Degenerate fibres in the Stone-Cech 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 Stone-Cech compactification functor fi to obtain a surjection fip : fiE ! fiB. 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 fiE = fiB x fiF , and fip is again projection on the first factor. In this paper, we apply fi 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 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 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 fip 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 fip. The proof of this theorem uses a weak form of the so-called 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 Stone-Cech 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 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 RP1 is a universal Z=2Z-bundle. 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 Stone-Cech 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 Hausdorff-ness, is illustrated by the upper half-plane 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 zero-set 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 cozero-set 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 zero-sets 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 non-empty family F of zero-sets of X is a z-filter if it is closed under fini* *te intersection, contains all zero-sets containing any one of its members, and does not contain the empty set. A maximal z-filter is called a z-ultrafilter. It is customary to denote ultrafilters by small letters u, v, etc. Note that two z-ultrafilters u and v differ iff there are z-sets A 2 u and B 2 v with A \ B = ;. The collection_of all z-ultrafilters on X is denoted fiX. For a zero-set 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 z-ultrafilter 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 )|f-1 (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 z-ultrafilter 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 Stone-Cech Compactification of G-Bundles For the remainder of this paper, unless otherwise stated, G denotes a non- trivial finite group of order n. By a G-bundle, we always mean a covering map p : E ! B where E is a G-space, B is E=G and p is the canonical surjection. 3 We shall apply the Stone-Cech functor to such a G-bundle. 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 z-ultrafilter u fixed by some non-trivial 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 non-trivial subgroup of G. Since the Stone-Cech functor is only well-behaved 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 p|U is a homeomorphism. Let {Ni}i2I be a neighborhood basis of closed sets for the point p(x) in p(U). Then {U \ p-1(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 \ p-1(Ni) is closed in E (and not just in U.) Take y 2 cl(N"i). Since "Ni p-1(Ni), and p-1(Ni) is closed, y 2 p-1(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 G-bundle. Then the fibres of the map fip are 4 orbits under the action of G. In particular, the map fip is at most n-to-one. Proof Let u; v be z-ultrafilters 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 g-1u. 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, p-1(p(C)) = gC. But as u is a z-ultrafilter 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 cozero-set U E is H-antipolar if U is disjoint from hU for some h 2 H. A zero-set F E is H-equatorial if F [ hF = E for some h 2 H, that is, if F is the complement of an H-antipolar set. Theorem 1 The following conditions are equivalent: 1. The z-ultrafilter u is fixed by all the elements of H. 2. u contains every H-equatorial set. Proof 1 ) 2: Suppose u is fixed by H. Let F be an H-equatorial set, so F [hF = E for some h 2 H. Then either F 2 u or hF 2 u, since u is a z-ultrafilter, 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 \ h-1C2 is a zero-set in u disjoint from hC. Let f 2 C(E) be a non-negative function with zero-set C. Define F = {x : f(x) f(hx)} [ {x : f(x) f(h-1x)}. Then F is a zero-set, but F \ C = ;, so F 62 u. Nevertheless F [ hF = E, so F is H-equatorial. | Corollary 1 H fixes points of fiE if and only if the the class of H-equatorial sets has the finite intersection property. 5 Proof If the H-equatorial sets satisfy the finite intersection property, then they generate a z-filter which may be extended to a z-ultrafilter. | Definition A cozero-set U E is H-sectional if U is disjoint from hU for all h 2 H. A zero-set F E is H-cosectional if F [ hF = E for all h 2 H, that is, if F is the complement of an H-sectional set. Remark: Let pH : E ! E=H be the natural quotient map. A set U E is H-sectional if and only if U is the image of a section of pH over a cozero-subs* *et of E=H. Theorem 2 The following conditions are equivalent: 1. The z-ultrafilter u is fixed by some element of H. 2. u contains all H-cosectional sets. Proof 1 ) 2: Suppose hu = u for some non-trivial 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 z-ultrafilter. But since hu = u, hF 2 u implies F 2 u. Hence F 2 u. 2 ) 1: Suppose that hu 6= u for every