The mathematical work of Everett Pitcher A talk given by Don Davis in July 2002, on the occasion of Pitcher's 90th birthday If you look at a History of Topology, such as the 1989 book by Dieudonne or the 1999 volume edited by Ioan James, you will find two significant citations to Everett's work. One is to a 1947 paper with J.L. Kelley in the Annals of Math entitled "Exact homomorphism sequences in homology," and the other is to a talk at the 1950 International Congress of Mathematicians entitled "Homotopy groups of the space of curves, with applications to spheres." I will focus on these two contributions, and then talk briefly at the end about his work during the past 24 years on the Poincare Conjecture. The paper with Kelley is the first place in which the notion of exact sequence appeared in print. Anyone who has any exposure to algebraic topology knows that exact sequences are absolutely fundamental to the subject. Briefly, a sequence of homomorphisms between groups is exact if at each position, the image of the homomorphism coming in equals the kernel of the homomorphism going out. The paper with Kelley is 28 pages long. It states and proves many of the properties of exact sequences that appear in a first course in algebraic topology. For example, they prove that the direct limit of a direct system of exact sequences is exact, and the fundamental result that a short exact sequence of chain complexes yields a long exact sequence of homology groups. Two weeks ago I was at a conference and mentioned to a friend who teaches at Ohio State University about the talk that I am giving now. He said that he always refers to this fundamental result as the Kelley-Pitcher theorem in his courses. According to Everett, here is how the paper came about. In 1939, Everett had written a short paper, "Identification of two subsets," in which he discussed the homology groups of a space obtained by identifying corresponding points of disjoint homeomorphic subspaces of a topological space. This is a generalization of the Mayer-Vietoris sequence, which at that time was not thought of as an exact sequence, but rather some complicated isomorphisms. Incidentally, Vietoris died this April at the age of 110, almost 111, as the oldest person in Austria. So, Everett, I think you have another 20 years to polish up your proof of the Poincare Conjecture. Everett and Kelley became close friends during World War II, during which both were doing ballistics research at Aberdeen Proving Ground. Kelley was aware of ongoing work of Eilenberg and Steenrod, published in 1945, in which they gave an axiomatic approach to homology theory, which is still very important. In their 1945 paper, Eilenberg and Steenrod did not use the word "exact." They just said that the kernels equaled the images. Kelley suggested to Everett that this point of view might simplify some of the things he had done in his 1939 paper. After the war, Everett and Kelley were both in Princeton for awhile, and developed these ideas into their influential Annals paper. Kelley had learned that Eilenberg and Steenrod planned to use the word "exact," so he and Everett incorporated this word, and developed its properties very nicely. Eilenberg and Steenrod were delayed in getting the word into print because they were working on a book "Foundations of Algebraic Topology," which was not published until 1952. Everett's PhD at Harvard was directed by Marston Morse in 1935. What is now called Morse Theory is a hybrid of analysis, topology, and geometry. One of its main concerns is the critical points of a real-valued function f on a smooth manifold M. These are the points where the first partial derivatives are zero. In a 1925 paper Morse had proved results which would later be interpreted to say that M can be built from cells which correspond to the critical points, with the dimension of each cell being the index of the critical point (number of negative eigenvalues of the Hessian matrix of second partials). He had also considered a similar situation where X=Omega S is the space of loops on a Riemannian manifold S and f(omega) is the length of omega. Then the critical points of f just correspond to geodesics on S. If S = S^n, then the geodesics are just great circles wrapping around the sphere any number of times, from which follows quite easily that Omega S^n can be built with one cell of dimension each multiple of n-1. This direction was not one of Morse's principal interests, and so it lay buried until Everett brought it to attention in 1950. At that time, algebraic topologists were learning how to compute some of the homotopy groups of spheres. In his talk at the ICM, Everett showed that Morse's result could be used to prove that pi_5(S^3) was cyclic of order 2. The proof begins by noting pi_5(S^3) = pi_4(Omega S^3) = pi_4(S^2 u e^4 u e^6 u ...). Using the known fact that pi_4(S^2)=Z/2 and the way in which the 4-cell was attached to the 2-cell, Everett made his deduction. Also in 1950 George Whitehead and Lev Pontryagin had proved that pi_5(S^3) = Z/2, by different methods. The significance of Everett's talk here was the bringing to public attention this approach to homotopy theory. Dieudonne's History credits Everett and Rene Thom with doing this. Several years later, Raoul Bott used related methods to obtain his celebrated Bott Periodicity Theorem about the homotopy groups of the unitary groups. In his obituary of Morse, Bott writes that "In the 50's Thom, Pitcher, and I used to formulate" Morse's theorem in purely homotopy theoretic terms. By 1960, Smale had built upon these methods to prove the higher-dimensional Poincare Conjecture. In 1954 Ioan James gave a combinatorial model for the loop space on any space, which turned out to be very important in homotopy theory. He credits Everett's 1950 ICM lecture, subsequent correspondence, and some lectures that Everett gave in Oxford in 1952, while on sabbatical from Lehigh, with being influential on his obtaining this result. During the 24 years since his retirement, Everett has been working on a proof of the classical 3-dimensional Poincare Conjecture, which states that a simply-connected 3-manifold must be homeomorphic to S^3. This is a very famous problem, and there is a million dollar prize awaiting the person who produces an accepted proof of it. This problem in geometric topology is very subtle. During the past 15 years there have been several well-publicized announcements of proofs which were found to be lacking. Since 1978, Everett has been refining a manuscript entitled "The Poincare Conjecture is True." He uses methods of classical Morse Theory with some new twists in his argument. I once found a small mistake in an early version of his proof, and he patched his argument to correct this gap. He has been extremely careful in refining and clarifying his argument, and the paper is now being considered by an editor and a referee. It would be wonderful if they would accept Everett's proof as valid. But even if that never happens, Everett has made his mark on topology via the two papers on which I focused. I wish you many more years of health and fruitful mathematics, Everett.