The equivariant category of proper G-Spaces

ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 31, Number 4, Winter 2001 THE EQUIVARIANT CATEGORY OF PROPER G-SPACES R. AYALA, F.F. LASHERAS AND A. QUINTERO Introduction. If Gis a Lie group, then by a G-space we mean a completely regular space Xtogether with a fixed action of Gon X. We follow the standard notation of the theory of transformation groups used in [5]and[9]. In this paper we deal with “minimax invariants” of a G-space. If one restricts consideration to compact Lie groups, then a substantial general theory of G-genus, G-index, as well as G-category in the sense of Lusternik-Schnirelmann has already been developed. See [1], [11]and[17]. In contrast, if Gis not compact, results on such invariants become scarce in the literature. The aim of this paper is to give a general overview of the invariants of type Lusternik-Schnirelmann for an interesting class of G-spaces without the assumption of compactness for the group G;thatis,G-spaces with proper actions. The crucial result that allows such a generalization is due to Palais [19]. Namely, Palais shows that “slices” still exist for proper G-spaces. This fact leads us to extend a remarkable amount of the theory of G-spaces with Gcompact to G-spaces with proper actions. With this paper we intend to point out that there is no real difficulty in extending to proper actions the notion of equivariant Lusternik- Schnirelmann category defined for compact transformation groups in [11]and[17]. In fact, we show that the basic properties of equivariant “minimax” invariants still hold for proper G-spaces and more generally for Cartan G-spaces.1See Section 1 for definitions. Moreover, in many cases the computation of the equivariant category of a proper G-space can be reduced to the compact case, see (2.6) below. The main result of this paper states a Kranosielski type theorem for proper actions of discrete groups on acyclic manifolds (Theorem 4.2). Incidentally, we give in addition an alternative proof of a Kranosielski type theorem for (co)homology spheres due to Marzantowicz (Theorem 4.4). Finally we Received by the editors on September 1, 1998, and in revised form on February 12, 2001. Copyright c 2001 Rocky Mountain Mathematics Consortium 1111 1112 R. AYALA, F.F. LASHERAS AND A. QUINTERO give a Lusternik-Schnirelmann theorem for critical points of G-invariant real functions on proper G-manifolds (Theorem 5.3). Most of the proofs of the basic results are formally similar to the corresponding ones for G compact already in the literature. See [7], [8], [11]and[17]. However, we include here several proofs to emphasize the crucial role played by Palais’s results as well as to avoid an excessively dull exposition. 1. Basic results on proper actions. In this section we collect the basic notations and results on proper actions contained in [3], [9], [16]and[19]. Let Gbe a Lie group (with trivial element e∈G) not necessarily compact. If Xis a space, an action of Gon Xis a homomorphism Tof Ginto the group of homeomorphisms of Xsuch that the map G×X→Xgiven by (g,x)→ T(g)xis continuous. We will write gx for T(g)xif the action Tis clear from the context. A G- space is a completely regular space together with a fixed action of Gon X. A map between two G-spaces f:X→Yis said to be G-invariant (or a G-map)iff(gx)=gf(x) for all g∈Gand x∈X.Areal function h:X→Ris called invariant if h(gx)=hx.AG-homotopy H:X×I→Yis a homotopy such that H(gx,t)=gH(x, t). The isotropy group of x∈Xis the subgroup Gx⊆G,Gx={g∈ G;gx =x}. The action is said to be free if Gx={e}for all x∈X. The set of fixed points is the set XG={x∈X;Gx=G}.Theorbit of x∈Xis the subset Gx ={gx;g∈G}. Moreover the