non-trivial h 2 H. Hence"there is a disjoint family of zero-sets Gh 2 hu; h 2 H. Then C = h-1Gh is h2H\{e} a zero-set in u disjoint from every hC; h 2 H \ {e}. Let f 2 C(E) be a non-negative function[with zero-set C. Define Fh = {x : f(x) f(hx)}. Define F = Fh. Then F is a zero-set, but F \ C = ;, so F 62 u. h2H\{e} Nevertheless F [ hF = E, so F is H-cosectional. | 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 G-bundle p : E ! B is principal iff G acts freely on E. A universal (principal) G-bundle is a numerable principal G-bundle p : EG ! BG having 6 the property that, for any numerable principal G-bundle 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 G-bundle with BG paracompact and H is a subgroup of G, then fiEG contains points fixed by a non-trivial 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 Lusternik-Schnirelmann category [4]. In light of Theorem 2 and its corollary, there exist points in fiE fixed by a non-trivial 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 H-cosectional sets. Now, as remarked in section 3, U E is H-cosectional iff the canonical quotient map pH : E ! E=H has a section over pH (U) = U=H. Since pH : E ! E=H is an H-bundle, we conclude that the H-sectional sets generate a non-trivial 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 G-bundle, G-category of p is the minimum cardinality of a covering of B by open sets over each of which p is trivial as a G-bundle. Since a principal G-bundle is trivial iff it has a section, sectional category and G-category coincide for principal G-bundles. The Lusternik-Schnirelmann category of a space X, (L-S category for short) is the minimal cardinality of an open cover {Ui} such that each Uiis contractible in X. The L-S category of X is a homotopy invariant (see Prop. (1.1) of [4]). Lemma 4 The G-category of any universal principal G-bundle p : E ! B coincides with the L-S 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 G-bundles have the same sectional category. The n-fold 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 non-negative 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 well-known construction due to Milnor of a universal G-bundle (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 n-fold 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 well-defined 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 non-zero, 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 G-category 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 non-trivial 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 H-bundle. Since H is finite, by Lemma 4 and Fact 2, pH : EG ! EG=H has infinite sectional category. Thus, as remarked above, the H-sectional sets generate a non-trivial filter. By Corollary 2, there are points of fiEG that are fixed by some non-trivial 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 H-Fixed 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 H-cosectional 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 H-equatorial sets had the finite intersection property, or equivalently, that EG can't be covered by finitely many H-antipolar 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 H-set F , we write F for the simplex generated by F . Since F carries an action by H, we may speak of H-antipolar subsets of F . We denote by F the subcomplex of F which is the union of all H-antipolar faces. Given an H-bundle p : E ! B, we may perform this construction on each fibre to obtain an H-bundle, 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(p-1(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(Q-1(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(Q-1(A)) then t(x) 2 A. Any function continuous on the closed set Uj and on each of the finitely many closed sets cl(Q-1(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(Q-1(A)) \ Uj to cl(Q-1(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(Q-1(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(Q-1(A)) ! [0; 1]A agreeing with t on cl(Q-1(A))\Uj. Finally, to extend t to cl(Q-1(A)), we compose s with a retraction from [0; 1]A onto A. | 10 Lemma 6 Let p : E ! B be an H-bundle, with H acting transitively on the fibres and the space B normal. Let V be an H-antipolar subset of E. Then there is a partial section t : p(V ) ! E of p : E ! B such that t(x) 2 (p-1(x) \ V ) for all x 2 p(V ). Proof The special case where E has the form B x F for a transitive H-set 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 H-bundle 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 : p0-1(p(V )) ! E0 xB E of p : E0 xB E ! E0 such that s(x) 2 (q-11(x) \ q-12(V ))for all x 2 p0-1p(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 H-equivariant. 