A History of Manifolds and Fibre Spaces: Tortoises and Hares

During the early 1930’s topology developed some of its most important notions. The first international conference on the young subject took place in Moscow 1935. Fibre spaces were introduced by H. Seifert (1907–1996). By 1950 the notions of fibre space and fibre bundle had become central in the study of algebraic topology. In that year in Bruxelles, and in 1953 at Cornell University, international conferences on topology focused on the study of these spaces. The 1949/50 Séminaire of H. Cartan (1904– ) in Paris, an influential seminar in the spread of new ideas in topology, was dedicated to fibre spaces. In 1951, N.E. Steenrod (1910–1971) published the first textbook on the subject—this was also the first textbook in algebraic topology to give complete accounts of homotopy groups and cohomology groups. In this paper we will discuss how fibre spaces came to become basic objects in algebraic topology. In his report and problem set from the Cornell University conference, W. S. Massey (1920– ) listed five definitions of fibre space [30]: (a) fibre bundles in the American sense; (b) fibre spaces in the sense of Ehresmann and Feldbau; (c) fibre spaces as defined by the French school; (d) fibre spaces in the sense of Hurewicz and Steenrod, and (e) fibre spaces in the sense of Serre. Each of these competing definitions developed out of a mix of examples and problems of interest to the research community in topology, often marked by a national character. We will consider the origins of each of these strands and the relations among them (see also [68]). This paper is about definitions, and about ordinary developments in twentieth century mathematics. The principal hare in the development of fibre spaces is H. Whitney