saturation of the subset S⊆Xis the subset GS ={gs;s∈S}. The subset S⊆X is said to be G-invariant (or a G-set)ifGS =S. Finally, X/G stands for the set of orbits of Xtopologized with the usual quotient topology. Following Bourbaki [3, Chapter 3], the action of Gon Xis said to be proper if the map ψ:G×X→X×Xdefined by ψ(g,x)=(gx,x) is proper. We recall that a continuous map f:Y→Zis proper if fis closed and f−1(z)iscompactforanyz∈Z. If the action of Gon Xis proper, then Xis called a proper G-space.Equivalently,aG-space Xis proper if for each pair of points x, y ∈Xthere exist neighborhoods Vx and Vyof xand yrespectively, such that the set {g∈G;gVx∩Vy=∅} has compact closure in G.See[9]and[19] for details. Obviously, any G-space with Gcompact is a proper G-space. Moreover, if Gacts properly on X, then the restriction of the action to any closed subgroup H⊆Gis also proper. Clearly if Xis a proper G-space, the isotropy group Gxis compact for all x∈X, and hence XG=∅if Gis not EQUIVARIANT CATEGORY OF PROPER G-SPACES 1113 compact. Moreover, each orbit Gx is closed in Xand the natural map G/Gx→Gx defined by [g]→ gx is a homeomorphism; see [9]. Remark 1.1. Palais [19] considers a more restrictive definition of a proper G-space Xby requiring that for each x∈Xthere exists a neighborhood Uxsuch that for all y∈Xthere exists a neighborhood Uysuch that the set {g∈G;gUx∩Uy=∅}has compact closure in G. Palais also introduces the weaker notion of Cartan G-space by requiring that for each x∈Xthere exists a neighborhood Uxsuch that the set {g∈G;gUx∩Ux=∅}has compact closure in G. Clearly one has Palais-proper ⇒Bourbaki-proper ⇒Cartan G-space. In general, the converses are not true. See [13]and[16], respectively. However, the three notions coincide for locally compact spaces and more generally if X/G is regular. See [19, p. 303]. Examples 1.2. Examples of proper actions are the following: (1) If Gis a Lie group, any closed subgroup H⊆Gacts properly on Gby left translations. Moreover, if K⊆Gis a compact subgroup, then the left translation action of Hon G/K is proper. See [9, Example 24]. (2) It follows from the previous observations that every closed (e.g., discrete) subgroup H⊆Gof a connected Lie group acts properly on some Euclidean space. The proof of this fact follows from a wellknown result which states that Gcontains a maximal compact subgroup K⊆G(unique up to conjugation) such that G/K is diffeomorphic to a Euclidean space. See [14]. (3) A G-space Mis called a G-manifold if Mis a manifold and the map x→ gx is differentiable for each g∈G.NotethatGacts on the tangent space TM via the differential; that is, gv =(dg)xvif v∈TxM. In addition, Mis called a Reimannian G-manifold if Mis endowed of a Riemannian metric Rfor which the maps x→ gx are isometries. That is, Ris a section of the dual bundle (TM ⊕TM)∗ such that for each x∈M,Rxis an inner product and for each g∈G,Rgx((dg)xv,(dg)xw)=Rx(v,w) for all v,w ∈TxM.IfMis a Riemannian G-manifold and Gis a closed subgroup of the isometry group Iso (M)ofM, then the action of Gis proper. Moreover, if M is a proper G-manifold, then there exists a G-invariant Riemannian metric on M(i.e., the action of Gis an isometric with respect to that 1114 R. AYALA, F.F. LASHERAS AND A. QUINTERO metric). Therefore, a G-manifold is proper if and only if there exists a Riemannian metric on Msuch that Gis a closed subgroup of Iso (M). See [20] for details. (4) The group of deck transformations of a covering p:˜ X→Xacts properly on ˜ X. More generally, if the completely regular space Xis the total space of a principal G-bundle, then Xis a Cartan G-space (1.1). In particular, Xis a proper G-space if Xis locally compact or, more generally, if X/G is