Degenerate fibres in the Stone - Čech compactification of the universal bundle of a finite group : An application of homotopy theory to general topology

If p : E → B is a continuous surjection between completely regular spaces E and B, we may apply the Stone-Čech compactification functor β to obtain a surjection βp : βE → βB. It is well-known that if E = B × F where F is a finite set and p is projection on the first factor, then βE = βB × βF , and βp is again projection on the first factor. In this paper, we apply β to an n-fold covering map, that is, a local homeomorphism p : E → B such that p−1(b) has cardinality n for any b ∈ B. We show that the fibres of βp, while never exceeding n points, may degenerate to sets whose cardinality properly divides n (in contrast with the more usual, explosive sort of Stone-Cech “pathology”).