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I have had difficulty locating the full name and story of the mathematician Puppe whose name adorns the beloved underlying long-exact sequence algebraic topology is built on. Does anyone know who they are?

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Siegmund Dieter Puppe (16. Dezember 1930 - 13. August 2005) has already been identified by Francois Ziegler. But since the question asks also for the story which is not yet available in English language, some additional information may be useful.

He studied physics and mathematics at Göttingen and Heidelberg and became a professor in 1960 at Saarbrücken. 1968 he returned to Heidelberg and stayed there for the rest of his career until 1996. He was guest professor at Princeton 1957/58, Chicago 1961 and Minneapolis 1966/67. He gave in 1962 a talk at the International Mathematical Congress at Stockholm.

Information from the German Wikipedia. There is also a photograph.

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