Lusternik-Schnirelmann invariants in proper homotopy theory

PACIFIC JOURNAL OF MATHEMATICS Vol. 153, No. 2, 1992 LUSTERNIK-SCHNIRELMANN INVARIANTS IN PROPER HOMOTOPY THEORY R. AYALA, E. DOMINGUEZ, A. MARQUEZ, AND A. QUINTERO We introduce and study proper homotopy invariants of the Lusternik-Schnirelmann type, p-cat (-), p-Cat(-), and cat e(-) in the category of Γ2-locally compact spaces and proper maps. As an application, Rn (n Φ 3) is characterized as (i) the unique open manifold X with p-Cat(ΛΓ) = 2, or (ii) the unique open manifold with one strong end and p-cat( c) = 2. Introduction. The category cat(JΓ) of a space X in the sense of Lusternik and Schnirelmann (L-S category) is the smallest number k such that there exists an open covering {X\, ... , Xk} of X for which each inclusion Xj c X is nullhomotopic in X. This concept was introduced by the quoted authors in their studies on calculus of variations [16] and they used it as a lower bound for the number of critical points of a differentiable real function on a manifold. The basic work on the homotopical significance of cat is due to Borsuk (see [5]). Borsuk's work was continued by Fox [10]. Here we present the definition and the basic properties of a new numerical topological invariant for Γ2-locally compact spaces which agrees with the notion of L-S category for ^-compact spaces. This invariant, denoted p-cat(X), is called the proper L-S category of X and turns out to be a proper homotopy invariant of X. Hence, p-cat(X) is a finer invariant than cat(X). In [10] several generalizations of L-S category are suggested. More explicitly, a general notion of L-S si -category with respect to a class si of spaces has been developed by Puppe and Clapp in [6]. Our work shares some common points with [6] but does not fit into the notion of L-S si -category since we entirely deal with proper maps instead of ordinary continuous maps. Another generalization of L-S category has been given in [1], where L-S category for pro-objects in pro-c%/* is defined. This idea is related to proper L-S category by the Edwards-Hastings embedding (see [8]) which provides a close link between proper homotopy theory and homotopy in 201 202 R. AYALA, E. DOMINGUEZ, A. MARQUEZ, AND A. QUINTERO We shall work entirely in the category <Poo of non-compact T2locally compact σ-compact spaces and proper maps. Notice that any ^-compact space X can be regarded in ^oo as the wedge Iv/, where / will stand for the half-line [0, oo). We recall that a proper map (p-map) is a continuous map /: X —• Y such that f~ι(K) is compact for each compact K c Y. Proper homotopy (p-homotopy), proper deformation (p-deformation), etc. can be defined in the natural way. The symbols "~", "~p" and "=" stand for homotopy equivalence, p-homotopy equivalence and homeomorphism respectively. Finally we also recall the notion of end in proper homotopy. A Freudenthal end of X e^oo is an element of the inverse limit ^(X) = lim πo(X-K) where K ranges over the family of compact subsets of X and πo stands for the set of connected components. The topology of X can be enlarged to a topology on IuF(I) in such a way that ^(X) turns out to be homeomorphic to a closed set of the Cantor set (see [11] for details). It is easy to check that any p-map /: X —• Y induces a continuous map f*: ^(X) —• &~(Y) such that /j= is a homeomorphism when / is a p-homotopy equivalence. 1. Basic properties. In [8] the following lemma is proven, 1.1. LEMMA ([8; 6.3.5]). Any space X in φoo admits a p-map r: X —• / unique up to p-homotopy. 1.2. DEFINITION. A closed subset C C X is said to be properly deformable to / in X if there exists a diagram in φoo C ^X commutative up to p-homotopy. Notice that we may use r as the restriction of a p-map X —• / given by Lemma 1.1. 