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Edwin Henry Spanier (1921–1996)

Morris W. Hirsch

Edwin Henry a Miller Research Fellow at Berkeley in 1961–62. His Spanier died visiting appointments include positions in Ar- of cancer in gentina, Brazil, Canada, Chile, France, Germany, Scottsdale, Italy, Spain, and Switzerland, and at UCSD and Arizona, on UCLA. October 11, Spanier was appointed professor of math- 1996. Born in ematics at Berkeley in 1959 at the beginning of a Washington, period of rapid expansion of the mathematics de- D.C., on Au- partment. An internationally recognized authority gust 8, 1921, in the swiftly developing field of topology, he at- he was grad- tracted first-class mathematicians as visitors and uated from new faculty members. He played a major role in the Univer- organizing new programs in geometry and topol- sity of Min- ogy, subjects in which Berkeley soon achieved pre- nesota in eminence. He served several times as vice chair and 1941 and acting chair of the department and directed four- then spent teen doctoral dissertations at Berkeley in addition three years to three at Chicago. In 1991 he became professor as a math- emeritus. ematician in Photograph courtesy of Jerome Spanier. the U. S. From his doctoral dissertation through the mid- Edwin Spanier Army Signal 1960s, and again in the last fifteen years of his life, Corps. He re- Spanier’s research was concentrated in algebraic ceived his doctorate in mathematics in 1947 from topology. This subject, founded a century ago by the under the direction of Henri Poincaré to aid in the qualitative analysis of Norman Steenrod. After a postdoctoral fellowship differential equations, was little known to most at the Institute for Advanced Study in Princeton, mathematicians when Spanier started his career. , he joined the faculty at the University His dissertation supervisor, Norman Steenrod, to- of Chicago in 1948. He was a Guggenheim Fellow gether with , had just set out sim- in Paris in the academic year 1952–53, a member ple and powerful axioms for the main tool, ho- of the Institute for Advanced Study in 1958–59, and mology theory; this cleared up much of the confusion endemic in a subject that combines Morris W. Hirsch is professor of mathematics at the Uni- geometry, algebra, and analysis. Spanier was at versity of California at Berkeley. His e-mail address is the forefront of the explosive development of al- [email protected]. He received his Ph.D. in gebraic topology during the next decade. The im- 1958 under the supervision of Ed Spanier. portance of his work was quickly recognized; in

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1950 he gave an Invited Address to the Interna- languages from this viewpoint and obtained a tional Congress of Mathematicians at Harvard. number of impressive results. In another approach, Spanier’s interest in topology continued through- they investigated families of languages generated out his career. Both his first paper in 1948 and one from a single grammar, machine, or structure using of his last, published in 1992, dealt with the Eilen- simple substitution operations. berg-Steenrod axioms. The importance of their work can be seen from Spanier’s first major contribution in topology the review1 of a 1983 article [2] by Spanier, S. Gins- was the theory of cohomotopy groups, which gave burg, and J. Goldstine, “On the equality of gram- an algebraic classification of matical families”. The review maps of polyhedra into states: “This paper presents spheres [4]. Together with Ph.D. Students of Edwin Spanier a major and difficult decid- S. S. Chern he pioneered the Clair Miller (1951) ability result in grammar form analysis of the Morris W. Hirsch (1958) theory. It is proved that, given groups of fibre spaces [1]. Elon Lima (1958) two arbitrary context-free In a series of papers with Alphonse Thomas Vasquez (1962) grammar forms, it is decid- the English topologist J. H. C. able whether or not the fam- Robert Emmett Williamson Jr. (1963) Whitehead, Spanier devel- ily of one is contained in the oped the new algebraic tool Samuel Feder (1964) other. This leads immediately of duality in homotopy the- John Ucci (1964) to the decidability of the ory [5]. In 1966 his long- Benson Brown (1965) equality of two context-free awaited book Algebraic Jose Alves (1965) grammar forms.” This result Topology [3] was published. is based on an earlier paper, David Kraines (1965) The first comprehensive “A prime decomposition the- modern treatment of the Denis Sjerve (1967) orem for grammatical fami- subject, it is still a funda- Jon Goldstine (1970) lies”, in which the same au- mental source. John Paul Alexander (1971) thors obtained a prime In all, Spanier published Mark Luker (1975) decomposition for formal lan- more than forty papers in Kenneth Klingenstein (1975) guage families, closely analo- gous in form to the decom- , con- Gerald Eisman (1977) tributing to most of the position of whole numbers Richard B. Hull (1980) major research areas in the into prime factors. field, including Spanier’s publications, as operations, obstruction the- were his lectures, are charac- ory, homotopy theory, imbeddability of polyhe- terized by unusual lucidity and precision and an dra in Euclidean spaces, and topology of function even rarer quality of naturalness and simplicity. No spaces. Many of his results are now standard tools matter how complex the subject, at the end the in all fields that utilize global geometrical rea- reader feels the theorems are the right ones, the soning. These include not only various subjects in hypotheses natural, and the methods as simple as pure mathematics, but also diverse areas in applied possible. mathematics, including computer science, math- Ed Spanier will be remembered as a gifted re- ematical physics, economic models, and game the- searcher, an inspiring teacher, an able adminis- ory. Interestingly, one of Spanier’s theories, now trator, and as a modest, friendly, wise, and help- called Alexander-Spanier homology, is currently ful colleague. being applied to analyze differential equations— References a return to Poincaré’s original use of algebraic topology. [1] S. S. CHERN and E. SPANIER, The homology structure of sphere bundles, Proc. Nat. Acad. Sci. U.S.A. 36 (1950), In 1961 Spanier began a fruitful collaboration 248–255. with Seymour Ginsburg of the University of South- [2] S. GINSBURG, J. GOLDSTINE, and E. SPANIER, On the equal- ern California, resulting in more than a score of pa- ity of grammatical families, J. Comput. System Sci. pers on the structure of formal languages. This sub- 26 (1983), 171–196. ject is of importance in several mathematical [3] E. SPANIER, Algebraic topology, McGraw-Hill, New disciplines, including theoretical computer sci- York, 1966. ence and foundations of mathematics. Recently it [4] E. SPANIER, Borsuk’s cohomotopy groups, Ann. Math. 50 (1949), 203–245. has been applied in dynamical systems theory. [5] E. SPANIER AND J. H. C. WHITEHEAD, Duality in homotopy Their work was driven by an operational perspec- theory, Mathematika 2 (1955), 56–80. tive on formal languages: What formal operations on these objects are reasonable, and what are their effects? This view led them to abstract and study families of languages closed under specific oper- ations. They also investigated abstract families of 1Mathematical Reviews 85g:68037.

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