Title: Degenerate fibres in the Stone-Cech compactification
of the universal bundle of a finite group:
An application of homotopy theory to general topology
Authors: David Feldman and Alexander Wilce
Abstract:
If p: E ----->B is a continuous surjection between completely regular
spaces E and B , we may apply the Stone-Cech compactification functor
\beta to obtain a surjection \beta p: \beta E ----->\beta B . It is
well-known that if E = B x F where F is a finite set and p is
projection on the first factor, then \beta E = \beta B x \beta F ,
and \beta p is again projection on the first factor. In this paper, we
apply \beta 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 \in B .
We show that the fibres of \beta p , 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'').
What is particularly striking about this phenomenon is that it depends on a
homotopy invariant, the sectional category, of the map p . In particular,
we show that if p: E ----->B is an H-bundle where H is a finite group,
then \beta p has degenerate fibres iff p has infinite sectional category.
In the special case where G is a p-group and p:EG ----->BG is the universal
G-bundle, we can show more precisely that every possible G-orbit occurs
somewhere as a fibre of \beta p . The proof of this theorem uses a weak
form of the so-called generalized Sullivan conjecture, which is now a
theorem of H.~Miller. It is interesting to see the structure group G
manifesting itself in this way even though it is not explicitly part of the
data fed to the Stone-Cech functor.
Algebraic and general topology have grown far appart in recent years.
Accordingly, we have tried to include enough detail to make the paper
essentially self contained. Regarding the Stone-Cech compactification, we
use few facts beyond the basic definitions. Readers unfamiliar with
universal G -bundles should bear in mind the simplest non-trivial example,
G= Z/2Z . The double cover of the infinite real projective space
RP^\infty is a universal Z/2Z -bundle. No other finite group has a
universal bundle which is so easily pictured; it is this case which
motivated some of our terminology.