Andrew Casson has had an enormous influence on geometric topology,
making fundamental contributions to the theory of ndimensional
manifolds over the course of his career for steadily decreasing
values of n.
In the period 196670 he was one of the leading actors in the dramatic
developments that were taking place in higherdimensional manifold
topology (n>=5). In 1967 he proved the first general positive result
on the Hauptvermutung, simultaneously with Dennis Sullivan. Around the
same time he gave the classification of fake tori (this was also done,
independently, by HsiangShaneson and by Wall), which was central to
the KirbySiebenmann triangulation program. In the early 1970's he
turned his attention to dimension 4. Here, he introduced his famous
Casson handles and used them to prove the existence of exotic smooth
noncompact 4manifolds. These Casson handles formed the basis of
Mike Freedman's later work on topological 4manifolds, in particular
his proof of the (topological) 4dimensional Poincare Conjecture. In
197475 Casson and Gordon showed that the classical knot concordance
group is larger than its higherdimensional analogs. In 1984
Casson defined what is now called the Casson invariant of a homology
3sphere, and used it to show that the Rohlin invariant of a homotopy
3sphere is zero. In the early 1990's, he made another important
contribution to 3dimensional topology by proving, with Jungreis, that
an irreducible 3manifold whose fundamental group contains an infinite
cyclic normal subgroup is Seifert fibered. (This was also proved by
Gabai.) Casson's introduction (with Gordon) of strongly irreducible
Heegaard splittings, and his work on normal surfaces in 3manifolds,
have also been very influential in 3dimensional topology.
Casson's work has been remarkable for its breadth as well as
its depth, and he is equally at home with geometric and algebraic
arguments. Also, no account of his career would be complete without
mentioning his skill as an expositor. His lectures are legendary for
their clarity; indeed it is wellestablished principle in topology
that the best way to really understand something is to persuade
Andrew to give a graduate course on it.
Andrew Casson spent the early part of his career at Trinity College,
Cambridge, first as an undergraduate (he received his BA in 1965),
then as a Research Fellow (196771), and Lecturer (197181). He was
Professor of Mathematics at the University of Texas at Austin from
198186, at the University of California at Berkeley from 198699, and
has been at Yale University since 2000. He was awarded the Veblen
Prize in Geometry by the American Mathematical Society in 1991,
and was elected to a Fellowship of the Royal Society in 1998.
The present volume arose out of two events that took place in 2003:
the 28th University of Arkansas Spring Lecture Series, April 1012,
at which Andrew was the principal lecturer, and the Conference on the
Topology of Manifolds of Dimensions 3 and 4, which was held at the
University of Texas at Austin from May 1921, in honor of Andrew's
60th birthday. The first was attended by over 120 participants, the
second by over 150, showing clearly the high esteem in which Andrew
is held by the topological community. We are pleased and honored to
dedicate these proceedings to him.
One final note. Andrew's PhD advisor was Terry Wall, although the
reader will observe that there is no mention above of Andrew actually
getting a PhD: in characteristic style, what might have been Andrew's
PhD thesis was never submitted as such, but instead became his Trinity
Fellowship dissertation. However, this omission was rectified at
the Austin conference, where Andrew Ranicki presented Andrew with a
"PhD" certificate, signed by all the participants.
Cameron Gordon and Yo'av Rieck
Austin and Fayetteville, 2005
