ON ANALYTICAL APPLICATIONS OF STABLE HOMOTOPY (THE ARNOLD CONJECTURE, CRITICAL POINTS) Yuli B. Rudyak 8 9 9 1 Abstract. We prove the Arnold conjecture for closed symplectic manifolds with n π2(M)=0 and catM =dimM. Furthermore, we prove an analog of the Lusternik– a Schnirelmann theorem for functions with “generalized hyperbolicity” property. J 6 2 Introduction 3 v 8 Here we show that the technique developed in [R98] can be applied to the Arnold 0 conjecture and to estimation of the number of critical points. For convenience of 0 8 the reader, this paper is written independently of [R98]. 0 7 Given a smooth (= C∞) manifold M and a smooth function f : M → R, we 9 denote by critf the number of critical points of f and set CritM = min{critf} / a where f runs over all smooth functions M → R. g - g The Arnold conjecture [Ar89, Appendix 9] is a well-known problem in Hamil- d tonian dynamics. We recall the formulation. Let (M,ω) be a closed symplectic : v manifold, and let φ : M → M be a Hamiltonian symplectomorphism (see [HZ94], i X [MS95] for the definition). Furthermore, let Fixφ denote the number of fixed points r of φ. Finally, let a Arn(M,ω) := minFixφ φ where φ runs over all Hamiltonian symplectomorphisms M → M. The Arnold conjecture claims that Arn(M,ω) ≥ Crit(M). It is well known and easy to see that Arn(M,ω) ≤ CritM. Thus, in fact, the Arnold conjecture claims the equality Arn(M,ω) = CritM. Let catX denote the Lusternik–Schnirelmann category of a topological space X (normalized, i.e., catX = 0 for X contractible). Given a symplectic manifold (M2n,ω), we define the homomorphisms I : π (M) → Q, I (x) = hω,h(x)i ω 2 ω I : π (M) → Z, I (x) = hc,h(x)i c 2 c where h : π (M) → H (M) is the Hurewicz homomorphism, c = c (τM) is the first 2 2 1 Chern class of M and h−,−i is the Kronecker pairing. 1991 Mathematics Subject Classification. Primary 58F05, secondary 55M30, 55N20,. TheauthorwaspartiallysupportedbyDeutscheForschungsgemeinschaft,the ResearchGroup “Topologie und nichtkommutativeGeometrie” 2 YULI B. RUDYAK Theorem A (see 3.6). Let (Mn,ω),n = dimM be a closed connected symplec- tic manifold such that I = 0 = I (e.g., π (M) = 0) and catM = n. Then ω c 2 Arn(M,ω) ≥ CritM, i.e., the Arnold conjecture holds for M. It is well known that CritM ≥ 1 + clM for every closed manifold M, where cl denotes the cup-length, i.e., the length of the longest non-trivial cup-product in H∗(M). So, one has the following weaker version of the Arnold conjecture: e Arn(M,ω) ≥ 1+cl(M), and most known results deal with this weak conjecture, see [CZ83], [S85], [H88], [F89-1], [F89-2], [LO96]. (Certainly, there are lucky cases when CritM = 1+clM, e.g. M = T2n, cf. [CZ83].) For example, Floer [F89-1], [F89-2] proved that Arn(M) ≥ 1+clM provided I = 0 = I , cf. also Hofer [H88]. So, my contribution ω c is the elimination of the clearance between CritM and 1 + clM. (It is easy to see that there are manifolds M as in Theorem A with CritM > 1+clM, see 3.7 below.) Actually, we prove that Arn(M,ω) ≥ 1+catM and use a result of Takens [T68] which implies that CritM = 1+catM provided catM = dimM. After submission of the paper the author and John Oprea proved that catM = dimM for every closed symplectic manifold (M,ω) with I = 0, see [RO97]. So, ω the condition catM = n in Theorem A can be omitted. Passing to critical points, we prove the following theorem. Theorem B (see 4.5). Let M be a closed orientable manifold, and let g : M × Rp+q → R be a C2-function with the following properties: (1) There exist disks D ⊂ Rp and D ⊂ Rq centered in origin such that + − int(M ×D ×D ) contains all critical points of g; + − (2) ∇g(x) points inward on M ×∂D ×intD and outward on M ×intD × + − + ∂D . − Then critg ≥ 1+r(M). In particular, if M is aspherical then critg ≥ 1+catM.