regular. Conversely, if Xis a proper G-space or, more generally, a Cartan G-space, and Gacts freely on X, the orbit map X→X/G is a locally trivial principal G-bundle. See (1.1) and [19, 4.1]. (5) In the language of dynamical systems (i.e., R-spaces) proper flows are called dispersive flows, and they coincide with parallelizable flows for metrizable locally compact spaces. We recall that a flow φ:X×R→Xon the space Xis said to be parallelizable if the orbit map X→X/Ris a trivial bundle. See [13]. Important objects in a transformation group theory are tubes and slices. We recall here some basic properties of them which will be used below. A tube about xin a G-space Xis an open G-invariant neighborhood U⊆Xof xin Xfor which there exists a G-invariant map π:U→Gx called a projection of U.Aslice at xis a subset S⊆Xwhich is Gx-invariant and such that the map G×GxS→X taking [g,s]togs yields a homeomorphism onto a tube Uabout x. We recall that, for a subgroup H⊆Gand an H-space X,thetwisted product G×HXis the orbit space of the H-action h(g,x)=(gh−1,hx) on G×X. Tubes and slices come in pairs. Actually, if π:U→Gx is a projection of the tube Uabout x,thenS=π−1(x) is a slice at x.Conversely,ifSis a slice at x,thenGS is a tube about xwith projection π(gs)=gx. Also, if Sis a slice at xin the G-space X,then the natural map S/Gx→X/G is a homeomorphism onto the open subspace GS/G.See[5, II.4.7]. The following existence theorem is crucial in dealing with proper G-spaces. See [5] for compact groups and [19] for the general case. Theorem 1.3 (Palais). Let Xbe a proper G-space. Then there exists a slice at each point x∈X. More precisely, Cartan G-spaces are exactly the G-spaces for which there is a slice at each point. EQUIVARIANT CATEGORY OF PROPER G-SPACES 1115 Theorem 1.3 allows one to generalize many results on compact transformation groups to proper actions. We next state some of them which will be used below. The first one is the path lifting theorem for proper G-spaces. We give a proof to illustrate the way of generalizing results already known for G-spaces with Gcompact to proper G-spaces. Theorem 1.4. Let Xbe a proper G-space and let α:I→X/G be any path. Then there exists a lifting α:I→Xof αwith respect to the orbit map p:X→X/G. Proof.IfSxis a slice at x∈X, the family {GSx/G}is an open cover of the orbit space X/G. Then by Lebesgue’s lemma one finds a positive integer nsuch that for each 0 ≤i≤n, the restriction αi=α|[i/n, (i+1)/n] is contained in some GSxi/G ∼ =Sxi/Gxifor some xi∈X. We now apply the path lifting theorem for compact groups [5, II.6.2] to the Gxi-space Sxi, and we get a lifting of αi, ˜αi:[i/n, (i+1)/n]→Sxi⊆GSxifor all 0 ≤i≤n. Moreover, p(˜αi((i+1)/n)) = p(˜αi+1((i+1)/n)) for 0 ≤i≤n−1. Hence there exists gi+1 ∈Gwith ˜αi((i+1)/n)=gi+1 ˜αi+1((i+1)/n), 0 ≤i≤n−1. Then ˜α(t)=g1...g i˜αi(t)ifi/n ≤t≤(i+1)/n is a lifting of α. As it was pointed out by Palais [19, 4.5], the covering homotopy theorem [5, II.7.3] can also be extended to proper G-spaces. Namely, the following theorem holds Theorem 1.5. Let Xand Ybe separable metrizable proper G-spaces, and let ˜ f0:X→Ybe a G-map. If F:X/G ×I→Y/G is any homotopy with F0the map induced by ˜ f0and F(p(X(H))×I)⊆p(Y(H))) for all orbit type (H), then there exists a G-homotopy ˜ F:X×I→Y with ˜ F0=˜ f0which covers F. We recall that for a subgroup H⊆G,theorbit type of x∈Xis said to be (H)ifGx=gHg−1for some g∈G. The subset X(H)⊆X is the union of all orbits in Xof type (H). A proof of Theorem 1.5 can be given along the lines of the proof in [5] for compact groups. Moreover, since for a proper G-space the set of fixed points XGis empty, one actually skips Part D in that proof. In addition, that 1116 R. AYALA, F.F. LASHERAS AND