Degenerate fibres in the StoneCech compactification
of the universal bundle of a finite group:
An application of homotopy theory to general topology
David Feldman
Department of Mathematics
University of New Hampshire
Alexander Wilce
Department of Mathematics
University of Pittsburgh at Johnstown
1 Introduction
If p : E ! B is a continuous surjection between completely regular spaces E
and B, we may apply the StoneCech compactification functor fi to obtain
a surjection fip : fiE ! fiB. It is wellknown that if E = B x F where F is
a finite set and p is projection on the first factor, then fiE = fiB x fiF , and
fip is again projection on the first factor. In this paper, we apply fi to an
nfold covering map, that is, a local homeomorphism p : E ! B such that
p1(b) has cardinality n for any b 2 B. We show that the fibres of fip, while
never exceeding n points, may degenerate to sets whose cardinality properly
divides n (in contrast with the more usual, explosive sort of StoneCech
"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 Hbundle where H is a finite group, then
fip has degenerate fibres iff p has infinite sectional category. In the special
case where G is a pgroup and p : EG ! BG is the universal Gbundle,
we can show more precisely that every possible Gorbit occurs somewhere as
a fibre of fip. The proof of this theorem uses a weak form of the socalled
generalized Sullivan conjecture [5, 1], 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 StoneCech functor.
1
Algebraic and general topology have grown far appart in recent years. Ac
cordingly, we have tried to include enough detail to make the paper essentially
self contained. Regarding the StoneCech compactification, we use few facts
beyond the basic definitions. Readers unfamiliar with universal Gbundles
should bear in mind the simplest nontrivial example, G = Z=2Z. The double
cover of the infinite real projective space RP1 is a universal Z=2Zbundle. No
other finite group has a universal bundle which is so easily pictured; it is th*
*is
case which motivated some of our terminology.
Acknowledgements: We are indebted to Mike Hopkins for introducing us
to the literature concerning sectional category and the Sullivan conjecture,
thereby making a crucial contribution to our work. We also wish to thank
Raoul Bott, Wis Comfort, Art Copeland and Bob Heath for helpful discus
sions.
2 Background on fi
We record here some notations and facts we use regarding StoneCech com
pactifications. For further information see [8].
A space X is completely regular if for any point x 2 X and any closed set A
X not containing x, there exists f 2 C(X) such that f(x) = 0 and f(A) = 1,
i.e., real valued functions separate points from closed sets. A completely
regular Hausdorff space is called Tychonoff space. It is not hard to see that
that a regular space in which every point has a Tychonoff neighborhood is
already a Tychonoff space. (That one needs complete regularity, as opposed
merely to Hausdorffness, is illustrated by the upper halfplane with the Half
Disc topology [7].
If X is a topological space, we write C(X) for the ring of real valued continu
ous functions on X. The zeroset of f 2 C(X) is the set Z(f) = {x : f(x) =
0}. The family of all zero sets is denoted Z[X]. The cozeroset Cz (f) of f
is X \ Z(f). Since Z(fg) = Z(f) [ Z(g) and Z(f + g) = Z(f) \ Z(g), the
family of all zerosets of X is closed under finite intersections and unions. It
is also useful to observe, for example, that {x : f(x) 0} = Z(f ^ 0).
2
A nonempty family F of zerosets of X is a zfilter if it is closed under fini*
*te
intersection, contains all zerosets containing any one of its members, and
does not contain the empty set. A maximal zfilter is called a zultrafilter.
It is customary to denote ultrafilters by small letters u, v, etc. Note that
two zultrafilters u and v differ iff there are zsets A 2 u and B 2 v with
A \ B = ;.
The collection_of all zultrafilters on X is denoted fiX. For a zeroset F
of X, set F = {u : u 2 fiX and F 2 u}. To_topologize_fiX, take as a
basis of closed sets the collection of sets F where F is closed in X. There
is a natural map i : X ! fiX assigning to x 2 X the principal zultrafilter
i(x) = {F : F 2 Z[X] and x 2 F }. When X is Tychonoff, the map i is a
homeomorphism onto its image, and we identify X with i(X). In this case,
fiX compactifies X: X is dense in fiX, and fiX is a compact Hausdorff space.
