Abstract. Applied to a continuous surjection π: E → B of completely regular Hausdorff spaces E and B, the Stone-Čech compactification functor β yields a surjection βπ: βE → βB. For an n-fold covering map π, weshow that the fibres of βπ, while never containing more than n points, may degenerate to sets of cardinality properly dividing n. In the special case of the universal bundle π: EG → BG of a p-group G, we show more precisely that every possible type of G-orbit occurs among the fibres of βπ. To prove this, we use a weak form of the so-called generalized Sullivan conjecture. 1
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