Celebratio Mathematica — Williams

R. F. Williams’ work on the Hilbert–Smith conjecture

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 $X Δ K$ . 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 $Δ$ 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.

References

[e1] D. Montgomery: “Topological groups of differentiable transformations,” Ann. of Math. (2) 46 (1945), pp. 382–387. MR 13162 Zbl 0063.04074 article

[e2] C.-T. Yang: “ $p$ -adic transformation groups,” Michigan Math. J. 7 (1960), pp. 201–218. MR 120310 Zbl 0094.17502 article

[e3] D. Repovš and E. Ščepin: “A proof of the Hilbert–Smith conjecture for actions by Lipschitz maps,” Math. Ann. 308 : 2 (1997), pp. 361–364. MR 1464908 Zbl 0879.57025 article

[e4] I. Maleshich: “The Hilbert-Smith conjecture for Hölder actions,” Uspekhi Mat. Nauk 52 : 2(314) (1997), pp. 173–174. In Russian; translated in Russian Mathematical Surveys (1997), 52:2, 407–408. MR 1480156 Zbl 0988.57021 article

[e5] G. J. Martin: “The Hilbert–Smith conjecture for quasiconformal actions,” Electron. Res. Announc. Amer. Math. Soc. 5 (1999), pp. 66–70. MR 1694197 Zbl 0979.30013 article

[e6] J. Pardon: “The Hilbert–Smith conjecture for three-manifolds,” J. Amer. Math. Soc. 26 : 3 (2013), pp. 879–899. MR 3037790 Zbl 1273.57024 article

Works

[1] F. Raymond and R. F. Williams: “Examples of $p$ -adic transformation groups,” Bull. Am. Math. Soc. 66 : 5 (1960), pp. 392–394. Full descriptions of these examples are given in an article published in Ann. Math. 78:1 (1963). MR 123634 Zbl 0096.17202 article

Our purpose here is to outline the construction of an $ n $-dimensional space $ X^n $, $ n\geq 2 $, upon which the $ p $-adic group $ A_p $ acts so that the orbit space $ X^n/A_p $ is of dimension $ n+2 $. Though such examples are new, it had been known [Williams et al. 1961; Yang 1960], that either they do exist or a certain long standing conjecture on transformation groups must be true. The conjecture states that every compact effective group acting on a (generalized) manifold must be a Lie group; it may well be false.

Another question concerns the amount, $ k $, by which the (cohomology) dimension of a compact space can be raised under the decomposition map \[ X \to X/A_p .\] By [Williams et al. 1961; Yang 1960], $ k\leq 3 $. (An example in which $ k = 1 $ is essentially due to Kolmogoroff [1937].) No example is known for which $ k = 3 $. The authors expect to have more to say on this subject.

[2] G. E. Bredon, F. Raymond, and R. F. Williams: “ $p$ -adic groups of transformations,” Trans. Am. Math. Soc. 99 : 3 (1961), pp. 488–498. MR 142682 Zbl 0109.15901 article

There is the important conjecture that every compact group $ G $ of transformations on a manifold $ M $ is a Lie group. As is well known, if $ G $ is not a Lie group then $ G $ must contain a $ p $-adic group which in turn acts effectively on $ M $.

Recently, Yang has shown [1960] that if a $ p $-adic group does act on $ M $, then the orbit map raises the integral cohomology dimension by 2. To obtain this result Yang extends the Smith special homology theory to maps of prime power period, using the reals mod 1 as coefficients.

In this paper the authors compute the cohomology of the universal classifying space $ B_G $, for $ G = A_P $, the $ p $-adic group, and $ G = \Sigma_p $, the $ p $-adic solenoid. These results are then used with methods fitting the general scheme of [Borel 1960] to prove the dimension theorems of Yang.

We conclude the paper with fixed point theorems for $ p $-adic solenoids

Frank Albert Raymond	Related
Glen Eugene Bredon	Related

[3] R. F. Williams: “A useful functor and three famous examples in topology,” Trans. Am. Math. Soc. 106 : 2 (1963), pp. 319–329. MR 146832 Zbl 0113.37803 article

The purpose of this note is to describe a functor which provides a framework for certain constructions in topology. It is related to the sets $ (E,\pi,B,X,q) $ described in [1960] and is particularly adapted to discussing the limit of repeated modifications of triangulable spaces. Roughly speaking, one forms a space $ X\Delta K $ by replacing each top dimensional simplex of a complex $ K $ with a copy of a space $ X $. If in addition there are mappings on the spaces $ X $, $ K $, these induce a mapping on the new space $ X\Delta K $.

It has been called to my attention that several authors have considered analogous functors (though not as far as I know, in written form). This is not surprising inasmuch as $ X \Delta K $ is defined just as the Whitney sum of two bundles.

Though the principal applications of this functor are to be found elsewhere, in a paper by Frank Raymond and the author [1960, 1963] and a forthcoming paper by the author, three famous examples [Pontrjagin 1930; Boltyanskii 1951; Kolmogoroff 1937] are given as applications in the last section. Two of these are in dimension theory proper, but the third is essentially about transformation groups.

It is hoped that the reader will find our description of Boltyanskii’s example easier than the original, as a simpler, more homogeneous version is given. In addition, in our version of Kolmogoroff’s example, the group acts without fixed points. This answers a question raised by Anderson [1957].

[4] F. Raymond and R. F. Williams: “Examples of $p$ -adic transformation groups,” Ann. Math. (2) 78 : 1 (July 1963), pp. 92–106. These examples originated in an article published in Bull. Am. Math. Soc. 66:5 (1960). MR 150769 Zbl 0178.26003 article