#### by Frank Raymond

The 1950s witnessed a surge in new ways to look at mathematics. New tools and methods offered a fresh approach to old problems and set the stage for much further development. The Institute for Advanced Study and the mathematics department at Princeton University were in the thick of this activity.

Bob Williams and I were very fortunate to be in Princeton from 1958–1960. Both Bob and I were “grandsons” of R. L. Moore — Bob (Ph.D. 1954) via G. T. Whyburn, and I (1958) via R. L. Wilder. We also benefited from fantastic teachers and fellow students at Virginia and Michigan, and so we had much in common. Our wives, too, became close friends because of a mutual interest in the arts.

Professor Armand Borel organized and ran a seminar on topological transformation groups which was in the spotlight owing to contemporaneous breakthroughs. D. Montgomery, L. Zippin, C. T. Yang, E. E. Floyd, P. A. Smith, G. D. Mostow, Pierre Conner and many others had created much of the notation and methods for studying transformation groups on manifolds, a subject that had grown out of the nineteenth-century work of such luminaries as Lie, Klein, and Poincaré. With the promulgation in 1900 of Hilbert’s famous list of 23 unsolved problems, the topic, as formulated in his fifth problem, gained wider mathematical attention.

It took over 50 years and much effort by many mathematicians to put
Hilbert’s original formulation of his fifth problem into modern form. The
structure of locally compact, finite dimensional topological groups as
generalized Lie groups was completed in the 1950s by
A. Gleason,
D. Montgomery,
L. Zippin,
and
H. Yamabe.
A generalized Lie group’s
identity component __\( G_0 \)__ contains arbitrarily small normal invariant
subgroups __\( N \)__ whose quotient __\( G_0 / N \)__ is a Lie group. For actions on
manifolds, if one could show that subgroups __\( N \)__ do not occur then one
need only concentrate on Lie group actions. __\( N \)__ always contains central
totally disconnected compact subgroups isomorphic to the __\( p \)__-adic
integers, __\( A_p \)__, for some prime __\( p \)__. So, the remaining question became
known as the *Hilbert–Smith conjecture*: The __\( p \)__-adic group, __\( A_p \)__,
does not
act effectively on any __\( n \)__-dimensional manifold. It is known to be true
for manifolds of dimension 1, 2, and recently 3
[e6].
It
is also true if all orbits are locally connected or when a
transformation group acts differentiably
[e1],
or as
Lipschitz homeomorphisms
[e3],
or
quasiconformal
homeomorphisms
[e5]
or as Holder actions
[e4].

P. A. Smith had shown that if a __\( p \)__-adic group acted on an __\( n \)__-dimensional
manifold then the dimension of the orbit space was not __\( n \)__. C. T. Yang
[e2],
using his extension of Smith theory, proved that the dimension
of such an orbit space is exactly __\( n+2 \)__ or infinity and the maximum on
any __\( n \)__-dimensional space was __\( n+3 \)__ or infinity.
Bredon,
Raymond
and
Williams
using the join construction calculated the group cohomology
of the __\( p \)__-adic group and the __\( p \)__-adic solenoid
[2].
Using this, and methods of the Borel Seminar notes,
we reproved Yang’s
theorems.
In
addition, we found that solenoidal actions would raise the dimension
of the orbit space by 1 or infinity and
that the components of their fixed
point sets would be cohomology __\( (n-r) \)__ manifolds over the rational field
with __\( n-r \)__ even, analogous to an action by a circle group.

No examples of
dimension-raising orbit maps were known at that time.
Deane Montgomery suggested that an old example, of
Kolmogorov,
mapping
a 1-dimensional space onto a 2-dimensional space could be an orbit
mapping. There were these older examples of
Pontryagin,
Boltyanski
and
Kolmogorov
that showed
that
the dimension
of a product of two spaces was
not necessarily the sum of the dimensions of each of the factors. The
technique used
there was to punch holes in a nice space like a sphere
__\( K \)__ and attach a new space __\( X \)__ along their boundaries to get a new space __\( X
\Delta K \)__. If one could do this equivariantly and in a controlled fashion
one might be able to construct a __\( p \)__-adic action if not on an
__\( n \)__-manifold then at least on some reasonable space whose orbit mapping
raised the dimension by exactly 2. Bob had studied the Russian examples
and found a functorial way to make and improve the Russian examples
[3].
In
our
papers
[1],
[4]
we utilized this __\( \Delta \)__ process
to construct examples of __\( n \)__-dimensional spaces with __\( p \)__-adic actions that
raised dimension by 2. While we tried to make our examples as close to
manifolds as possible, it seemed impossible to get a manifold by this
method. Perhaps on a cohomology manifold but that has not worked
either. Nevertheless, the orbit spaces that we did construct exhibited
all the bizarre
failures of the dimension of a product of these orbit
spaces to be additive on the dimension of the factors. This reflected
exactly what must occur if these where actions of __\( A_p \)__ on __\( n \)__-manifolds.
For me this problem has never been put to rest as I am sure that Bob
cannot escape from thinking about it from time to time. Whenever we
met we exchanged our thoughts about this conjecture. Our wonderful
friendship has lasted now for 65 years and some of the glue holding it
together has been our experience working together on this problem.

*Frank Raymond is a Professor Emeritus at the University of Michigan.*