Moreover, fiX is universal among compactifications of X in the sense that
if Y compactifies X, then there is a unique continuous map from fiX to Y
which is constant on X.
More generally, if X and Y are spaces and f : X ! Y is continous, we can
extend f to a continous map fif : fiX ! fiY by defining fif(u) = {F 2
Z(Y )f1 (F ) 2 u}. It is straightforward that fi(f O g) = fif O fig.
Lemma 1 Let E and B by Tychonoff. If p : E ! B is surjective, then so
is the map fip.
Proof Every zultrafilter on B lifts to E. Since fiE is compact, the image
of fip is also compact, hence closed. The image of fip contains B which is
dense in fiB, therefore fip is surjective. 
3 StoneCech Compactification of GBundles
For the remainder of this paper, unless otherwise stated, G denotes a non
trivial finite group of order n. By a Gbundle, we always mean a covering
map p : E ! B where E is a Gspace, B is E=G and p is the canonical
surjection.
3
We shall apply the StoneCech functor to such a Gbundle. As we shall see,
G continues to act transitively on the fibres of fip : fiE ! fiB. A fibre will
therefore have fewer than n points iff it contains a zultrafilter u fixed by
some nontrivial element of G. The main results of this section (Theorems
1 and 2) give conditions under which an ultrafilter u 2 fiE is fixed by a
nontrivial subgroup of G.
Since the StoneCech functor is only wellbehaved on Tychonoff spaces, we
want both spaces E and B to be Tychonoff. In fact, in the context of
Hausdorff spaces it is only necessary to assume that B is Tychonoff:
Lemma 2 Let p : E ! B be a local homeomorphism. Suppose B is Ty
chonoff and E is Hausdorff. Then E is Tychonoff.
Proof The space E is locally Tychonoff, so it will suffice to show that E is
regular. A space is regular if every point has a neighborhood basis consisting
of closed sets. Let x be a point of E. Let U be an open neighborhood of
x such that pU is a homeomorphism. Let {Ni}i2I be a neighborhood basis
of closed sets for the point p(x) in p(U). Then {U \ p1(Ni)}i2I is clearly a
neighborhood basis of of x.
Showing that E is regular is now just a matter of showing that each set
"Ni= U \ p1(Ni) is closed in E (and not just in U.) Take y 2 cl(N"i). Since
"Ni p1(Ni), and p1(Ni) is closed, y 2 p1(Ni)), that is, p(y) 2 Ni.
Thus there is a point x 2 N"i such that p(x) = p(y). We claim x = y.
Otherwise, being Hausdorff, E contains disjoint open sets V1 and V2 contain
ing x and y respectively. Since y 2 cl(N"i), y 2 cl(V1) [ cl(N"i\ V1). But
clearly y 62 cl(V1). Also, as in the previous paragraph, y 62 cl(N"i\ V1), since
p(y) 62 p(N"i\ V1). Thus we must have y = x 2 "Ni, so N"iis closed. 
Comment To see that the hypothesis that E be Hausdorff is essential
consider the real line with the origin "doubled."
Henceforth, we suppose that B is Tychonoff.
Lemma 3 Let p : E ! B be a Gbundle. Then the fibres of the map fip are
4
orbits under the action of G. In particular, the map fip is at most ntoone.
Proof Let u; v be zultrafilters in fiE. Suppose v 6= gu for any g 2 G.
We wish to show that (fip)(v) 6= (fip)(u)." For each g 2 G, choose Cg 2 v
such that Cg 62 g1u. Then C = Cg 2 v, so p(C) 2 (fip)(v). On the
g2G
other hand, gC 62 u[for any g 2 G. Since G acts transitively on the fibres
of p, p1(p(C)) = gC. But as u is a zultrafilter not containing any of
g2G
the (finitely many) sets gC, u does not contain this last set either. Thus,
p(C) 62 (fip)(u). 
Let H be a subgroup of G.
Definition A cozeroset U E is Hantipolar if U is disjoint from hU for
some h 2 H. A zeroset F E is Hequatorial if F [ hF = E for some
h 2 H, that is, if F is the complement of an Hantipolar set.
