Degenerate fibres in the Stone-Cech compactification of the universal bundle of a finite group

Abstract:

Applied to a continuous surjection $\pi : E \rightarrow B$ of completely regular Hausdorff spaces $E$ and $B$, the Stone-Čech compactification functor $\beta$ yields a surjection $\beta \pi : \beta E \rightarrow \beta B$. For an $n$-fold covering map $\pi$, we show that the fibres of $\beta \pi$, while never containing more than $n$ points, may degenerate to sets of cardinality properly dividing $n$. In the special case of the universal bundle $\pi :EG \rightarrow BG$ of a $p$-group $G$, we show more precisely that every possible type of $G$-orbit occurs among the fibres of $\beta \pi$. To prove this, we use a weak form of the so-called generalized Sullivan conjecture.