Norman Steenrod

Norman Steenrod (1910-1971) was one of the leading topologists of the twentieth century. Together with Samuel Eilenberg, he axiomatized homology theory and established the modern framework of algebraic topology, a subject invented by Poincaré. The 1952 text Foundations of Algebraic Topology, known as "Eilenberg-Steenrod" to generations of topologists, codified the language and philosophy of twentieth-century topology. Steenrod also developed the theory of fiber bundles, and introduced the "Steenrod algebra" of cohomology operations.

Steenrod was born and raised in Dayton, Ohio. Both his parents were teachers. He started his college education at Miami University (in Oxford, Ohio) in 1927, but was severely hampered by lack of funds and several times had to leave school to earn money. He transferred to the University of Michigan and graduated in 1932. It was at Michigan that he met R. L. Wilder, who encouraged his interest in topology and gave him his first research problem. By 1933 he had finished his first paper, which helped him obtain a fellowship at Harvard. After receiving a master's degree from Harvard in 1934, he enrolled at Princeton (on Wilder's advice); he completed his doctorate there in 1936 under the supervision of Lefschetz. He held positions at the University of Chicago and at Michigan before returning to Princeton in 1947, where he was promoted to full professor in 1952.

The basic ideas of homology theory go back to Poincaré, but it took the subject a long time to reach its modern form. In the first half of the twentieth century, it gradually became clear that the best way to express the homological invariants of a space (such as its Betti numbers and Euler characteristic) was through the definition of homology groups, which are a series of commutative groups, one for each nonnegative dimension, which may be over any (commutative) ring of coefficients. The other difficulty was that homology was originally only understood for very restricted types of spaces (such as simplicial complexes); a theory that would work for general spaces was a natural objective. By the early 1940's, many homology theories had been defined, including those of Alexander and Veblen, Lefschetz, Vietoris, and Cech; it was unclear how these theories related to each other, and the subject was in a chaotic state. Eilenberg and Steenrod set down seven axioms and showed that all homology theories that satisfied them gave the same results, at least for compact spaces.

It turns out that the seventh Eilenberg-Steenrod axiom--the "dimension axiom", which asserts that the positive-dimensional homology groups of a one-point space are trivial--can be discarded without losing most of the properties of homology. Eilenberg and Steenrod probably included it because it was a property shared by all the homology theories then known. But in the 1950's and 1960's, many interesting examples of "extraordinary theories" that satisfy all the axioms except the dimension axiom were developed. These include the stable homotopy and cohomotopy theories of Spanier and G. W. Whitehead; the K-theories of Atiyah and Hirzebruch; and the bordism theories of Conner and Floyd.

Steenrod's introduction of cohomology operations provided additional structure on the cohomology groups of a space. Thanks to Lefschetz and others, the cup-product structure of cohomology was understood by the early 1940's. Steenrod was able to define operations from one cohomology group to another that generalized the cup-square. The additional structure made cohomology a finer invariant, and allowed Steenrod and J. H. C. Whitehead to obtain new results on the problem of counting the number of linearly independent vector fields on a sphere.

The Steenrod cohomology operations form a (noncommutative) algebra under composition, which is the well-known Steenrod algebra. Adem studied the relations between the Steenrod operations (the "Adem relations") and discovered secondary cohomology operations. Using these secondary operations, J. F. Adams obtained the definitive answer to the vector-fields-on-spheres problem (though a much easier proof using K-theory emerged later).

Steenrod played a crucial role in the development of the theory of fiber bundles, which can be thought of as generalized Cartesian products. His 1951 book The Topology of Fiber Bundles not only gave a clear exposition of that subject, but also of homotopy groups and obstruction theory, which until then had no treatment in book form. The spectacular development of spectral sequences in the 1950's made the book's results somewhat dated within a decade, but on many topics it remains a useful reference.

Steenrod also did an important service to the discipline of algebraic topology by compiling, classifying, and cross-referencing a collection of reviews of papers in topology. The "Steenrod volumes" became an essential reference on their appearance in 1968, and topologists of subsequent generations have bemoaned the lack of a suitable successor to this work.

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