Theorem 1 The following conditions are equivalent:
1. The zultrafilter u is fixed by all the elements of H.
2. u contains every Hequatorial set.
Proof
1 ) 2: Suppose u is fixed by H. Let F be an Hequatorial set, so F [hF = E
for some h 2 H. Then either F 2 u or hF 2 u, since u is a zultrafilter, But
since u is fixed by H, hF 2 u implies F 2 u. Hence F 2 u.
2 ) 1: Suppose that hu 6= u for some h 2 H. Hence there are disjoint zero
sets C1 2 u and C2 2 hu. Then C := C1 \ h1C2 is a zeroset in u disjoint
from hC. Let f 2 C(E) be a nonnegative function with zeroset C. Define
F = {x : f(x) f(hx)} [ {x : f(x) f(h1x)}. Then F is a zeroset, but
F \ C = ;, so F 62 u. Nevertheless F [ hF = E, so F is Hequatorial. 
Corollary 1 H fixes points of fiE if and only if the the class of Hequatorial
sets has the finite intersection property.
5
Proof If the Hequatorial sets satisfy the finite intersection property, then
they generate a zfilter which may be extended to a zultrafilter. 
Definition A cozeroset U E is Hsectional if U is disjoint from hU for
all h 2 H. A zeroset F E is Hcosectional if F [ hF = E for all h 2 H,
that is, if F is the complement of an Hsectional set.
Remark: Let pH : E ! E=H be the natural quotient map. A set U E is
Hsectional if and only if U is the image of a section of pH over a cozerosubs*
*et
of E=H.
Theorem 2 The following conditions are equivalent:
1. The zultrafilter u is fixed by some element of H.
2. u contains all Hcosectional sets.
Proof
1 ) 2: Suppose hu = u for some nontrivial h 2 H. Let F be an H
cosectional set, so F [ hF = E. Then either F 2 u or hF 2 u, since u is a
zultrafilter. But since hu = u, hF 2 u implies F 2 u. Hence F 2 u.
2 ) 1: Suppose that hu 6= u for every nontrivial h 2 H. Hence"there is
a disjoint family of zerosets Gh 2 hu; h 2 H. Then C = h1Gh is
h2H\{e}
a zeroset in u disjoint from every hC; h 2 H \ {e}. Let f 2 C(E) be a
nonnegative function[with zeroset C. Define Fh = {x : f(x) f(hx)}.
Define F = Fh. Then F is a zeroset, but F \ C = ;, so F 62 u.
h2H\{e}
Nevertheless F [ hF = E, so F is Hcosectional. 
Corollary 2 H fails to act freely on fiH if and only if the the class of H
cosectional has the finite intersection property. 
4 Fibre Degeneration and Sectional Category
A Gbundle p : E ! B is principal iff G acts freely on E. A universal
(principal) Gbundle is a numerable principal Gbundle p : EG ! BG having
6
the property that, for any numerable principal Gbundle q : E ! B there
exists a map OE : B ! BG, unique up to homotopy, such that q = OE*p. This
is equivalent to the condition that EG be contractible. (A locally trivial
bundle is numerable if it is trivialized by a covering which supports a locally
finite partition of unity; see [3]. In particular, all locally trival bundles o*
*ver
a paracompact base are numerable.)
In this section, we shall prove the following
Theorem 3 If pG : EG ! BG is a universal principal Gbundle with BG
paracompact and H is a subgroup of G, then fiEG contains points fixed by a
nontrivial subgroup of H.
Our main tool here is the notion of the sectional category of a bundle p : E !
B. This is the minimum cardinality of a covering of B by open sets over each
of which p has a section. The standard reference on sectional category is [6],
where the term used is genus.1 The term sectional category originates with
James' survey of LusternikSchnirelmann category [4].
In light of Theorem 2 and its corollary, there exist points in fiE fixed by a
nontrivial element of H G iff the collection of H sectional sets has the
finite intersection property  equivalently, iff there is no covering of E by
finitely many Hcosectional sets. Now, as remarked in section 3, U E is
Hcosectional iff the canonical quotient map pH : E ! E=H has a section
over pH (U) = U=H. Since pH : E ! E=H is an Hbundle, we conclude that
the Hsectional sets generate a nontrivial filter iff pH has infinite section*
*al
category.