1.3. DEFINITION. Given a space X in φoo, A c X is said to be properly categorical (p-categorical) in X if there is a closed neighbourhood of A properly deformable to / in X. An open covering {Ua} of X is said to be p-categorical if each Ua is p-categorical in X (i.e. Ua is properly deformable to / in X). The p-category of X, p-cat(X), is the least number n such that X admits a p-categorical open covering with n elements. If no finite p-categorical covering exists then ρ-cat(Z) = oo. LUSTERNIK-SCHNIRELMANN INVARIANTS 203 LEMMA 1.4 [4; 3.4]. Let P be a locally compact metrizable space and Q a locally compact ANR. Suppose X is a closed subset of P and f', g: X —• Q are p-homotopic maps. If f ,~g are extensions off and g respectively, there exists a closed neighbourhood U of X such that f\U and ~g\U are p-homotopic. Hence, if we deal with locally finite polyhedra we have 1.5. PROPOSITION. Given a locally finite polyhedron P, ρ-cat(P) is the smallest number n such that P can be covered by n p-categorical subpolyhedra. Proof. Given a p-categorical open covering {Wi} let Hι: TFZ x / -* P be a p-deformation of Wi. Since P is ANR by Lemma 1.4 there is a closed neighbourhood Ω; of Wι and a p-extension Hι: Ω; xl —• P. Now by [23; 3.5] we may take a closed subpolyhedron P/ such that Wi C Pi C intΩ/ and thus {Pi} is a covering of P with P/ pcategorical in P. Conversely, if P is covered by m p-categorical subpolyhedra we may use regular neighbourhoods to obtain a p-categorical open covering with m elements. The following properties of p-cat(-) are straightforwardly checked: 1.6. PROPOSITION, (i) cat(X) < ρ-cat(X). If X is compact the equality holds. (ii) p-cat(X) = 1 if and only if X ~p J. (iii) Let f: X —>Y and g\Y-*X bep-mapssuch that fg ~p idy. Then ρ-cat(Γ) < ρ-cat(X). In particular, ρ-cat(-) is a p-homotopy invariant. (iv) If A is a p-retract of X then p-cat(^) < p-cat(X). (v) If A and B are closed and X = int^Uinti?, then p-cat(X) < p-cat(Λ) + p-cat(£). The next proposition gives us an elementary relation between the set of Freudenthal ends of X and ρ-cat(X). 1.7. PROPOSITION. Given a space X in φoo we have card(^(X)) < ρ-cat(X). Proof. If p-cat(X) = oo there is nothing to prove. Otherwise, if C\, Cι, ... , Cm is a p-categorical closed covering of X, the natural inclusions Kj\ Cj —• X induce continuous maps kj\ 204 R. AYALA, E. DOMINGUEZ, A. MARQUEZ, AND A. QUINTERO . Moreover, it is easy to check that ^(X) = U Im ^/* and since Cj is p-deformable to / we get that each Im fc,* is an one-point set. Hence card(^(X)) < p-cat(X). 1.8. EXAMPLES, (a) p-cat(Rπ) = 2 > 1 = cat(RΛ). (b) For any Γ2-compact space X, cat(X) = p-cat(X x /) = ρ-cat(X V /). (c) If Sn is the space obtained from / by attaching one copy of Sn at each natural number n e / we have p-cat(*SΛ) = 2. (d) If X is a space in <Poo and r: X -> / is a p-map we can define (up to p-homotopy) the proper cone CPX oϊ X as the push-out of the diagram X x I <— X -^ J and the proper mapping cone Cpf of a p-map / is defined in the natural way (see [2] for details). If /= r, Cpr turns to be the proper suspension Σp X of X. As in the ordinary case it follows from 1.6(iii) and (iv) p-cat(Q/) < p-cat(Γ) + 1. In particular p-cat(^ X) < 2. (e) As a consequence of (d), p-cat(X) < dimX+1, if X is a locally finite CW-complex with only one Freudenthal end. (f) The notion of categorical sequence due to Fox (see [10]) can be translated into proper terms. Namely, given a space X in φ^ a sequence of open sets V\ c c Vn — X for which each difference Vι - Vι_\ is p-categorical (Vo = 0), is called a p-categorical sequence. It is easy to check that p-catX < n if and only if X admits a pcategorical sequence of length n . By using this result, one shows the inequality p-cat(X xF)< p-cat(ΛT) + p-cat(Γ) - 1. (g) The inequality max{catX, cat Y} < cat(X x Y) holds in ordinary L-S category but not in proper L-S category as the following example shows. Let X = R2 and Y = J then ρ-cat(X x Y) = 1 < p-cat(R2). The following definition provides a new proper L-S invariant which is the translation of Ganea's strong L-S category into proper homotopy (see [15]). DEFINITION 1.9. given a space X in φ^, the strong L-S category of X is the smallest integer p-Cat(X) such that there exists a space Y in *Poo p-homotopically equivalent to X which can be covered LUSTERNIK-SCHNIRELMANN INVARIANTS 205 