† Notice that functions g as in Theorem B are related to the Conley index theory, see [C76]. I remark that Cornea [Co98] have also estimated the number of crirical points of functions as in Theorem B. We reserve the term “map” for continuous functions of topological spaces, and we call a map inessential if it is homotopic to a constant map. The disjoint union of spaces X and Y is denoted by X ⊔ Y. Furthermore, X+ denotes the disjoint union of X and a point, and X+ is usually considered as a pointed space where the base point is the added point. We follow Switzer [Sw75] in the definition of CW-complexes. A CW-space is defined to be a space which is homeomorphic to a CW-complex. Given a pointed CW-complex X, we denote by Σ∞X the spectrum E = {E } n where E = SnX for every n ≥ 0 and E = pt for n < 0; here SnX is the n-fold n n †Recall that a connected topological space is called aspherical if π (X)=0 for every i>1. ON ANALYTICAL APPLICATIONS OF STABLE HOMOTOPY 3 reduced suspension over X. Clearly, Σ∞ is a functor from pointed CW-complexes to spectra. Given any (bad) space X, the cohomology group Hn(X;π) is always defined to be the group [X,K(π,n)]where [−,−] denotes the set of homotopy classes of maps. “Smooth” always means “C∞”. “Fibration” always means a Hurewicz fibration. “Connected” always means path connected. The sign “ ≃” denotes homotopy of maps (morphisms) or homotopy equivalence of spaces (spectra). §1. Preliminaries on the Lusternik–Schnirelmann category 1.1. Definition. (a) Given a subspace A of a topological space X, we define cat A to be the minimal number k such that A = U ∪···∪U where each U X 1 k+1 i is open in A and contractible in X. We also define cat A = −1 if A = ∅. X (b) Given a map f : X → Y, we define catf to be the minimal number k such that X = U ∪ ··· ∪ U where each U is open in X and f|U is inessential for 1 k+1 i i every i. (c) We define the Lusternik–Schnirelmann category catX of a space X by setting catX := cat X = cat1 . X X Clearly, catf ≤ min{catX,catY}. The basic information concerning the Lusternik–Schnirelmann category can be found in [Fox41], [J78], [Sv66]. Let X be a connected space. Take a point x ∈ X, set 0 PX = P(X,x ) = {ω ∈ XI ω(0) = x } 0 (cid:12) 0 (cid:12) and consider the fibration p : PX → X, p(ω) = ω(1) with the fiber ΩX. Given a natural number k, we use the short notation (1.2) p : P (X) → X. k k for the map p ∗ ···∗ p : PX ∗ ···∗ PX −−−−→ X X X X X X X | k t{imzes } | k t{imzes } where ∗ denotes the fiberwise join over X, see e.g. [J78]. In particular, P (X) = X 1 PX. 1.3. Proposition. For every connected compact metric space X and every natural number k the following hold: (i) p : P (X) → X is a fibration; k k (ii) catP (X) < k; k (iii) The homotopy fiber of the fibration p : P (X) → X is the k-fold join k k (ΩX)∗k; (iv) If catX = k and X is (q−1)-connected then p : P (X) → X is a (kq−2)- k k equivalence; 4 YULI B. RUDYAK (v) If X has the homotopy type of a CW-space then P (X) does. k Proof. (i) This holds since a fiberwise join of fibrations is a fibration, see [CP86]. (ii) It is easy to see that cat(E ∗ E ) ≤ catE +catE +1 for every two maps 1 X 2 1 2 f : E → X and f : E → X. Now the result follows since catP (X) = 0. 