Contemporary Mathematics Volume 58, Part I, 1986 A PAGE OF MATHEMATICAL AUTOBIOGRAPHY BY SOLOMON LEFSCHETZ INTRODUCTION As my natural taste has always been to look forward rather than backward this is a task which I did not care to undertake. Now, however, I feel most grateful to my friend Mauricio Peixoto for having coaxed me into accepting it. For it has provided me with my first opportunity to cast an objective glance at my early mathemati- cal work, my algebro-geometric phase. As I see it at last it was my -lot to plant the harpoon of algebraic topology into the body of the whale of algebraic geometry. But I must not push the metaphor too far. The time which I mean to cover runs from 1911 to 1924, from my doctorate to my research on fixed points. At the time I was on the faculties of the Universities of Nebraska (two years) and Kansas (eleven years). As was the case for almost all our scientists of that day my mathematical isolation was complete. This circumstance was most valuable in that it enabled me to develop my ideas in com- plete mathematical calm. Thus I made use most uncritically of early topology a la Poincar~, and even of my own later developments. Fortunately someone at the Acad~mie des Sciences (I always sus- pected :£mile Picard) seems to have discerned "the harpoon for the whale" with pleasant enough consequences for me. To close personal recollections, let me tell you what made me turn with all possible vigor to topology. From the Po formula of Picard, applied to a hyperelliptic surface 4} (topologically the product of 4 circles) I had come to believe that the second Betti number R2(4}) = 5, whereas clearly Rt(4}) = 6. What was wrong? After considerable time it dawned upon me that Picard only dealt with finite 2-cycles, the only useful cycles for calculating periods of certain double integrals. Missing link? The cycle at infinity, that is the plane section of the surface at infinity. This drew my attention to cycles carried by an algebraic curve, that is to algebraic cycles, and · · · the harpoon was in! An address delivered at Brown University on April 14, 1967. Submitted by invitation of the editors received by the editors September 7, 1967. Reproduced with permission from Bulletin of the American Mathematical Society 74 (1968), 854-879, copyright © American Mathematical Society 1968.
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