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 H-equivariant partial section from p-1(p(V )) to E0xB E. While the partial section s : p-1(p(V )) ! E0xB E may not be H-equivariant, the fibrewise barycenter __sof its translates hsh-1, h running over H, will be. Clearly __s(x) 2 (q-11(x) \ q-12(V )). Letting t : p(V ) ! E be the quotient of __sby H we get t(x) 2 (p-1(x) \ V ) for all x 2 p(V ) as desired. | Theorem 4 Let p : E ! B be an H-bundle, 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 H-antipolar sets. 11 Proof 1 ) 2: Suppose that E is covered by finitely many H-antipolar 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 H-antipolar 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 H-translates cover EH. | Comment One might bypass Theorem 3 by working work with a different auxiliary bundle, one whose fibres are finite non-Hausdorff spaces with one point representing each H-antipolar set in p-1(x), the closure of that point consisting of the points that represent the various subsets. 6 p-Groups 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 p-group, this proves feasible. We construct a bundle EH ! BH as follows. For a finite H-space F , let F denote the subcomplex of F consisting of all but the cell of high- est dimension. Being H-invariant, F is itself an H-space and F indeed contains F as a subcomplex since the face F is certainly not antipolar. Let EH ! BH denote the sub-bundle 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 n-fold 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 n-fold join of F may be represented by weighted n-tuple of the form (t1(s1;1; : :;:s|F|;1); : :;:tn(s1;n; : :;:s|F|;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 n-tuple 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 n-tuple, 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 p-group, 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 G-bundle with BG a normal, paracompact space. Then for each p-subgroup 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 p-groups, 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 zero-sets. In particular, call a cozero-set U EH H-incomplete if " hU = ; ; h2H that is, if U misses at least one element in each fibre over BH. Call a zero-set H-coincomplete if its complement is H-incomplete. We have just seen that for p-groups H, the H-incomplete sets generate a non-trival filter. The H- incomplete filter is at least as large as the H-equatorial 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 H-incomplete 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 H-antipolar 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 j-1x 62 S, whence j = x(x-1j) 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 H-incomplete sets miss only a few points from those fibres that they meet at all, the H-coincomplete and the H-equatorial filters will be equal. Remark: Properties intermediate between H-antipolarity and H-incompleteness 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 G-bundles are equivalent up to homotopy, the Stone-Cech 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 G-bundle, 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 G-bundle 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 p-groups G, we may associate to each conjugacy class of subgroups of G a non-empty locus in BG. We emphasize that the action of G on EG is not part of the data available to the Stone-Cech functor; rather the compactification process directly detects the symmetry of the bundle. Theorem 6 Let EG ! BG be Milnor's model of the universal G-bundle. 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 zero-sets in a space X, let F0 denote the collection of zero-sets that have a non-empty intersection with every member of F. If F is a filter, then F0 is simply the union of the z-ultrafiltersSextendingTF. 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. 16 Conversely, suppose that C 62 F00. Then C is disjoint from some zero-set D that meets every element of F. Being zero-sets, 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 H-equatorial filter is non-trivial, so for every H-stable point u there is a group Ki such that u is Ki-stable. That is, any ultrafilter extending the H-equatorial filterSH extends at least one of the Ki-equatorial 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 g-1kUk so that U is disjoint from gU for any g 62 H. Since U is Ki-antipolar, 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 H-antipolar 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 H-bundle 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 H-antipolar sets. | 17 References [1]E. Dror Farjoun and A. Zabrodsky, Fixed points and homotopy fixed points, Comment. Math. Helvetici 63 (1988), 286-295 [2]S. Eilenberg and T. Ganea, On the Lusternik-Schnirelmann category of abstract groups, Ann. Math. 65 (1957), pp. 517-518 [3]C. Husemoller, Fibre Bundles 3d edition, Springer-Verlag, 1993 [4]I. M. James, On category, in the sense of Lusternik-Schnirelmann, Topol- ogy 17 (1978), pp.331-348 [5]H. Miller, The Sullivan conjecture on maps from classifying spaces. An- nals of Math. 120 (1984), 39-87. Erratum: Annals of Math. 121 (1985), 605-609. [6]A.S. Svarc, The genus of a fibre space, Am. Math. Soc. Transl. 55 (1966), pp. 49-140 [7]L. 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