1 2 2 2 1 (iii) This holds since the homotopy fiber of p is ΩX. 1 (iv) Recall that A ∗ B is (a + b + 2)-connected if A is a-connected and B is b-connected. Now, ΩX is (q − 2)-connected, and so the fiber (ΩX)∗k of p is k (kq −2)-connected. (v) It is a well-known result of Milnor [M59] that ΩX has the homotopy type of a CW-space. Hence, the space (ΩX)∗k has it. Finally, the total space of any fibration has the homotopy type of a CW-space provided both the base and the fiber do, see e.g. [FP90, 5.4.2]. (cid:3) 1.4. Theorem([Sv66,Theorems3and19′]). Let f : X → Y be a map of connected compact metric spaces. Then catf < k iff there is a map g : X → P (Y) such that k p g = f. (cid:3) k §2. An invariant r(X) Consider the Puppe sequence P (X) −p−m→ X −−jm→ C (X) := C(p ) m m m where p : P (X) → X is the fibration (1.2) and C(p ) is the cone of p . m m m m 2.1. Definition. Given a connected space X, we set r(X) := sup{m|j is stably essential}. m (Recall that a map A → B is called stably essential if it is not stably homotopic to a constant map.) 2.2. Proposition. (i) r(X) ≤ catX for every connected compact metric space X. (ii) Let X be a connected CW-space, let E be a ring spectrum, and let u ∈ i E∗(X)),i = 1,... ,n be elements such that u ···u 6= 0. Then r(X) ≥ n. In other 1 n weords, r(X) ≥ cl (X) for every ring spectrum E. E It makes sense to remark that r(X) = catX iff X possesses a detecting element, as defined in [R96]. Proof. (i) This follows from 1.4. (ii) Because of 1.3(v), without loss of generality we can and shall assume that C (X) is a CW-space. We set u = u ···u ∈ Ed(X). Because of the cup- n 1 n e length estimation of the Lusternik–Schnirelmann category, and by 1.3(ii), we have p∗(u) = 0. Hence, there is a homotopy commutative diagram n Σ∞X −Σ−−∞−j→n Σ∞C (X) n u    y y ΣdE ΣdE. Now, if r(X) < n then Σ∞j is inessential, and so u = 0. This is a contradic- n tion. (cid:3) ON ANALYTICAL APPLICATIONS OF STABLE HOMOTOPY 5 2.3. Lemma. Let f : X → Y be a map of compact metric spaces with Y connected, and let j : Y → C (Y) be as in 2.1. If the map j f is essential then catf ≥ m. m m m In particular, if r(Y) = r and the map f# : [Y,C (Y)] → [X,C (Y)] r r is injective then catf ≥ r. Proof. Consider the diagram X f  y P (Y) −−p−m−→ Y −−j−m−→ C (Y). m m If catf < m then, by 1.4, there is a map g : X → P (Y) with f = p g, and hence m m j f is inessential. This is a contradiction. (cid:3) m 2.4. Theorem. Let Mn be a closed oriented connected n-dimensional PL manifold such that catM = dimM ≥ 4. Then r(M) = catM. Proof. Let MSPL (−) denote the oriented PL bordism theory. By the definition ∗ of r(X), it suffices to prove that (j ) : MSPL (M) → MSPL (C (M)) is a non- n ∗ ∗ ∗ n zero homomorphism. Hence, it suffices to prove that (p ) : MSPL (P (M)) → n ∗ n n MSPL (M) is not an epimorphism. Clearly, this will be proved if we prove that n [1 ] ∈ MSPL (M) does not belong to Im(p ) . M n n ∗ Suppose the contrary. Then there is a map F : W → M with the following properties: (1) W isacompact(n+1)-dimensionalorientedPLmanifoldwith∂W = M⊔V; (2) F|M = 1 , F|V : V → M lifts to P (M) with respect to the map p : M n n P (M) → M. n Without loss of generality we can assume that W is connected. Suppose for a moment that π (W,M) is a one-point set (i.e., the pair (W,M) is 1 simply connected). Then (W,M) has the handle presentation without handles of indices ≤ 1, see [St68, 8.3.3, Theorem A].By duality, the pair(W,V) has the handle presentation without handles of indices ≥ n. In other words, W ≃ V ∪e ∪···∪e 1 s where e ,... ,e are cells attached step by step and such that dime ≤ n − 1 for 1 s i every i = 1,... ,s. However, the fibration p : P (M) → M is n − 2 connected. n n Thus, F : W → M can be lifted to P (M). In particular, p has a section. But n n this contradicts 1.4. So, it remains to prove that, for every membrane (W,F), we can always find a membrane (U,G) with π (U,G) = ∗ and G|∂U = F|∂W. Here ∂U = ∂W = M ⊔V 1 andG : U → M. Westartwithanarbitraryconnected membrane (W,F). Consider a PL embedding i : S1 → intW. Then the normal bundle ν of this embedding is trivial. Indeed, w (ν) = 0 because W is orientable. 1 Since M is a retract of W, there is a commutative diagram 0 −−−−→ π (M) −−−−→ π (W) −−−−→ π (W,M) −−−−→ 0 1 1 1 (cid:13) F∗ (cid:13)  (cid:13) y π (M) π (M) 6 YULI B. RUDYAK where the top line is the homotopy exact sequence of the pair (W,M). Clearly, if F is monic then π (W,M) = ∗. ∗ 1 Letπ (W)begeneratedbyelementsa ,... ,a . Wesetg := F (a )a−1 ∈ π (W) 1 1 k i ∗ i i 1 where we regard π (M) as the subgroup of π (W). Then KerF is the smallest 1 1 ∗ normal subgroup of π (W) contained g ,... ,g . Now we realize g ,... ,g by 1 1 k 1 k PL embeddings S1 → intW and perform the surgeries of (W,F) with respect to these embeddings, see [W70]. The result of the surgery establishes us a desired membrane. (cid:3) 2.5. Corollary. Let M be as in 2.4, let X be a compact metric space, and let f : X → M be a map such that f∗ : Hn(M;π (C (M))) → Hn(X;π (C (M))) is n n n n a monomorphism. Then catf ≥ catM. Notice that, in fact, catf = catM since catf ≤ catM for general reasons. Proof. We set π = π (C (M)). It is easy to see that C (M) is simply connected. n n n Hence, by 1.3(iv) and the Hurewicz theorem, C (M) is (n−1)-connected. Thus, n [M,C (M)] = Hn(M;π). Let ι : C (M) → K(π,n) denote the fundamental class. n n Then f∗ can be decomposed as f∗ : Hn(M;π) = [M,C (M)] −f−→# [X,C (M)] −ι→∗ [X,K(π,n)]= Hn(X;π). n n Since f∗ is a monomorphism, we conclude that f# is. Thus, by 2.3 and 2.4, catf ≥ r(M) = catM. (cid:3) §3. The invariant r(M) and the Arnold conjecture Recall(seetheintroduction)thattheArnoldconjectureclaimsthatArn(M,ω) ≥ CritM for every closed symplectic manifold (M,ω). 3.1. Recollection. A flow on a topological space X is a family Φ = {ϕ },t ∈ R t whereeach ϕ : X → X isaself-homeomorphism andϕ ϕ = ϕ forevery s,t ∈ R t s t s+t (notice that this implies ϕ = 1 ). 