We now gather some facts concerning sectional category which we will use
in the sequel.
______________________________
1Many arguments about fibre bundles hinge at some point on the availability *
*a partition
of unity (or something similar) and therefore require suitable hypotheses. Auth*
*ors deal
with this technicality in different ways. The approach of Svarc in [6] employs *
*a nonstandard
definition of open covering more restrictive than the usual one. Sectional cate*
*gory in our
sense thus only bounds sectional category in his sense from below, in general. *
*The notion
of a numerable bundle serves a similar purpose. The most common solution is jus*
*t to take
all base spaces to be paracompact.
7
If p : E ! B is a Gbundle, Gcategory of p is the minimum cardinality of
a covering of B by open sets over each of which p is trivial as a Gbundle.
Since a principal Gbundle is trivial iff it has a section, sectional category
and Gcategory coincide for principal Gbundles.
The LusternikSchnirelmann category of a space X, (LS category for short) is
the minimal cardinality of an open cover {Ui} such that each Uiis contractible
in X. The LS category of X is a homotopy invariant (see Prop. (1.1) of [4]).
Lemma 4 The Gcategory of any universal principal Gbundle p : E ! B
coincides with the LS category of B.
Proof Let U be an open set in B. If U is contractible in B, p has a section
over U.
If p has a section over U, then by the uniqueness aspect of the universal prop
erty of the bundle (pullbacks of numerable bundles always being numerable),
the inclusion map from U to B is homotopic to a constant map. That is, U
is contractible in B. 
Thus, all universal principal Gbundles have the same sectional category.
The nfold fibre join of a bundle p : E ! B is the bundle p(n): E(n)! B,
where E(n)consists of points of the form x = (t1e1; : :;:tnen). Here t1; : :;:tn
are nonnegative real numbers (which we shall refer to as weights), such that
t1 + . .+.tn = 1, ei 2 E with p(e1) = : :=:p(en) := p(n)(x), and we take tiei
to be independent of ei when ti = 0.
There is a wellknown construction due to Milnor of a universal Gbundle
(for G discrete), obtained by taking for EG the infinite fibre join G * G * . .*
* .
of G (i.e., the direct limit of the fibre joins G(n)), and for BG, the quotient
EG=G.
Fact 1 (Prop. (8.1) in [4 ]; see also Theorem 3 in [6 ])
Let B be paracompact. Then the sectional category of p : E ! B is n if
and only if the nfold fibre join E(n)admits a section.
8
For the reader's convenience we recall the short proof.
Suppose U1; :::; Un is a covering of B by open sets over each of which p has a
section oei. If {ti} is a partition of unity subordinate to {Ui}, then we obtain
a welldefined section of p(n)by setting oe(b) = (t1(b)oe1(b); : :;:tn(b)oen(b)*
*) for
b 2 B.
Conversely, if oe is a section of p(n), then we have open sets Un of B where the
nth weight is nonzero, and over each Ui a section of p given by oei = ssiO oe.

Fact 2 (Prop. 50 in [6 ]) If a discrete group G contains elements of finite
order, then the Gcategory of the bundle p : EG ! BG is infinite. 2
Proof of Theorem 3: Let p : EG ! BG be a universal principal G
bundle. Let H be a nontrivial subgroup of G, possibly G itself. Since
EG is contractible and the action of H on EG is free, the natural map
pH : EG ! EG=H is a universal principal Hbundle. Since H is finite, by
Lemma 4 and Fact 2, pH : EG ! EG=H has infinite sectional category.
Thus, as remarked above, the Hsectional sets generate a nontrivial filter.
By Corollary 2, there are points of fiEG that are fixed by some nontrivial
subgroup of H. 
Note that in particular, if H is prime cyclic, then H itself must fix some
points of fiEG. Thus for each such H, fip has degenerate fibres containing
no more than [G : H] points.
5 HFixed Points and an Auxiliary Bundle
So far we have seen that the infinite sectional category of the universal H
bundle entails the finite intersection property for the family of Hcosectional
sets which in turn guarantees the existence of points in fiEG fixed by some
subgroup of H, by Corollary 2.