A. QUINTERO proof also shows that the previous theorem also holds for hereditarily paracompact spaces Xand Y. 2. The equivariant category of proper G-spaces. We now proceed to extend the definition and basic properties of the equivariant category as introduced in [11]and[17] for compact Lie groups to proper actions of arbitrary Lie groups. This extension is completely natural, and it will be carried out more or less by way of showing what can be done. Definition 2.1. Let Xbe a proper G-space or, more generally, a Cartan G-space in the sense of Remark 1.1. A G-set A⊆Xis said to be of G-category 1 if there exists an orbit Gxsuch that the inclusion iA:A⊆Xis G-homotopic in Xto a G-map A→Gx ⊆X. Given a G-set A⊆X,theG-category (or equivariant category), of Ain X,cat G(A;X), is the smallest number ksuch that there exists a cover A⊆∪ k j=1Ajconsisting of closed G-sets in Xof G- category 1. If no such cover exists for all k,thencat G(A, X)=∞. As usual, catG(X) will replace catG(X, X). The classical Lusternik- Schnirelmann category, cat (−), defined with closed covers [15]is obtained as cat (A, X)=cat G(A, X) for the trivial group G={e}. Remark 2.2. As usual, the invariant catG(−) can be extended to G- maps f:X→Yin such a way that catG(idX)=cat G(X). Namely, catG(f) is the smallest ksuch that there exists a cover {X1,... ,X k} of Xconsisting of closed G-sets such that for each ione has an orbit Gyi⊆Ysuch that the restriction f|Xiis G-homotopic in Yto a G-map Xi→Gyi⊆Y. At this point we should mention the connection of the above definition with the general definition of equivariant category introduced by Clapp andPuppein[7]. That definition starts by fixing a class Aof G-spaces for some compact Lie group G. Then they define A−cat (f)≤kif there exist spaces Ai∈Asuch that the restriction f|Xiis G-homotopic to a composition of G-maps Xi→Ai→Y.IfGis now an arbitrary Lie group, we can consider the corresponding definition by choosing a family Aproper of proper G-spaces. We recover the definition of catG(−) above if Aproper ={G/H;H⊆Gis a compact subgroup}. EQUIVARIANT CATEGORY OF PROPER G-SPACES 1117 Notions of relative equivariant category as well as G-genus and A- genus can also be studied for proper G-spaces. See [21], [8], [1]and [7]. As one expects, the fundamental properties of “minimax” invariants extend to proper G-spaces. Namely, Proposition 2.3. Let A, B be closed G-sets of the proper G-space X.Then (1) (Subadditivity). catG(A∪B,X)≤catG(A, X)+cat G(B,X). (2) (Deformation monotonicity). Assume that there exists a G- homotopy H:A×I→Xsuch that H0=id Aand H1(A)⊆B.Then catG(A, X)≤catG(B,X).InparticularcatG(X)is a G-homotopy invariant of X. (3) (Continuity). If Xis a G-ANR, then there exists a G-neighborhood Uof Ain Xwith catG(A, X)=cat G(U, X). Properties (1) and (2) are easily checked following the classical arguments, and they are left to the reader. Compare [15]and[7]. Similarly for property (3), but some further remarks are needed in this case since G-ANR spaces are usually studied in the literature only for compact groups (see [18]). In Section 3 we include the results on proper G-ANR needed for the proof of (3). We now continue with some results concerning proper free actions. Proposition 2.4. If Xis a proper G-space and Ais a closed G- set, then cat (A/G, X/G)≤catG(A, X). Moreover, if Xis separable metrizable and Gacts with only one orbit type, in particular freely, then catG(A, X)=cat(A/G, X/G). Proof.Letp:X→X/G betheorbitmap. Thefirstpart is obvious since given a cover A⊆∪ m i=1Aiof closed G-sets with catG(Ai,X) = 1 one has that {p(Ai)}is a closed categorical cover of X/G and hence the first inequality is proved. Assume now the extra hypothesis, then we