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.