______________________________
2The proof of Prop 50 in [6] contains a misprint: The reference to Theorem 1*
*0 should
be to Theorem 17.
9
We could apply Corollary 1 to prove the existence of points in fiEG fixed
by H itself if we knew that the family of Hequatorial sets had the finite
intersection property, or equivalently, that EG can't be covered by finitely
many Hantipolar sets. In this section, we show that this is in turn equivalent
to a certain auxiliary bundle's having infinite sectional category.
For a finite Hset F , we write F for the simplex generated by F . Since
F carries an action by H, we may speak of Hantipolar subsets of F . We
denote by F the subcomplex of F which is the union of all Hantipolar
faces. Given an Hbundle p : E ! B, we may perform this construction on
each fibre to obtain an Hbundle, which we denote by p : E ! B.
Lemma 5 Let p : B x F ! B be projection onto B, with B normal and
F = {v1; : :;:vn} discrete. Let V be an open set in BxF . For x 2 B, let Q(x)
denote the set p(p1(x) \ V ) F . Then there exists a map t : p(V ) ! F
such that t(x) 2 Q(x).
Proof Set Uj = {x 2 p(V ) : Q(x) j}. For x 2 U1, let t(x) be the unique
element of Q(x). Now we proceed by induction on j. Thus we suppose we
have defined t on Uj so that for every A F with A = j, if x 2 cl(Q1(A))
then t(x) 2 A. We must show that it is possible to extend t continuously
to Uj+1 so that for every A F with A = j + 1, if x 2 cl(Q1(A)) then
t(x) 2 A.
Any function continuous on the closed set Uj and on each of the finitely many
closed sets cl(Q1(A)) \ Uj (A F with A = j + 1) is continuous on all
of Uj+1. Thus it is sufficient to show that for each A F with A = j + 1,
t may be extended continuously from cl(Q1(A)) \ Uj to cl(Q1(A)) with
values in A.
For each vi 2 F \ A, the coordinate function of t associated with vi is iden
tically zero on cl(Q1(A)) \ Uj, so we simply extend by zero.
Now we apply the Tietze extension theorem to the remaining coordinate
functions, those associated to vertices in A. This yields a preliminary function
s : cl(Q1(A)) ! [0; 1]A agreeing with t on cl(Q1(A))\Uj. Finally, to extend
t to cl(Q1(A)), we compose s with a retraction from [0; 1]A onto A. 
10
Lemma 6 Let p : E ! B be an Hbundle, with H acting transitively on
the fibres and the space B normal. Let V be an Hantipolar subset of E.
Then there is a partial section t : p(V ) ! E of p : E ! B such that
t(x) 2 (p1(x) \ V ) for all x 2 p(V ).
Proof The special case where E has the form B x F for a transitive Hset
F follows immediately from Lemma 5. We now reduce the general case to
this special case. Take E0 to be another copy of E. Consider the pullback
q2
E0xB E ! E
# q1 # p :
p0
E0 ! B
We regard the map q1 : E0 xB E ! E0 as an Hbundle by letting H act
in the usual way on E, but trivially on E0. Moreover E0 xB E is a trivial
bundle since it admits the diagonal map as a section. Thus there is a par
tial section s : p01(p(V )) ! E0 xB E of p : E0 xB E ! E0 such that
s(x) 2 (q11(x) \ q12(V ))for all x 2 p01p(V ).
Letting H now act in the usual way on E0, but trivially on E, we may
view the map q1 : E0xB E ! E0 as Hequivariant. Observe that the quo
tient of the bundle q1 by this action is canonically isomorphic to the bundle
p : E ! B . Thus, giving a partial section t : p(V ) ! E amounts to giving
an Hequivariant partial section from p1(p(V )) to E0xB E.
While the partial section s : p1(p(V )) ! E0xB E may not be Hequivariant,
the fibrewise barycenter __sof its translates hsh1, h running over H, will be.
Clearly __s(x) 2 (q11(x) \ q12(V )). Letting t : p(V ) ! E be the quotient
of __sby H we get t(x) 2 (p1(x) \ V ) for all x 2 p(V ) as desired. 