will show catG(A, X)≤cat (A/G, X/G). Let {Uj}1≤j≤nbe a closed cover of X/G with cat (Uj,X/G)=1. Let Hj:Uj×I→X/G be a deformation of the inclusion Uj⊆X/G 1118 R. AYALA, F.F. LASHERAS AND A. QUINTERO toapointp(xj)∈X/G. Clearly, the inclusion kj:p−1(Uj)⊆Xis an equivariant lifting of the inclusion Hj 0:Uj⊆S/G.SinceGacts with only one orbit type, we can apply the covering homotopy theorem (1.5), and one finds an equivariant homotopy ˜ Hjwith covers Hjand with ˜ Hj 0=kj.Inparticular,˜ Hj 1(x)∈Gxjfor all x∈p−1(Uj). Hence catG(p−1(Uj),X) = 1, and the proof is finished. We now consider the relationship between catGand catHwhen H⊆Gis a closed subgroup. The following general result is available. Proposition 2.5. Let Xbe a proper G-space and H⊆Gbe a closed subgroup. Then catH(X)≤catG(X)max{catH(Gx):x∈X}. The proof is similar to that of [1, 2.14]. In particular, if Gacts freely and Xis separable metrizable, then (2.4) yields catH(X)≤ catG(X)cat(G/H). Moreover, if His a maximal compact subgroup of the connected Lie group G,thenG/H ∼ =Rmfor some m(see (1.2)) and one gets catH(X)≤catG(X). The next result shows that the latter inequality actually holds for any G-space, and furthermore it turns out to be an equality in many cases. Proposition 2.6. Let Gbe a connected Lie group, and let Xbe a proper G-space. If H⊆Gis a maximal compact subgroup in G,then catH(X)≤catG(X). Moreover, if the action is free, the space Xis metrizable and separable, and the orbit spaces X/G and X/H have the homotopy type of CW-complexes, then catH(X)=cat G(X). Proof. Assume that Hcontains the isotropy group Gxfor some x∈X. Then the bundle G/Gx→G/H of fiber H/Gxand group His trivial since G/H ∼ =Rmfor some m. Hence, the composition of the trivialization G/Gx∼ =G/H ×H/Gxwith the second projection π2:G/H ×H/Gx→H/Gxis a homotopy H-equivalence Gx ≃Hx. Now let Abe a G-set such that the inclusion A⊆Xis G-homotopic (and hence H-homotopic) to a G-map A→Gz ⊆X.AsGzis compact and the maximal compact subgroups are uniquely determined up to conjugation, there exists g∈Gsuch that gGzg−1=Ggz ⊆H. Therefore, Hgz is homotopy H-equivalent to Gz =Ggz. Hence, EQUIVARIANT CATEGORY OF PROPER G-SPACES 1119 catH(A, X) = 1, and it readily follows that catH(X)≤catG(X). Assume now that Gacts freely. According to (1.2)(4), X→X/G and X→X/H are principal bundles of fibers Gand H, respectively. Then the exact sequences of homotopy groups of these bundles as well as the bundle G→G/H yield isomorphisms πn(X/H)∼ =πn(X/G) for all n. Hence X/H and X/G have the same homotopy type by Whitehead’s theorem [24, V.3.5] and so catG(X)=cat(X/G)=cat(X/H)= catH(X). Corollary 2.7. Let Gbe a contractible Lie group acting properly on a separable metrizable space X.ThencatG(X)=cat(X). Proof.SinceGis contractible, then Gcontains no compact subgroup other than the trivial subgroup {e}. Therefore, any proper action of G is necessarily free and hence the result follows from (2.6). Let Gnow be an arbitrary Lie group acting properly on X. Assume that H⊆Gis a closed and open subgroup. Then the H-orbit Hx is closed and open in the G-orbit Gx for all x∈X. This follows from the fact that the canonical maps H→Hx and G→Gx are both closed and open. For this we use the homeomorphisms Hx ∼ =H/Hx and Gx ∼ =G/Gxsince Gacts properly. After these observations, one can prove Proposition 2.8. Assume that Xis a path connected separable and metrizable G-space. Assume that His a closed and open subgroup such that the action restricted to Hhas only one orbit type. Then catH(X)≤catG(X). Proof.Let{U1,... ,U k}be a cover of Xby closed G-sets such that for all ithere is a G-deformation (and so an H-deformation) φiin X to G-orbits Gxi.LetGxi=∪Hyi jbe the decomposition of