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.