by p-Cat(X) closed sets each with the same p-homotopy type as /. Obviously p-cat(X) < p-Cat(X). The relation between p-Cat(X) and the 1-LC at oo condition is given in the following theorem. We firstly recall that a space X in φoo is 1-LC at oo if given a sequence of compact subsets {Lj} with Lj C intLj+ι and X = \JLj, the inverse sequence (1.9.1) πι(X-Lι,a{tx))<^-πι{X-L2,a{t2))X is trivial, where a: J —• X is a proper map with a{[tι?, oo)) C X - Li (/ > 1) and θj is the inclusion induced homomorphism followed by the change of basepoint isomorphism given by α|[ί, , ί, +i]. THEOREM 1.10. Let X be a space in ty^ with one Freudenthal end. If p-Cat(X) = 2, X is l-LC at oo if and only if the inverse sequence (1.10.1) H^X - Lό+-Hi(X - I*) < *-Hι(X-Ln)< is trivial. This theorem is an immediate corollary of PROPOSITION 1.11. Let X be as in Theorem 1.10. Assume that X can be covered by two one-ended p-categorical closed subsets U and V. Then X is 1 - LC at 00 if and only if the inverse sequence (1.10.1) is trivial. Proof. Let {£/, V) be a p-categorical open covering with U and V one-ended. We claim that UnV also is one-ended or equivalently that the inverse sequence (1.11.1) HoiUnV-Li)* <r-HQ(UnV-Ln) is trivial. In the commutative diagram Hχ{X-Ln) « Hχ{X-Lnχ) H0(U nV - Ln) H0(U-Lnι)QHQ(V-Lnι) « H0(U-Ln where the vertical arrows are provided by the respective MayerVietoris sequences. We may choose nι > n\ > n such that the upper 206 R. AYALA, E. DOMINGUEZ, A. MARQUEZ, AND A. QUINTERO and lower horizontal arrows are trivial. This implies that the composition of the morphisms (1) and (2) is trivial and so the inverse sequence (1.11.1) is trivial. We have the following facts: (i) Since U and V are p-categorical, we may choose n?>> n such that any loop either in U - Ln orin FLn^ is null-homotopic in X-Ln. (ii) On the other hand, since U n V is one-ended there is n4 > n-$ such that all the connected components of U Π V - Ln^ are included in the same connected component of U Π V - Ln^. Given any loop /: (/, {0, 1}) —> (X-Ln^, *), by Lebesgue's Lemma we may find a partition O = ίo<ίi <•••<*„ = 1 such Λat f\[ti> U~\] is included either in U-Ln^ orin V -Ln^ Therefore, by (ii), / is a loop in X-Ln^ that can be expressed as a product of loops either in U - Ln^ or in V - Ln^ and each factor is null-homotopic in X - Ln by (i). Thus the morphism LnJ ^ πx(X - Ln) is trivial and X is 1-LC at oo. COROLLARY 1.12. Let X be a homologically trivial open n-manifold (n > 3) with p-Cat(JΓ) = 2. Then X is \-LCat oo. Proof. As n > 3, Poincare-Lefschetz Duality arguments (see [7; 3.2]) show that the inverse sequence (1.10.1) is semistable (i.e. satisfies the Mittag-Leffler condition) and its inverse limit is trivial. Then by [18; Π.6.2.2] the inverse sequence (1.10.1) is trivial. Now the result follows from Theorem 1.11. 2. Proper L-S category and L-S category in pro-Top. We recall that cat(/) < n for a continuous map / from X to Y, if there is an open covering {V\, ... , Vn} of X such that f\Vj is homotopically trivial for 1 < j < n . In [1] a notion of L-S category for inverse systems is defined. Namely, given an inverse systems χ = {Xa Paβ} > the L-S category of χ, cat(χ), is < n if for each a there exists β > a such that C3t(paβ) < n. If pro S0/1, is the category of topological inverse systems and promorphisms, Edwards and Hastings (see [8]) have proven that there exists a natural functor ε: φoo —> pro-<5ζ^, ε(X) being the inverse LUSTERNIK-SCHNIRELMANN INVARIANTS 207 system {U\ <— U2 *— ••} where £4 = X - Q and Q is an increasing sequence of compacts with Q C int Q+1 and X = |J Q . A relation between ρ-cat(X) and cat(ε(X)) is given by 2.1. THEOREM. 7f X is a space in φ^, ρ-cat(X) > cat(ε(X)). Proof. If {Wγ, ... ,Wn} is a p-categorical open covering of X, we consider the covering {WjΠUk} of U^. Let r7: PF7 —• / and α;:/->Ibe p-maps such that there are p-homotopies W: WjXl —• X between α7 o r7 and the inclusion W7 c X. For each k there exists s(k) > k such that Hj(Wj Π C/J(fc) x /) Q Uk. But since / is contractible it follows that Us^ is nuU-homotopic in each Wj n J75 So cat(β(JΓ)) < p-cat(X). 