0 X A flow is called continuous if the function X ×R → X,(x,t) 7→ ϕ (x) is contin- t uous. A point x ∈ X is called a rest point of Φ if ϕ (x) = x for every t ∈ R. We denote t by RestΦ the number of rest points of Φ. A continuous flow Φ = {ϕ } is called gradient-like if there exists a continuous t (Lyapunov) function F : X → R with the following property: for every x ∈ X we have F(ϕ (x)) < F(ϕ (x)) whenever t > s and x is not a rest point of Φ. t s 3.2. Definition (cf. [H88], [MS95]). Let X be a topological space. We define an index function on X to be any function ν : 2X → N ∪ {0} with the following properties: (1) (monotonicity) If A ⊂ B ⊂ X then ν(A) ≤ ν(B); (2) (continuity) For every A ⊂ X there exists an open neighbourhood U of A such that ν(A) = ν(U); (3) (subadditivity) ν(A∪B) ≤ ν(A)+ν(B); (4) (invariance) If {ϕ },t ∈ R is a continuous flow on X then ν(ϕ (A)) = ν(A) t t for every A ⊂ X and t ∈ R; (5) (normalization) ν(∅) = 0. Furthermore, if A 6= ∅ is a finite set which is contained in a connected component of X then ν(A) = 1. ON ANALYTICAL APPLICATIONS OF STABLE HOMOTOPY 7 3.3. Theorem. Let Φ be a gradient-like flow on a compact metric space X. Then RestΦ ≥ ν(X) for every index function ν on X. Proof. The proof follows the ideas of Lusternik–Schnirelmann. For X connected see [H88], [MS95, p.346 ff]. Furthermore, if X = ⊔X with X connected then i i RestΦ = XRest(Φ|Xi) ≥ Xν(Xi) ≥ ν(X). (cid:3) 3.4. Corollary. Let Φ be a gradient-like flow on a compact metric space X, let Y be a Hausdorff space which admits a covering {U } such that each U is open and α α contractible in Y, and let f : X → Y be an arbitrary map. Then RestΦ ≥ 1+catf. Proof. Given a subspace A of X, we define ν(A) to be the minimal number m such that A ⊂ U ∪···∪U where each U is open in X and f|U is inessential. It is easy 1 m i i to see that ν is an index function on X (normalizationfollowsfrom the properties of Y). But ν(X) = 1+catf, and so, by 3.3, we conclude that RestΦ ≥ 1+catf. (cid:3) 3.5. Theorem. Let (M,ω) be a closed connected symplectic manifold with I = ω 0 = I , and let φ : M → M be a Hamiltonian symplectomorphism. Then there c exists a map f : X → M with the following properties: (i) X is a compact metric space; (ii) X possesses a gradient-like flow Φ such that RestΦ = Fixφ; (iii) The homomorphism f∗ : Hn(M;G) → Hn(X;G) is a monomorphism for every coefficient group G. Proof. Thiscan beprovedfollowing[F89-2, Theorem 7]. (Notethat theformulation ofthistheoremcontainsamisprint: there istypedz∗[P] = 0, whileitmust betyped z∗[P] 6= 0. Furthermore, the reference [CE] in the proof must be replaced by [F7].) In fact, Floer denoted by z : S → P what we denote by f : X → M, and he showed that the homomorphism z∗ : Z = Hn(P) → Hn(S) is monic. He did it for Z-coefficients, but the proof for arbitrary G is similar. Also, cf. [H88] and [HZ94, Ch. 6]. In fact, Floer considered Alexander–Spanier cohomology, but for compact met- ric spaces it coincides with H∗(−). In greater detail, you can find in [Sp66] an isomorphism between Alexander–Spanier and Cˇech cohomology and in [Hu61] an isomorphism between Cˇech cohomology and H∗(−). (cid:3) Recall that every smooth manifold turns out to be a PL manifold in a canonical way, see e.g [Mu66]. 3.6. Theorem. Let (M,ω) be a closed connected symplectic manifold with I = ω 0 = I and such that catM = dimM. Then Arn(M,ω) ≥ CritM. c Proof. The case dimM = 2 is well known, see [F89-1], [H88], so we assume that dimM ≥ 4. Consider any Hamiltonian symplectomorphism φ : M → M and the corresponding data Φ and f : X → M as in 3.5. Then, by 3.5, Fixφ = RestΦ, 8 YULI B. RUDYAK and hence, by 3.4 and 2.5, Fixφ ≥ 1 + catM, and thus Arn(M,ω) ≥ 1 + catM. Furthermore, by a theorem of Takens [T68], CritM ≤ 1+dimM. Now, 1+catM ≤ CritM ≤ 1+dimM = 1+catM, and thus