Subject: Obituary for Shoro Araki From: Atsushi Yamaguchi Date: Wed, 12 Oct 2005 20:38:11 +0900 Shoro Araki died on July 11th at the age of 75. He was ill seriously since the end of the last year. Araki was born on Jan. 28th 1930 in Yamanashi prefecture. After graduated from Nagoya University in 1952, he became an assistant professor of Kyushu University. He moved to Osaka City University in 1961, where he spent most of his career until 1993. He was a visitor of Institute for Advanced Study from 1958 to 1960, a visiting professor of Stanford University from 1965 to 67 and Swiss Federal Institute of Technology from 1970 to 1971. He served as dean of Faculty of Science of Osaka City University from 1986 to 1988. After his retirement of Osaka City University, he moved to University Marketing Distribution Sciences in Kobe. He worked there for five years and retired in March of 1998. His wife passed away in the early winter of 1995 by stomach cancer. He has two sons, one daughter and one grandson. He contributed to advences of very wide arias of algebraic topology, especially, (co)homology of Lie groups, homology operations, generalized cohomology theories, equivariant theories. Araki's first paper is "On the triad excision theorem of Blakers and Massey" published in 1953 (Nagoya Math. J. 6 (1953), 129-136). In this paper a new proof is given of a theorem on triad homotopy groups due to Blakers and Massey. In 1956 Kudo and Araki published two papers which opened up a new world in the algebraic topology (Proc. Japan Acad. 32 (1956), 333-335, Mem. Fac. Sci. Kyusyu Univ, Ser. A. 10(1956), 85-120). Their operation is known as Kudo-Araki operation. On the some class of spaces they define homology operations by a procedure dual to Steenrod operations. This was later generalized by Dyer-Lashof and now it plays an important role in the theory of infinite loop spaces. Next he worked on the cohomology of exceptional Lie groups in early sixties. It has been well known that if the integral cohomology of a compact connected Lie group G is p-torsion free, the mod p cohomology of G is an exterior algebra with genrators whose dimensions are the same as the dimensions of generators of the rational cohomology of G. However, since exceptional simple Lie groups, especially of E-type, do not have good linear representations, very few was known about the mod p cohomology of exceptional Lie groups of E-type till 1960. A. Borel showed that the integral cohomologies of E_6 and E_7 (resp. E_8) are p-torsion free if p is greater than 3 (resp. 5) and determined the structures of mod 3 cohomology of E_6 and mod 5 cohomology of E_8 (Tohoku Math. J. (2) 13 (1961), 216-240). Araki determined the mod p cohomologies of E_k for the rest of the all cases, namely, for (p,k)=(2,6),(2,7),(2,8),(3,7),(3,8), by a deep considerations on results of Bott-Samelson making use of a theorem of R.Bott which claims that the integral cohomology of the loop space of a simply connected compact Lie group is torsion free (J. Math. Osaka City Univ. 12 (1961), 43-65, Proc. Japan Acad. 37 (1961), 619-622, Ann. of Math. (2) 73 (1961), 404-436). These results are the first step of the later studies of the cohomology theory of Hopf spaces. On the other hand, Araki showed that the mod 3 Pontryagin ring of F_4 is not commutative and also showed a similar result for E_6, E_7 and E_8 (Nagoya Math. J. 17 (1960), 225-260). This is the first example of Lie group whose Pontryagin ring is not commutative. His result leads to a theorem of W.Browder which asserts that, for an odd prime p and a finite connected homotopy associative Hopf space X, the mod p Pontryagin ring of X is commutative if and only if the integral cohomology group of X is p-torsion free, and a theorem of J.R.Hubbuck which claims that if X is a finite connected homotopy commutative Hopf space, X is homotopy equivalent to a torus. In order to determine the cohomologies of exceptional Lie groups using the theory of Bott-Samelson, Araki studied the root systems of Lie algebras in detail and he obtained an important result on symmetric space (J. Math. Osaka City Univ. 13 (1962), 1-34). As an application, he proved a theorem on the Brouwer degrees of some maps of compact symmetric spaces (Topology 3 (1965), 281--290). Araki made remarkable contributions on generarized cohomology theories. In a joint work with H.Toda (Osaka J. Math. 2 (1965), 71-115, 3 (1966), 81-120), they made a deep analysis on the multiplicative structures on generalized cohomology theory with coefficients in finite cyclic groups. He studied "Hopf structure" of the Bockstein spectral sequence associated with mod p K-theory of homotopy associative Hopf spaces (Ann. of Math. (2) 85 (1967), 508-525, Osaka J. Math. 8 1971, 151-206 with Z. Yoshimura) and showed that a theorem of L.Hodgkin "the complex K-theory of compact simple simply-connected Lie group is torson free" without using the classification of simple Lie groups. Araki and Z.Yoshimura constructed certain spectral sequence associated with a generalized cohomology theory of CW-complexes with filtrations (Osaka J. Math. 9 (1972), 351-365) which generalize the well-known short exact sequence of J.W.Milnor (Pacific J. Math. 12 (1962), 337-341). Araki wrote a lecture note "Typical Formal Groups in Complex Cobordism and K-theory" (Lectures in Mathematics, Department of Mathematics, Kyoto University, No.6. Kinokuniya Book-Store Co., Ltd. Tokyo, 1973) which came out from a series of lectures given at Kyushu, Osaka City, Kyoto Universities (Dec. 1972 - July 1973). In this lecture note, he made use of the theory of P.Cartier on typical curves over formal groups to do a unified treatment in showing the Quillen decomposition of localized complex cobordism theory and the Adams decomposition of localized complex K-theory. He showed that the coefficients of the p-series of BP-theory give new polynomial generators (called "Araki generators") of the coefficient ring. As a continuation of the above work, he considered a p-typical formal group which is universal for p-typical formal groups over arbitrary ground rings (Osaka J. Math. 11 (1974), 347-352). He determined the ring structure of the ground ring of such formal group and showed that this ring is isomorphic to the image of the canonical map from the coefficient ring of the complex cobrodism theory to that of the oriented cobordism theory if p=2. He also studied stable multiplicative operations in BP-cohomology theory (Osaka J. Math. 12 (1975), 343-356) and showed that such operation is always invertible. He introduced Adams operations in BP-theory and proved that the center of the group of the stable multiplicative operations is given by such operations. Araki also wrote a textbook on generalized cohomology theories (in Japanese) in 1975. This is one of the indispensable books for graduate students who study algebraic topology and can read Japanese. It shuold be mentioned that Araki is one of the pioneers in equivariant homotopy theory. He wrote eight papers from 1978 to 1988. First he laid the foundation for equivariant spectrum and (co-)homology theories, and considered equivariant K-theory and cobordism theory. In the subsequent papers he showed that equivariant homotopy theory is also a fruitful area in algebraic topology. Araki had so nice personality that many people around him liked him. He was rigorous on mathematics and took very good care of his students. He liked to play go game and to drink alcoholic beverages very much. We miss him so bad and wish that we could have drunk with him again even if we must take care of him after he got drunk as before. ----------------------------- Acknowledgement I deeply acknowledge Akira Kono for writing the part of Araki's work on the cohomology of exceptional Lie groups and pointing out some errors, and Kouyemon Iriye for writing the part of Araki's works in 1950's and works on equivariant theories. I also thank Steve Wilson for a helpful comment.