sequence z [Vie22] he defined on the is called homologous to zero if for all >0 there collection of all nonempty n ∼ ≥ exists an N such that zk 0 for all k N. closed subsets CL(S) of a The quotient group topological space (S,T ) a topology as follows (com- Hn(X,G)=Zn(X,G)/{null sequences}, Photograph courtesy of Heinrich Reitberger. pare Nadler [Nad78]). For Leopold Vietoris each finite collection the n-th homology group, was the central object for the further studies of Vietoris6 (cf. Hirzebruch U1,... ,Un ∈T, let [Hir99] and Mac Lane [ML86]). U1,... ,Un denote the set of all A in CL(S) such ⊂∪n ∩ ∅ First an anecdote: Until World War Two, the that A i=1Ui and A Ui = for each Vietoris complex and V(ietoris)-cycles were part of i =1,... ,n; the sets U1,... ,Un form the base of a topology on CL(S). In the case of a compact con- the standard knowledge of all topologists (cf. nected metric space X this “Vietoris topology”on [Lef42]). Twenty years ago the complex was rein- CL(X) coincides with topology induced by the so- vented by E. Rips while studying hyperbolic met- called “Hausdorff metric”. Recently, this metric ric groups, and M. Gromov used it in his funda- has again become important, for example in frac- mental works on these groups. Finally in 1995, tal image compression—so to speak, a jump from J.-Cl. Hausmann saw that this concept goes back “hyperspace to cyberspace”! to Vietoris, and so he called it the Vietoris-Rips complex. Algebraic Topology An important tool to determine the homology Algebraic topology develops methods for deciding groups of a space is a process to compute them with algebraic tools whether two topological spaces from simpler pieces. This is given by the Mayer- are homeomorphic. The applications range from Vietoris sequence—the best-known result connected simple-sounding questions, such as whether a prod- with the name Vietoris. uct like the one in the complex numbers exists also in higher dimensions, to the theory of knots Theorem (Mayer-Vietoris sequence). Let K1 and K2 and its use in particle physics and biochemistry. be subcomplexes of a simplicial complex K. Then the Since Poincaré’s time topologists have tried to find sequence of homology groups appropriate invariants—first for simplicial com- ...→ H (K ∩ K ) → H (K ) ⊕ H (K ) plexes, then more generally for metric spaces, as q 1 2 q 1 q 2 Vietoris showed us [Vie27]5, and finally for general → Hq(K1 ∪ K2) → Hq−1(K1 ∩ K2) →··· spaces by means of coverings. is an exact sequence. Now we come to the so-called Vietoris complex. Concerning the genesis of this theorem, we may Let X be a metric space. An (ordered) n-dimen- let the principals speak for themselves. Mayer sional -simplex σ n of X is an (n +1)-tuple of points [May29]: “I was introduced to topology by my col- e ,e ,... ,e in X such that the distance of any two 0 1 n league Vietoris, whose lectures in 1926–7 I at- is less than . Let G be an abelian group. A formal tended at the local university. In many talks about linear combination g σ n of -simplices with co- i i i this field Vietoris gave me a lot of hints for which efficients gi ∈ G is called an -chain in X. The n I am very grateful.” Vietoris [Vie30]: “W. Mayer, boundary of an -simplex σ =[e0,... ,en] is defined by whom I told about the problem as well as the con- jectured result and a way to its solution, has solved n i ∂σ := (−1) [e0,... ,ei,... ,en]. the question, as far as it concerns Betti numbers, i in a somewhat different way in these Monatshefte. In what follows, I will return to my original idea and 2 4For R the regularity property results already from an use it for the solution in the general case.” Thus axiom of R. L. Moore, as mentioned by Chittenden. 5A short version appeared in Proc. Amsterdam 29 (1986), 6Vietoris commented that these studies were inspired by 1008-13. an oral remark of Brouwer.