Arn(M,ω) ≥ CritM. (cid:3) 3.7. Example (catM > clM). LetM beafour-dimensional asphericalsymplectic manifold described in [MS95, Example 3.8]. It is easy to see that H1(M) = Z3. Furthermore, H∗(M) is torsion free, and so a2 = 0 for every a ∈ H1(X). Hence, clM = 3. However, catM = 4 because catV = dimV for every closed aspherical manifold V, see [EG57]. Moreover, for every closed symplectic manifold N we have cl(M ×N) < cat(M ×N) because cat(M ×N) = dimN +4 accoring to [RO97]. §4. The invariant r(M) and critical points Let X be a CW-space and let A,B be two CW-subspaces of X. Then for every spectrum E we have the cap-product ∩ : E (X,A∪B)⊗Πj(X,A) → E (X,B), i i−j see [Ad74], [Sw75]. Here Π∗(−) denotes stable cohomotopy, i.e., Π∗(−) is the cohomology theory represented by the sphere spectrum S. In particular, if D = Dk isthe k-dimensional disk then for every CW-pair (X,A) we have the cup-product ∩ : E (X ×D,X ×∂D∪A×D)⊗Πk(X ×D,X ×∂D) → E (X ×D,A×D). i i−k Let a ∈ Πk(D,∂D) = Z be a generator. We set t = p∗a ∈ Πk(X × D,X × ∂D) where p : (X ×D,X ×∂D) → (D,∂D) is the projection. 4.1. Lemma. For every CW-pair (X,A) the homomorphism ∩t : E (X ×D,X ×∂D∪A×D) → E (X ×D,A×D) i i−k is an isomorphism. In fact, it is a relative Thom–Dold isomorphism. Proof. If A = ∅ then ∩t is the standard Thom–Dold isomorphism for the trivial Dk-bundle (or the suspension isomorphism, if you want), see e.g. [Sw75]. In other words, for A = ∅ the homomorphism in question has the form ∩t : E (Tα) → i E (X×D) where Tα is the Thom space of the trivialDk-bundle α. Furthermore, i−k the homomorphism in question has the form E (Tα,T(α|A)) → E (X ×D,A×D). i i−k Considering the commutative diagram ··· → E (T(α|A)) −−−−→ E (Tα) −−−−→ E (Tα,T(α|A)) → ··· i i i ∩(t|A) ∩t ∩t    y y y ··· →E (A×D) −−−−→ E (X ×D) −−−−→ E (X ×D,A×D)→ ··· ON ANALYTICAL APPLICATIONS OF STABLE HOMOTOPY 9 with the exact rows, and using the Five Lemma, we conclude that the homomor- phism in question is an isomorphism. (cid:3) 4.2. Definition ([CZ83], [MO93]). Given a connected closed smooth manifold M, we define GH (M) to be the set of all C2-functions g : M ×Rp+q → R with the p,q following properties: (1) There exist disks D ⊂ Rp and D ⊂ Rq centered in origin such that + − int(M ×D ×D ) contains all critical points of g; + − (2) ∇g(x) points inward on M ×∂D ×intD and outward on M ×intD × + − + ∂D . − 4.3. Definition ([CZ83], [MO93]). Given g ∈ GH (M), consider the gradient p,q flow x˙ = ∇g(x). Let x•R denote the solution of the flow through x. We choose D+ andD− asin4.2,setB := M×D+×D− anddefineSg = Sg,B := {x ∈ B|x•R ⊂ B}. 4.4. Theorem (cf [MO93, 4.1]). For every function g ∈ GH (M), there is a p,q subpolyhedron K of intB such that S ⊂ K and critg ≥ 1+cat K. g B Proof. We set S = S . Because of of 4.2, S is a compact subset of intB. Fur- g thermore, S is an invariant set of the gradient flow x˙ = ∇g(x), and S contains all critical points of g. Given A ⊂ S, we define ν(A) = 1 + cat A. Clearly, ν is an B index function onS. Thus, by 3.3, ν(S) ≤ critg. Now, let V ,...V bea covering 1 ν(S) of S such that every V is open and contractible in B. Choose any simplicial trian- i gulation of B. Then, by the Lebesgue Lemma, there exists a simplicial subdivision of B with the following property: every simplex e with e∩ S 6= ∅ is contained in some V . Now, we set K to be the union of all simplices e with e∩S 6= ∅. Clearly, i 1+cat K ≤ ν(S), and thus critg ≥ 1+cat K. Finally, we can find K ⊂ intB B B because of the collar theorem. (cid:3) Let r(M) be the invariant defined in 2.1. 