Gxinto H-orbits. According to the previous observations, each Hyi jis open and closed in Gxi, and hence Gxi/H is a discrete space. Since X/G is path connected, we consider paths γi j:I→X/H running from [yi j]to some fixed point [x0]. Since the H-action has only one orbit type, we 1126 R. AYALA, F.F. LASHERAS AND A. QUINTERO From this we get a commutative diagram Hn(G;ZG) u ˜ f∗ w ˜ f∗ ∼ =Hn(G;ZG)∼ =Hn c(X;Z)∼ =Z Hn(C(f∗E); ZG)=0 h h h hj ˜g∗ where the isomorphism Hn c(X,Z)∼ =Hn(G;ZG)isgivenbytheproof of [6, VIII.7.5] since X/G is compact and Cm(X)=0form≥n+1. Finally Hn(C(f∗E); ZG)=0sincedimf∗E=dimL≤n−1. This finishes the proof. Remark 4.3. a) In case Xis contractible, the orbit space X/G is an Eilenberg-MacLane space K(G, 1) (i.e., X/G is an aspherical n-manifold with π1(X/G)=G) and Theorem 4.2 follows, in fact, from a well-known theorem by Eilenberg and Ganea [12]whichstates cat (K(G, 1)) = cd (G)+1 except possibly for the case cat (K(G, 1)) = 3 and cd (G) = 2. Here “cd” stands for cohomological dimension; that is, cd (G) is the smallest nsuch that Hq(G;M) = 0 for all q>nand all ZG-modules M.See[6]. Moreover, for Gas above, one gets cd (G)=n by [6, VIII.8.1], and so catG(X)=n+1whenn=2. Forn=2,X/G is an orientable surface (=S2) and it is well known that cat (X/G)=3. b) The group Gin Theorem 4.2 is necessarily torsion free. More explicitly, let Gbe a discrete group acting properly on an acyclic nmanifold Xwith X/G compact. Then the action is free if and only if Gis torsion free. Indeed, if the action is free, then we have cd (G)=n as in a) above, and so Gis torsion free [6, VIII.2.5]. Conversely, since the action is proper all isotropy groups Gxare finite and so trivial since Gis torsion free. Theorem 4.2 above can be regarded as the noncompact counterpart of the result by Marzantowicz [17, 2.6] on free actions of finite groups on (co)homology spheres. Actually, the arguments used in the proof of (4.2) allow us to give an alternative proof of Marzantowicz’s result. Namely, we now prove Theorem 4.4. Let Xbe a (co)homology n-sphere, and let Gbe a finite group acting freely on X.ThencatG(X)=cat(X/G)=n+1. EQUIVARIANT CATEGORY OF PROPER G-SPACES 1127 Here a (co)homology n-sphere is an n-manifold whose integral (co)- homology coincides with the (co)homology of the n-sphere Sn.The universal coefficient theorem [23, 5.5.3] shows that homology and cohomology spheres coincide. As in the original, Marzantowicz’s proof of (4.4), we shall use the following crucial result on degrees of G-maps between (co)homology spheres also due to Marzantowicz [17, 2.9]. Theorem 4.5 (Marzantowicz). Let f:X→Ybe a Zp-map between (co)homology spheres. If the action of Zpis free, then deg (f)=0 (mod p). In the proof of (4.4) we shall also use the following elementary fact for (co)homology spheres of even dimension. Lemma 4.6. If dim X=nis even, then Z2is the only finite group acting freely on X. This result is an immediate consequence of the formula L(f)= 1+(−1)ndeg (f) for any map f:Sn→Snwhere L(f) stands for the Lefschetz number of f.See[23, 4.7.9]. Proof of 4.4. Let pbe a prime divisor of the order of G.ThenZpis a subgroup of Gand according to (2.9) and (3.8) we have catZp(X)≤ catG(X)≤n+ 1. Therefore, it will suffice to show catZp(X)=n+1. Assume on the contrary that catZp(X)=cat(X/Zp)≤n.Then by (4.1) there exist an (n−1)-dimensional polyhedron Land a map g:X/G →Lwhich induce an isomorphism g∗:π1(X/G)→π1(L). Next we consider the diagram X u π w ˜ fE(Zp,1) = Ep u qp X/Zp w fK(Zp,1) = Kp defined by the classifying map fof the Zp-bundle π:X→X/Zp.Now we construct a map h:L2→X/Zprealizing the inverse g−1 ∗as in the proof of (4.2). Moreover, by using the asphericity of Kp,wecanextend