Subject: for the list From: "W. Stephen Wilson" Date: Mon, 21 Aug 2006 11:05:01 -0400 >From Japan: Dear Steve, I visited Kudo's house with Masayoshi Kamata, Mitsuyoshi Kato, Yasumasa Hirashima, Nobuyuki Oda and Kazuyuki Fujii and met his wife. The following is our short obituary for Tatsuji Kudo. This obituary is based on Minoru Tomita's article "Tatsuji Kudo --- professional career and works" (Mem. Fac. Sci. Kyusyu Univ, Ser. A. 38 (1984), 1-4) and Fank Adams' review on the paper of Dyer-Lashof "Homology of iterated loop spaces" (Amer. J. Math. 84 (1962), 35-88). Best Regards, Norio Iwase -- Tatsuji Kudo died on 7th of August at the age of 86. His wife told us that he had cancers of stomach and lung for several years. Until very recently, he visited hospital four or five days a week for the treatment. But he could not survive the hottest days of this summer. Kudo was born on 18th of November 1919 in Nagano prefecture, Japan and graduated Osaka University. Under the supervision of Atuo Komatu, Kudo published his first paper on problem of stability of complexes raised by H. Hopf and E. Pannwitz as "Contribution to the problem of stability" (Osaka Math. J. 1 (1949), 62-72). Kudo published two papers on his idea to calculate homology groups of fibre-bundles as "Homological properties of fibre bundles" (J. Inst. Polytechn. Osaka City U. Ser. A 1 (1950), 101-114) and "Homological structures of fibre bundles" (J. Inst. Polytechn. Osaka City Univ. Ser. A 2 (1952), 101-140)). His idea is calculating the subquotients of relative homology groups associated with the filtration of total space which is induced from the skeletal filtration of the base space, which can now be understood as the Leray spectral sequence and was established independently. The next important work of Kudo is a theorem known as Kudo's Transgression Theorem which is an important tool to calculate mod p cohomology of Eilenberg-Mac Lane spaces (Mem. Fac. Sci. Kyushu Univ. Ser. A 9 (1956), 79-81). With Shoro Araki, Kudo investigated homotopy property of iterated loop spaces and published two papers introducing an operad which measures higher structures of homotopy commutativity, together with homology operations which are known as Kudo-Araki operations (Proc. Japan Acad. 32 (1956), 333-335, Mem. Fac. Sci. Kyusyu Univ, Ser. A. 10 (1956), 85-120). Later by W. Browder, Kudo-Araki operations are defined more generally on homotopy commutative Hopf spaces and applied to study torsion in such spaces. On the other hand, following the pattern established by Kudo and Araki, E. Dyer and R. K. Lashof introduced a mod p homology operation for p odd, which plays now an important role in studying infinite loop spaces. From his early careers of mathematics, Kudo was interested in category theory and axiomatic homotopy theory. One goal of the study is on additive relations in abelian category (Rep. Fac. Sci. Engrg. Saga Univ. Math. 11 (1983), 9-25). Kudo also showed how to construct his spectral sequence using additive relations in a course lecture for graduate students. Kudo encouraged many young mathematicians as a professor of Kyushu University. Those who know him well might remember that he enjoyed alcohol and cigarettes quite a lot as well as mathematics in his life. But his wife told us that he did not have alcohol much and had no cigarettes in these days. We are sure he is enjoying alcohol and cigarettes a lot as well as mathematics in the heaven. Nobuhiro Ishikawa Masayoshi Kamata Mitsuyoshi Kato Yasumasa Hirashima Nobuyuki Oda Kazuyuki Fujii Norio Iwase --