4.5. Theorem. For every function g ∈ GH (M), the number of critical points p,q of g is at least 1+r(M). In particular, critg ≥ 1+catM if M is aspherical. Proof. Here we follow McCord–Oprea [MO93]. However, unlike them, here we use certain extraordinary (co)homology instead of classical (co)homology. Let r := r(M). We choose K as in 4.4 and prove that cat K ≥ r. Consider the B Puppe sequence P (M) −p→r M −j→r C (M). r r Let e : M+ → C (M) be a map such that e|M = j and e maps the added point to r r the base point of C (M). Let h : C (M) → C be a pointed homotopy equivalence r r such that C is a CW-complex. We set E = Σ∞C and let u ∈ E0(M) be the stable r homotopy class of the map he : M+ → C. Then u 6= 0 since j is stably essential. r r We define f : K ⊂ B = M ×Rp+q −p−r−oj−ec−t−io→n M. 4.6. Lemma. If f∗u 6= 0 then cat K ≥ r. r B Proof. Since (p ,j ) is a Puppe sequence, p∗u = 0. Hence, the map f can’t be r r r r lifted to P (M), and therefore the inclusion K ⊂ B can’t be lifted to P (B). So, r r cat K ≥ r. The lemma is proved. B 10 YULI B. RUDYAK We continue the proof of the theorem. Let j : K ⊂ B be the inclusion. By 4.6, it suffices to prove that j∗ : E∗(B) → E∗(K) is a monomorphism. Notice that if Y is a CW-subspace of RN then there is a duality isomorphism E0(Y) ∼= E (RN,RN \Y) := E (RN ∪C(RN \Y)) −N −N see e.g. [DP84]. So, it suffices to prove that the dual homomorphism D(j∗) : E (RN,RN \B) → E (RN,RN \K) ∗ ∗ is monic for a certain (good) embedding B → RN. We have the following commutative diagram: E (RN,RN \B) E (RN,RN \B) ∗ ∗ h∼= D(j∗)   y y E (RN,RN \intB) −−−−→ E (RN,RN \K) ∗ ∗ ex∼= e′x∼=     E (B,∂B) −−−a∗−→ E (B,B \K) ∗ ∗ where all the homomorphisms except D(j∗) are induced by the inclusions. Here h is an isomorphism since the inclusion intB → B is a homotopy equivalence (the space B\ {collar} is a deformation retract of intB). Furthermore, e is an isomorphism since (B,∂B) and (RN,RN \intB) are cofibered pairs, while e′ is an isomorphism by Lemma 3.4 from [DP84]. So, D(j∗) is monic if a is. Since B \K ⊂ B \S, it ∗ suffices to prove that E (B,∂B) → E (B,B \S) is a monomorphism. ∗ ∗ Let B = M × ∂D × D , and let B = M × D × ∂D . Furthermore, + + − − + − let A+ := {x ∈ B (cid:12) x•R− ∈ B} and let A− := {x ∈ B (cid:12) x•R+ ∈ B}. Then (cid:12) (cid:12) B ∩A = ∅ = B ∩A , and so there arethe inclusions i : (B,B ) → (B,B\A ) + − − + + + − and i : (B,B ) → (B,B \ A ). It turns out to be that both i and i are − − + + − homotopy equivalences, [CZ83, Lemma 3]. Let t ∈ Πm(B,B ) be the class as in 4.1, and let t′ := ((i )∗)−1(t). Since − − S = A ∩A , we have the commutative diagram + − E (B,∂B) −−−−→ E (B,B \S) i i ∼=∩t ∩t′   y y ∼= E (B,B ) −−−−→ E (B,B \A ) i−q + i−q − wheretheleftmapisanisomorphismby4.1andthebottommapistheisomorphism (i ) . (Generally, (B,B \ S) is not a CW-pair, but nevertheless in our case the + ∗ map ∩t′ is defined, see [DP84, 3.5].) Thus, the top homomorphism is injective. Finally, if M is aspherical then catM = dimM, EG57], and so r(M) = catM by 2.4. (cid:3) References [Ad74] J.F. Adams: Stable Homotopy and Generalised Cohomology. The Univ. of Chicago Press, Chicago 1974