Theorem 4 Let p : E ! B be an Hbundle, with H acting transitively on
the fibres and the space B normal. Then following are equivalent:
1. p : E ! B has infinite sectional category.
2. E can not be covered by finitely many Hantipolar sets.
11
Proof
1 ) 2: Suppose that E is covered by finitely many Hantipolar sets Vi. Then
the open sets Ui = p(Vi) cover B. Lemma 6 allows us to find a partial section
ti of p : E ! B over each Ui.
Comment The hypothesis of normality permits the use of the Tietze exten
sion theorem; it will play no role in the reverse implication.
2 ) 1: Suppose p : EH ! BH has finite sectional category. Then
there is a finite open cover {U1; : :;:Un} of BH and a partial section si of
p : EH ! BH over each Ui. Each si determines an Hantipolar open
subset Vi of EH as follows. If Fx is the fibre of EH over x 2 BH, Vi\ Fx
shall be the set of vertices of the smallest face containing si(x). The Vi and
all their Htranslates cover EH. 
Comment One might bypass Theorem 3 by working work with a different
auxiliary bundle, one whose fibres are finite nonHausdorff spaces with one
point representing each Hantipolar set in p1(x), the closure of that point
consisting of the points that represent the various subsets.
6 pGroups and the Sullivan Conjecture
Applied to a universal bundle p : EH ! BH, the results of the previous
section tell us that fiEH (and hence, fiEG ) has points fixed by H iff p
has infinite sectional category. Unfortunately, because of its complexity, the
bundle p : EH ! BH is difficult to work with directly; we do not yet
know if these bundles have infinite sectional category for every H. On the
other hand, it is sufficient for our purposes to show that a larger bundle has
infinite sectional category. If H is a pgroup, this proves feasible.
We construct a bundle EH ! BH as follows. For a finite Hspace F ,
let F denote the subcomplex of F consisting of all but the cell of high
est dimension. Being Hinvariant, F is itself an Hspace and F indeed
contains F as a subcomplex since the face F is certainly not antipolar.
Let EH ! BH denote the subbundle of p : EH ! BH obtained by
performing this construction on each fibre.
12
Now assume that BH is paracompact. We claim that p has infinite sectional
category. In light of Fact 1 from section 4, it is sufficient to show that the
nfold fibre join of p : EH ! BH has no section for any n.
Consider a particular fibre F of EH ! BH. Enumerate the points of F
from 1 to F . By definition, each point of the nfold join of F may be
represented by weighted ntuple of the form
(t1(s1;1; : :;:sF;1); : :;:tn(s1;n; : :;:sF;n)
satisfying
(i) tj;Psi;j2 [0; 1] for j = 1; : :;:n, i = 1; : :;:F ;
(ii) nj=1tj = 1;
P F
(iii) i=1si;j= 1 for j = 1; : :;:n;
(iv) for each j there exists an i such that si;j= 0;
with two such expressions identified precisely when they differ only at coor
dinates of weight 0.
Now to each such weighted ntuple associate the n x F  matrix ai;j where
ai;j= tjsi;j. One easily checks, first, that this matrix depends only on the
point of F (n), not on the particular representation of the point by a weighted
ntuple, and second, that the matrix determines the point. The n x F 
matrices which arise in this fashion are exactly those such that
(i) each ai;j2 [0; 1];
(ii) each row contains at least one entry equal to 0;
P n P F
(iii) j=1 i=1ai;j= 1.
Of course H acts on the set of such matrices by permuting columns. This
action can have no fixed points. Indeed, since the action of H on F is
transitive and every row contains a 0, a fixed matrix would have to vanish
identically, contrary to (iii).
Pulling (p)(n) : (EH)(n) ! BH back along p : EH ! BH gives the
bundle p xBH (p)(n): EH xBH (EH)n ! EH. Since EH is contractible,
this bundle must be trivial, hence isomorphic to p1 : EH x (H)n ! EH,
where the map is projection onto the first factor. In other words, (p)(n):
(EH)(n)! BH is isomorphic to p1=H : (EH x(H)n)=H ! EH=H. Thus
a section of (p)(n): (EH)(n)! BH is equivalent to an equivariant map
EH ! (H)(n).