1234 NOTICES OF THE AMS VOLUME 49, NUMBER 10 Vietoris calculated the homology groups and not for a complex function A(x) = exp{u(x)+iϕ(x)}, just the Betti numbers (i.e., the ranks of the groups). where u and ϕ are real functions of a real variable In May 1946 J. Leray introduced today’s central x, satisfying notions of sheaf, sheaf cohomology, and spectral sequence. His motivation was the following situation (2) u(x + ξ)=u(x)+u(ξ), [Die89]: Let X and Y be topological spaces and f : X → Y a continuous map. The main problem is (3) ϕ(x + ξ) ≡ ϕ(x)+ϕ(ξ) (mod 2πZ). (after Leray) to relate the homology of X and the homology of Y, perhaps under some restrictions on f. By using a Hamel basis, he found a new solution Apparently Leray was unaware in 1946 that to (3), simpler than an earlier one by van der Vietoris had obtained such a result twenty years Corput. In 1957 Vietoris used the Cauchy func- earlier together with the definition of homology tional equation (3) to give a remarkable proof—as groups for the case of compact metric spaces Aczél says—of the limit limx→0(sin x)/x =1. [Vie27]. In a series of papers, Vietoris treated the solu- Indeed, Leray first turned his attention to topol- tion of ordinary differential equations by me- ogy in earnest during World War Two, when access chanical means, beginning with a modification of to the literature was problematic. The story, as re- the Picard method of successive iterations. lated in [BHL00], is that Leray was imprisoned during the war as a French officer in a camp in the Probability Waldviertel (Lower Austria) as head of the “prisoner The object of Vietoris’s papers on probability the- university” Edelbach-Allentsteig. To avoid being ory was the introduction of nine axioms governing forced to work for the German army, Leray the “eher”-relation ≤, where aA ≤ bB corresponds remained silent about his knowledge of fluid to the intuitive idea “outcome a in trial A is not dynamics and instead represented himself as a more probable than outcome b in trial B”, and the topologist. derivation of the laws of classical probability from We now come to the second part of Vietoris’s these. His approach to probability is the same as paper: the mapping theorem (in the formulation of B. O. Koopman’s. Alluding to the problem of ob- S. Smale, who generalized it in 1957). taining original sources in wartime, Vietoris pointed Theorem (Vietoris-Begle). Let X,Y be compact met- out in a footnote that he had not seen Koopman’s ric spaces, f : X → Y surjective and continuous. Sup- paper and only learned about it later from a review. pose that for all 0 ≤ r ≤ n − 1 and all y ∈ Y the re- −1 Positive Trigonometric Sums duced homology groups H˜r (f (y)) vanish (in [Vie27] Vietoris used the coefficient group G := Z/2Z). Vietoris proved important inequalities in three pa- Then the induced homomorphism pers Über das Vorzeichen gewisser trigonometrischer Summen [Vie58, Vie59, Vie94], the last hav- f- : H˜r (X) → H˜r (Y ) ing been written at the youthful age of 103 years! is an isomorphism for r ≤ n − 1 and an epimor- Theorem. Let a0,a1,...,an and t be real numbers. phism for r = n. If The fibers are assumed to be acyclic (“without (1) a ≥ a ≥···≥a > 0 and holes”), in other words, they have the same ho- 0 1 n mology as a single point. For further explanation ≤ 2k−1 ≤ ≤ n we consider the situation in topological vector (2) a2k 2k a2k−1 (1 k 2 ), then spaces: For a nonempty subset A we have n convex =⇒ starshaped =⇒ contractible (3) ak sin kt > 0 and k=1 =⇒ acyclic =⇒ connected. n In 1950, E. G. Begle extended the theorem to com- ak cos kt > 0(0 0 spondences, see [Rei01]. k=1 k (0 0(0

NOVEMBER 2002 NOTICES OF THE AMS 1235 of the hypergeometric summation formula and to [Car37a] H. Cartan, Théorie des filtres, C. R. Acad. Sci. Paris other summation formulas. 205 (1937), 595–8. [Car37b] ——— , Filtres et ultrafiltres, C. R. Acad. Sci. Paris Applications 205 (1937), 777–9. [Die89] J. DIEUDONNÉ, A history of algebraic and differen- By conferring honorary doctor degrees of techni- tial topology. 1900–1960, Birkhäuser Boston Inc., cal sciences, the Technical University of Vienna in Boston, MA, 1989. 