2.2. REMARKS, (a) We now consider the category (pro-,%^, whose objects are arrows f:χ—>A where χ is an object in and A is a space regarded as the constant inverse system. Morphisms are pairs (φ, h): where φ: χ -^ ξ is a pro-morphism and h: A -+ 5 is a continuous map compatible with 0 via the bonding maps. Edwards and Hastings proved that ε: <Poo —• (ρro-,%^, 3fc/ι) given by e(X) = X «— C/i *- C/2 is a full embedding (see [8]). We can also prove that cat(ε(X)) < p-cat(X) in this case. (b) There exist spaces with cat(ε(X)) < p-cat(X). Indeed, let X be the punctured torus. Then p-cat(X) = 3 according to Corollary 3.3 but obviously cat(ε(X)) = cat^S1) = 2. 2.3. PROPOSITION. If M is a compact connected triangulable manifold with boundary and W = M - dM then cat(<9M) < p-cat(fΓ) < cat(9Jl/)+cat(Af). Furthermore, if M is contractible then p-cat(W^) = cat(<9M). Proof. The inverse system ε(W) in (pro-^/^, 9*/ι) is Now we have cat[/: dM x [0, 00) -• W] < cat(dM x [0? 00)) = cat(0Λ/) and cat(ε(X)) = max{cat(/), cat(9M)} = cat(ΘM). So, by Remark 2.2(a) caX(ΘM) < p-cat(FF). On the other hand if we take a copy of M, M' C W then p-cat( W) < p-cat(M; V /) + v-cdi\{dMr x J) = cat(M) + cat(ΘM) by 1.6(viii) and 1.8(b). 208R. AYALA, E. DOMINGUEZ, A. MARQUEZ, AND A. QUINTERO If we assume that M is contractible, we get cat(<9M) < p-cat(FΓ) < cat(dM) + 1. But actually p-cat(W) < caX(dM). Indeed, if {Wj}j<n is a categorical covering of dM, we may assume that W \ is a subpolyhedron (see Proposition 1.5). By [22; 6.30] we extend the proper deformation of Wx x J into {*} x / to a proper deformation of Wι x J UM. Thus {W x J uM,W2x J, ... , Wn x J} is a pcategorical covering of W. The invariant catε(X) can be used to give some results on the behaviour of the inverse sequence (2.3.1) πx(X - Kx) <- π1 (X - K2) < <-πx(X - Kn) < where {Ki} is a sequence of compact subsets with Kt C mtKi+x, and 2.4. THEOREM. Let X be a locally finite polyhedron with one Freudenthal end and catε(X) = 2. Assume that the inverse sequence of abelian groups HX{X - Kx) - K2)Hx{X-Kn) is trivial. Then the inverse sequence (2.3.1) is pro-isomorphic in to an inverse sequence of finitely generated groups. THEOREM 2.5. Under the hypotheses of Theorem 2.4, X is ί-LC at oc if and only if the inverse sequence (2.3.1) is semistable, i.e. \imι πx(X - Kj) = *. Proof of Theorem 2.4. We may choose the sequence {Kj} with Uj = X - Kj subpolyhedron of X. Up to pro-isomorphism we may replace nx{X-Kj) by Gj = πλ(Uj) in (2.3.1). By using the 1-skeleton of X, it is easy to check that there is a commutative diagram F(L2) (2.4.1) where F(Ln) denotes the corresponding free group of basis Ln , the differences Ln - LΛ+1 are finite, and the bonding morphisms are the natural inclusions. LUSTERNIK-SCHNIRELMANN INVARIANTS 209 Since the inverse sequence Gf ^Gf^Qf, <_ Gf < is trivial, by abelianizing (2.4.1) we readily check that G*b is finitely generated for n > 1. Now, since catε(X) = sup{cat[/« : Un - Un^]} = 2 we get from [15; 7.3] that for any n > 2, there is μn such that Un > Un^VUn^ is commutative up to homotopy. This diagram induces a commutative diagram Gn • Gn-\ * Gn-\ (2.4.2) j/.. [in_u Gn-\ • Gn-\ x ^Λ-I where /„* induces the natural epimorphism a*b —• (α, δ). From the Kurosh Subgroup Theorem (see [17]) the group l~\uΔim{Gn) is a free group. Therefore, μm : (?„ -• Im//^* is an epimorphism onto a free group and so (Imμn*)ab is a finitely generated free abelian group. Hence, Im/^* is a finitely generated free group. On the other hand, Δ is injective and so the commutativity of (2.4.2) yields a natural epimorphism Imμrt* -» Im/W*. Thus, \vs\im is a finitely generated group. It is a well-known fact that (2.3.1) is pro-isomorphic to the inverse sequence (2.4.3) Imiu ^-Im/2*^ ^- Im/„*<—••-. Proof of Theorem 2.5. If in addition lim1 G}• = *, we know that the inverse sequence (2.3.1) satisfies the Mittag-Leffler condition (see [18; p. 174]) and we may assume that the bonding morphisms in (2.4.3) are onto. We shall denote Im im by Hn .