Celebratio Mathematica — Williams

An Autobiography

Works connected to Frank Albert Raymond

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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

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