Celebratio Mathematica

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.

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

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].

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

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Robert Fones Williams

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

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