fh toamapf:L→Kpsuch that fg≃f. 1128 R. AYALA, F.F. LASHERAS AND A. QUINTERO At this point we consider two cases: a) p=2. ThenE2=S∞and K2=RP∞. Moreover, by the cellular approximation theorem [23, 7.6.18], we can assume that f(X/Z2)⊂RPnand ˜ f(X)⊆Sn. Now we apply (4.5) to the Z2- equivariant map ˜ f:X→Snand we get deg ( ˜ f) = 1 (mod 2). Then the differentiable (or simplicial) definition of degree yields that deg (f)=1 (mod 2), and so f∗:Hn(X/Z2;Z2)∼ =Z2−→ Hn(RPn;Z2)∼ =Hn(RP∞;Z2)∼ =Z2 is an isomorphism. This leads to a contradiction since f∗factorizes through Hn(L;Z2)=0. b) p= 2. Then by (4.6) dim X=n=2k+ 1 is odd. Moreover, Ep=S∞and Kp=L∞(p) is the infinite lens space. It is known that L∞(p)isaCW-complex such that the lens space L2m+1(p)is its (2m+ 1)-skeleton for each m.See[24, p. 91]. As in case a) we can assume f(X/Zp)⊆L2k+1(p)and ˜ f(X)⊆S2k+1, and by (4.5), deg ( ˜ f)=0(modp). As H2k+1(X/Zp;Zp)∼ =Zp[5, Example III.3], we argue as above to get that f∗:H2k+1(X/Zp;Zp)∼ =Zp−→ H2k+1(L2k+1(p); Zp) ∼ =H2k+1(L∞(p); Zp)∼ =Zp is an isomorphism, and we conclude as in case a). 5. Critical points on proper G-manifolds. In this final section we illustrate how the equivariant category of proper G-manifolds provides an upper bound for the number of critical orbits of invariant smooth functions. In the action-free case, one requires the Palais- Smale condition (PS). Namely, if Mis a complete Riemannian manifold, the smooth function f:M→Rsatisfies condition (PS) if any sequence {xn}⊆Mfor which |f(xn)|is bounded and ∇fxnconverges to zero admits a convergent subsequence. Then the Lusternik- Schnirelmann theorem states that a smooth function bounded below satisfying condition (PS) has at least cat (M)critical points. Recall that ∇f:M→TM is the gradient vector field defined as the dual of the differential df . Explicitly, for any vector field Yand any x∈M, df x(Yx)=Rx(Yx,∇fx). Here Ris the Riemannian structure of M. EQUIVARIANT CATEGORY OF PROPER G-SPACES 1129 In the equivariant setting any invariant function f:M→Rgives rise to an equivariant gradient ∇f:M→TM;thatis,∇fgx =dgx∇fx. Moreover, if the Riemannian metric on Mis G-invariant, then ∇f: M→Ris an invariant function and the flow {φt}generated by ∇f commutes with the action (i.e., φt(gx)=gφt(x)). In addition, if xis a critical point of f, then the whole orbit Gx consists of critical points, and so if the group Gis not compact, then the usual condition (PS) is not satisfied. In such a case we replace condition (PS) by the following orbitwise Palais-Smale condition (OPS). Definition 5.1. Let Mbe a complete Riemannian proper G- manifold and f:M→Ra smooth invariant function. The function fsatisfies condition (OPS) if, given a sequence {xn}⊆Msuch that |f(xn)|is bounded and ∇fxnconverges to zero, then the sequence of orbits {Gxn}contains a convergent subsequence in the orbit space M/G. Remark 5.2. (1) Let Sf⊆Mdenote the set of critical orbits of f.SinceSfis a G-set, the restriction f|Sfyields a continuous map ˜ f:Sf/G →R. Then one readily checks that condition (OPS) implies that ˜ fis proper. In particular, ˜ f(Sf/G)=f(Sf)⊆Ris a closed set. (2) Since the orbit space is not in general a smooth manifold, in order to avoid the possibly singular space M/G, condition (OPS) is often replaced by the Palais-Smale condition modulo G(G-PS): Given the sequence {xn}⊆Mfor which |f(xn)|is bounded and ∇fxn→0, there exists a sequence {gn}⊆Gsuch that the sequence {gnxn}has a convergent subsequence in M. Clearly (G-PS) implies (OPS) and in case Gis compact, (PS) implies (G-PS). Theorem 5.3. Let Mbe a complete Riemannian G-manifold and f:M→Ra smooth invariant function satisfying condition (OPS). Then fhas at least catG(M)critical orbits. The