13
On the other hand, an equivariant map from a point to (H)(n)is the same
thing as a fixed point. But we just saw that there are no fixed points for
the action of H on (H)(n). When H is a pgroup, it is a consequence of
Miller's version of the Sullivan conjecture that the homotopy fixed point space
of (H)(n), that is, Hom H (EH; (H)(n)) is homotopy equivalent to the fixed
point space of (H)(n)[[1], Theorem A]. Thus the homotopy fixed point space
must be empty and there can be no equivariant map EH ! (EH)(n). The
sectional category of p : EH ! BH is thus infinite and we have proved:
Theorem 5 Let EG ! BG be a universal Gbundle with BG a normal,
paracompact space. Then for each psubgroup H of G there are points in
fiEG fixed by H. 
A desire to extend this theorem to more general subgroups leads, by the
considerations above, to the following problem:
Problem 1 For what groups H, not pgroups, is the homotopy fixed point
space Hom H(EH; (H)(n)) empty for all n?
7 Remarks on More General Groups
Passing to a larger bundle is equivalent to considering the finite intersection
property for a larger family of zerosets. In particular, call a cozeroset
U EH Hincomplete if
"
hU = ; ;
h2H
that is, if U misses at least one element in each fibre over BH. Call a zeroset
Hcoincomplete if its complement is Hincomplete. We have just seen that
for pgroups H, the Hincomplete sets generate a nontrival filter. The H
incomplete filter is at least as large as the Hequatorial filter, so it is pos*
*sible
a priori that the former is trivial even when the latter is not. The following
proposition gives a sufficient condition for these two filters to coincide.
14
Proposition Assume that U is Hincomplete and misses not more than k
points from any fibre that it meets, where k2  k + 1 < n. Then U can be
covered with finitely many Hantipolar sets.
Proof Let
S = {S  S H and S > n  k} :
T
For each S 2 S, set US = h2S hU. Note that if p 2 U and S = {h  p 2 hU},
then p 2 US. Thus, the sets US cover U. We now show that each US is H
antipolar. Fix S 2 S and suppose that j 2 H is such that jS [ S 6= H. Then
there is some x 2 S with j1x 62 S, whence j = x(x1j) 2 (H \ S)(H \ S)1.
Since S > n  k, the set (H \ S)(H \ S)1 has no more than k2  k + 1 < n
elements. Thus,Sthere is some j 2 H with S [ jS = H. Since U is H
incomplete, h2HhU is empty and US is disjoint from jUS, as desired. 
Thus if maximal Hincomplete sets miss only a few points from those fibres
that they meet at all, the Hcoincomplete and the Hequatorial filters will
be equal.
Remark: Properties intermediate between Hantipolarity and Hincompleteness
may be useful if Theorem 4 is to be extended to a larger class of groups. Some
candidatesTmight be
a) T h2KhU = ; for some cyclic proper subgroup K;
b) h2KhU = ; for some proper subgroup K;
c) U meets no more than half the points in any orbit.
This last may be the most workable.
8 Loci of Degeneration
While all universal Gbundles are equivalent up to homotopy, the StoneCech
functor will be sensitive to topological differences between such bundles. We
will obtain our most precise results by concentrating on one particular model
of the universal Gbundle, that constructed by Milnor. Recall that Milnor's
EG is the infinite join G * G * G * . .;.the quotient by the natural left action
of G gives his BG.
15
Milnor's universal Gbundle has two advantages for us. First, it is metrizable.
Second, if H is a subgroup of G, Milnor's EH = H * H * H * . .s.its inside
EG as a closed subspace whose quotient is canonically of copy of Milnor's
BH sitting inside Milnor's BG as a closed subspace. Actually there will be
G=H copies of EH sitting over BH: For each left coset gH the infinite join
gH * gH * gH * . .w.ill be one such.
The degeneration of a fibre of fiEG ! fiBG is measured by the conjugacy
class of the stabilizer of any point in that fibre. When the hypotheses of
Theorem 6 below are satisfied, as they are for pgroups G, we may associate
to each conjugacy class of subgroups of G a nonempty locus in BG. We
emphasize that the action of G on EG is not part of the data available to
the StoneCech functor; rather the compactification process directly detects
the symmetry of the bundle.
