The Hare and the Tortoise

J.P. MAY It is a pleasure to be here to help celebrate Mike Boardman’s 60th birthday. I have just finished writing a history of stable algebraic topology from the end of World War II through 1966 [18]. The starting point was natural enough. The paper of Eilenberg and Mac Lane [6] that introduced the categorical language we now all speak appeared in 1945, and so did the paper of Eilenberg and Steenrod [7] that announced the axiomatic treatment of homology and cohomology. The ending point was more artificial, at first dictated by constraints of time and energy and the fact that Steenrod’s compendium of Mathematical Reviews in topology contained all reviews published through 1967 and thus all papers published through 1966. It also made it easy for me to be modest and impersonal. Although I got my PhD in 1964, I only plugged into the circuit and began to know what was going on when I arrived at Chicago, at the end of 1966. Mike also got his PhD in 1964. Since he is two years older than I am, I guess he was a little slow. But then, his thesis was a lot more important than mine was, although people at the time didn’t seem to understand that. Its results became available in a mimeographed summary in 1966. So maybe 1966 wasn’t such a bad stopping point mathematically. It is amazing how much we didn’t know then, how many familiar names had not yet made their mark. In fact, a complete list of the people who made sustained and important contributions to the development of stable algebraic topology in the years 1945 through 1966 would have no more than around 40 names on it. On the other hand, the caliber of the people working in the field was extraordinary. The towering figures were Adams, Atiyah, Borel, Bott, Cartan, Eilenberg, Hirzebruch, Mac Lane, Milnor, Serre, Steenrod, Thom, and J.H.C. Whitehead. Others who made major contributions were Adem, Araki, Barratt, Brown, Conner, Dold, Dyer, Eckmann, Floyd, Heller, Hilton, James, Kan, Kudo, Lashof, Liulevicius, Moore, Peterson, Pontryagin, Postnikov, Puppe, Spanier, Stong, Swan, Toda, Wall, G.W. Whitehead, and Wu. Several people attempted to build good stable homotopy categories. In 1953, Spanier and J.H.C. Whitehead [22] gave a start by introducing the “S-category”, which we now understand to be the full subcategory of finite CW complexes in the stable homotopy category. In 1962, G.W. Whitehead [24] gave the definitive category of Ω-spectra equivalent to the category of cohomology theories on spaces. It is definitely not triangulated, however, and therefore cannot be the stable homotopy category as we know it today. Adams [1] and Puppe [19, 20] gave constructions with roughly the same starting point in the early 1960’s. They consider CW prespectra, which are sequences of CW complexes Tn and inclusions of subcomplexes ΣTn → Tn+1, subject to certain