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