1984 and the Technical Faculty of the Innsbruck [Hir99] F. HIRZEBRUCH, Emmy Noether and topology, The her- University in 1994 acknowledged the contributions itage of Emmy Noether (Ramat-Gan, 1996), Bar-Ilan of Vietoris to practical applications. These con- Univ., Ramat-Gan, 1999, pp. 57–65. cern his works on orientation in mountainous ter- [Lef42] S. LEFSCHETZ, Algebraic topology, American Math- rain by differential geometric means, the strength ematical Society Colloquium Publications, Vol. 27, of the alpine ski, and the physics of block glaciers. Chapter IV, American Mathematical Society, New York, 1942, 389 pp. Final Remarks [LR82] R. LIEDL and H. REITBERGER, Leopold Vietoris—90 Jahre, Yearbook: Surveys of Mathematics 1982, Bib- Leopold Vietoris’s fundamental contributions to liographisches Inst., Mannheim, 1982, pp. 169–70. general as well as algebraic topology, and also to [May29] W. MAYER, Über abstrakte Topologie. I, II., Monatsh. other branches of the mathematical sciences, have Math. 36 (1929), 1–42, 219–58. made him immortal in the world of science. As a [ML86] S. MAC LANE, Topology becomes algebraic with person, he was outstandingly humble and grateful Vietoris and Noether, J. Pure Appl. Algebra 39 (1986), for his well-being, which he also wished and granted no. 3, 305–7. his fellow humans. He devoted his spare time to [MS22] E. H. MOORE and H. L. SMITH, A general theory of his large family, religious meditation, music, and limits, American J. 44 (1922), 102–21. his beloved mountains. On the other hand, ad- [Nad78] S. B. NADLER JR., Hyperspaces of sets, Marcel Dekker Inc., New York, 1978, Monographs and Textbooks in ministrative duties were not Vietoris’s favorite Pure and Applied Mathematics, Vol. 49. tasks, as he pointed out in a letter to L. E. J. Brouwer [Rei97] H. REITBERGER, The contributions of L. Vietoris and in 1947: “As dean I am overwhelmed with admin- H. Tietze to the foundations of general topology, istrative matters to such an extent that I often have Handbook of the History of General Topology, to hold my lectures inadequately prepared and Vol. 1, Kluwer Acad. Publ., Dordrecht, 1997, don’t have any time for scientific research. Luck- pp. 31–40. ily, the term will soon be over and then I hope to [Rei01] ——— , Vietoris-Beglesches Abbildungstheorem, Vi- be a scientist again and not a bureaucrat.” In re- etoris-Lefschetz-Eilenberg-Montgomery-Beglescher search Vietoris was a “lone fighter”: Only one of his Fixpunktsatz und Wirtschaftsnobelpreise, Jahresber. Deutsch. Math.-Verein. 103 (2001), no. 3, 67–73. more than seventy mathematical papers has a coau- [Rei02] ——— , Die Beiträge von L. Vietoris zu den Grund- thor. Half of the papers were written after his six- lagen der Topologie. I, Wiss. Nachrichten 119 (2002), tieth birthday. in press. A long life has fulfilled itself. Beside the grief [Vie21] L. VIETORIS, Stetige Mengen, Monatsh. Math. 31 comes our thankfulness! (1921), 173–204. [Vie22] ——— , Bereiche zweiter Ordnung, Monatsh. Math. Acknowledgements 32 (1922), 258–80. I thank Ottmar Loos, who, following G. Lochs, cur- [Vie27] ——— , Über den höheren Zusammenhang kom- rently occupies Vietoris’ professorial chair, and pakter Räume und eine Klasse von zusammen- hangstreuen Abbildungen, Math. Ann. 97 (1927), the editors for their help with the English version 454–72. of this obituary and many other helpful sugges- [Vie30] ——— , Über die Homologiegruppen der Vereinigung tions. A German obituary that includes a complete zweier Komplexe, Monatsh. Math. 37 (1930), 159–62. list of Vietoris’s publications will appear in the [Vie31] ——— , (with H. Tietze) Beziehungen zwischen den Jahresbericht der Deutschen Mathematiker- verschiedenen Zweigen der Topologie, Enc. Math. Vereinigung 104 (2002). Wiss. III.1.2 (1931), AB13. [Vie44] ——— , Zur Kennzeichnung des Sinus und ver- References wandter Funktionen durch Funktionalgleichungen, [Ask98] R. ASKEY, Vietoris’s inequalities and hypergeo- J. Reine Angew. Math. 186 (1944), 1–15. metric series, Recent progress in inequalities (Nisˇ, [Vie58] ——— , Über das Vorzeichen gewisser 1996), Kluwer Acad. Publ., Dordrecht, 1998, trigonometrischer Summen, Sitzungsber. Österreich. pp. 63–76. Akad. Wiss. 167 (1958), 125–35. [BHL00] A. BOREL, G. M. HENKIN, and P. D. LAX, Jean Leray [Vie59] ——— , Über das Vorzeichen gewisser (1906–1998), Notices Amer. Math. Soc. 47 (2000), trigonometrischer Summen. II, Anz. Österreich. Akad. no. 3, 350–9. Wiss. 10 (1959), 192–3. [Vie94] , Über das Vorzeichen gewisser [Bir35] G. BIRKHOFF, A new definition of limit, Bull. Amer. ——— Math. Soc. 41 (1935), 635. trigonometrischer Summen. III, Österreich. Akad. Wiss. Math.-Natur. Kl. Sitzungsber. II 203 (1994), [Bir37] ——— , Moore-Smith convergence in general topology, Ann. of Math., II. Ser. 38 (1937), 39–56. 57–61.

1236 NOTICES OF THE AMS VOLUME 49, NUMBER 10