proof of (5.3) just mimics the proof of the classical Lusternik- Schnirelmann theorem in [20]. Compare also with [11]forGcompact. More precisely, one considers the (equivariant) flow {φt}generated by −∇f. Then one shows that for all t>0, φtis defined on all of M,and 1130 R. AYALA, F.F. LASHERAS AND A. QUINTERO for any x∈M, limt→∞ φt(x) is a critical point of M. The proof is the same as [20, 9.1.6]. Next one proceeds to use the flow {φt}to prove the following equivariant deformation theorem. For this, let Scdenote the union of all critical orbits in f−1(c)andMc=f−1((−∞,c]), c∈R. Lemma 5.4 (Deformation theorem). For any G-neighborhood Uof Scin M,thereexistsε>0such that φ1(Mc+ε−U)⊆Mc−ε.In particular, if cis a regular value, (i.e., Sc=∅),thenφ1(Mc+ε)⊆ Mc−ε. Proof.Letddenote the (invariant) distance on Minduced by its Riemannian structure. Then the family Nof G-sets Nδ(Sc)= {x∈M;d(x, Sc)<δ},δ>0, is a basis of G-neighborhoods of Sc in M. Indeed, by [19, 4.3.4], if p:M→M/G is the orbit map ˜ d(p(x),p(y)) = inf {d(z,w); z∈Gx, w ∈Gy}is a distance on M/G. Moreover, by (5.2)(1), the set p(Sc)iscompactinM/G and so the sets ˜ Nδ(p(Sc)) = {˜x∈M/G;˜ d(˜x, p(Sc)) <δ}form a basis of neighborhoods of p(Sc)inM/G.Nowtheinclusionp(Nδ(Sc)⊆˜ Nδ(p(Sc))) easily implies that Nis the required basis. At this point the proof is carried out formally in the same way as in [20, 9.2.3]. Now we have the ingredients for a proof of (5.3) along the lines of the classical proof in [20, 9.2.9]. For this, one applies the properties of catGin (2.3). As usual, one defines for m≤catG(M)theminimax number cm(f)offwith respect to the family Fmof G-sets F⊆M with catG(F,M)≥m.Namely, cm(f)=inf{sup f(F); F∈F m}=inf{a∈R;cat G(Ma,M)≥m}. Then, clearly c1(f)=inff(M)andcm(f)≤cm+1(f). Moreover, the property that limt→∞ φt(x) is always a critical point yields that c1(f) is actually the minimum of f. Lemma (5.4) is used to check that the rest of minimax numbers cm(f) are critical values of f. Therefore, Theorem 5.3 will follow if one shows that c=cm(f)=cm+k(f) implies that Sccontains at least k+ 1 critical orbits. Suppose not. Then Sc is the union ∪{Gxj;1 ≤j≤r}of r≤kcritical orbits. Since Mis a G-ANR space by (3.3), one finds by (2.3)(3) a G-neighborhood Uof Sc with catG(U, M)=cat G(Sc,M)≤r. EQUIVARIANT CATEGORY OF PROPER G-SPACES 1131 Moreover, by (5.4) there exists ε>0 such that catG(Mc+ε−U, M)≤ catG(Mc−ε,M)≤m−1. For this we use property (2.3)(2) of catG and c=cm(f). Now the equality c=cm+1(f) yields the contradiction m+k≤catG(Mc+ε,M)≤catG(Mc+ε−U, M)+catG(U, M)≤m+k−1. For this, we use property (2.3)(1) of catG. This finishes the proof of (5.3). Acknowledgment. This work was partially supported by the project DGICYT PB96-1374. ENDNOTES 1. After having finished the manuscript we learned about the thesis of E. Shaw [22] where the equivariant LS category of proper G-spaces is also considered. In fact, (2.3) and (2.4) also appear in Shaw’s thesis. 2. This result also appears in the thesis of E. Elfving [10]. REFERENCES 1. T. Bartsch, Topological methods for variational problems with symmetries, Lecture Notes in Math. 1560, Springer, New York, 1993. 2. K. Borsuk , Theory of retracts, Polish Academic Publishers, Warsaw, 1967. 3. N. 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Departamento de Geometr´ ıa y Topolog´ ıa, Facultad de Matem´ aticas, Universidad de Sevilla, Apartado 1160, 41080 - Sevilla, Spain Departamento de Geometr´ ıa y Topolog´ ıa, Facultad de Matem´ aticas, Universidad de Sevilla, Apartado 1160, 41080 - Sevilla, Spain Departamento de Geometr´ ıa y Topolog´ ıa, Facultad de Matem´ aticas, Universidad de Sevilla, Apartado 1160, 41080 - Sevilla, Spain E-mail address: [email protected]