Theorem 6 Let EG ! BG be Milnor's model of the universal Gbundle.
Suppose that for each subgroup H of G, there are fibres of fipH : fiEH !
fiBH consisting of a single point, at least whenever BH is normal. Then for
each such H, there are points in fiEG whose stabilizer is exactly H.
We shall need a preliminary lemma. Given a family F of zerosets in a space
X, let F0 denote the collection of zerosets that have a nonempty intersection
with every member of F. If F is a filter, then F0 is simply the union of the
zultrafiltersSextendingTF. Note that if F G, then G0 F0, and that
( iFi)0= iF0i.
Recall that a space is perfectly normal if every closed set is a zero set. Metr*
*ic
spaces (e.g., Milnor's EG) are perfectly normal.
Lemma 7 Let F be a filter on a perfectly normal space X. Then a closed
set C belongs to F00if and only if every closed neighborhood of C belongs to
F.
Proof Suppose C 2 F00. Let N be a closed neighborhood of C. Let D denote
the closure of X \ N. As C \ D = ;, D itself must be disjoint from a set E
in F. Since E N and F is a filter, N 2 F.
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Conversely, suppose that C 62 F00. Then C is disjoint from some zeroset
D that meets every element of F. Being zerosets, C and D are completely
separated, so C has a closed neighborhood N disjoint from D. In particular
N cannot be in F. (Note that the converse makes no use of the hypothesis
on X). 
Proof of Theorem 6: Suppose the theorem were false. Then there would
be a subgroup H which was not the stabilizer of any point of fiEG. Let
K1; : :;:Kn be an enumeration of the subgroups strictly larger than H. The
Hequatorial filter is nontrivial, so for every Hstable point u there is a
group Ki such that u is Kistable. That is, any ultrafilter extending the
Hequatorial filterSH extends at least one of the Kiequatorial filters Ki  in
other words, H0 iK0i. Since Ki K00iwe will obtain a contradiction by
showing show that "
Ki 6 H00:
i
As noted above, over the canonical copy of BH in BG sits a canonical copy
of EH together with its translates gkEH by elements of G (we assume g1 is
the identity). By normality, the gkEH haveTdisjoint open neighborhoods Uk.
If necessary, U = U1 may be replaced by g1kUk so that U is disjoint from
gU for any g 62 H. Since U is Kiantipolar, its complement belongs to Ki for
each i.
We claim that the complement of U is not an element of H00. By Lemma 7,
any closed neighborhood of an element of H00must be an element of H. We
will obtain our contradiction by producing an open set W , whose closure is
contained in U, such that W cannot be covered by finitely many Hantipolar
sets. Let W be the open neighborhood of EH obtained by using normality
to separate EH from the complement of U. By hypothesis, there are fibres
of any universal Hbundle consisting of a single point. Therefore the H
equatorial filter is not trivial, as it would have to be if W were covered by
finitely many Hantipolar sets. 
17
References
[1]E. Dror Farjoun and A. Zabrodsky, Fixed points and homotopy fixed
points, Comment. Math. Helvetici 63 (1988), 286295
[2]S. Eilenberg and T. Ganea, On the LusternikSchnirelmann category of
abstract groups, Ann. Math. 65 (1957), pp. 517518
[3]C. Husemoller, Fibre Bundles 3d edition, SpringerVerlag, 1993
[4]I. M. James, On category, in the sense of LusternikSchnirelmann, Topol
ogy 17 (1978), pp.331348
[5]H. Miller, The Sullivan conjecture on maps from classifying spaces. An
nals of Math. 120 (1984), 3987. Erratum: Annals of Math. 121 (1985),
605609.
[6]A.S. Svarc, The genus of a fibre space, Am. Math. Soc. Transl. 55 (1966),
pp. 49140
[7]L. Steen and J. A. Seabach, CounterExamples in Topology, Holt, Rein
heart & Winston, 1970
[8]Russell C. Walker, The StoneCech Compactification, SpringerVerlag,
1974.
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