Minimal covers of open manifolds with half-spaces and the proper L-S category of product spaces

Minimal covers of open manifolds with half-spaces and the proper L-S category of product spaces M. C´ardenas F.F. Lasheras A. Quintero Abstract Classical results about the Lusternik-Schnirelmann category of product spaces have their analogues in the category of proper maps. By comparing the proper Lusternik-Schnirelmann category of an open manifold Xwith the smallest number of closed half-spaces needed to cover X, we obtain a proper analogue of Singhof’s theorem on the category of X×S1. Introduction The Lusternik-Schnirelmann category (L-S category) cat(X) of a space Xis the smallest number ksuch that there exists an open cover {X1,...Xk}for which each inclusion Xj⊆Xis nullhomotopic in X. The L-S category turns to be a homotopy invariant of the space X. See [9] for a survey on L-S category. Ordinary homotopy invariants do not take care of the behaviour of spaces at infinity. So, “proper” homotopy invariants are needed for the study of non-compact spaces. Proper analogues of Lusternik-Schnirelmann numerical invariants were introduced in [2] and [3]. The crucial point in the definition of the proper L-S category is the fact that the half-line [0,∞) plays in proper homotopy part of the role played by the point in ordinary homotopy. The parallelism between both roles breaks down on the Received by the editors January 2001. Communicated by Y. F´elix. 1991 Mathematics Subject Classification : 55M30, 55P57, 57Q30(Secondary). Key words and phrases : proper map, Lusternik-Schnirelmann category, open manifold, proper collapse. Bull. Belg. Math. Soc. 9 (2002), 419–431 420 M. C´ardenas – F.F. Lasheras – A. Quintero fibration-side of homotopy theory since product and fibration projections are not always proper maps. However, the class of proper maps still keeps the basic properties on the cofibration-side of homotopy theory which lead to a “combinatorial” homotopy in the sense of J.H.C. Whitehead [19]; see [1]. This paper continues the study of the proper L-S category of non-compact spaces. Here, we focus our interest on the behaviour of the proper L-S category on product spaces. More explicitly we work out the proper L-S category of spaces of the form W×Skand M×Rkwith Wand Mopen and closed manifolds respectively. These results can be regarded as proper analogues of a theorem due to Singhof [17] in relation with a classical question posed by Ganea; see Remark 3.9. In this paper we follow closely the combinatorial proof of Singhof’s theorem given by Montejano [12] since proper collapses [16] and a suitable Engulfing Theorem for open manifolds [10] are available; see Appendix A for details. This way we can compare the proper L-S category of an open manifold Xwith the smallest number of properly embedded half-spaces needed to cover X. Several new improvements of Singhof’s theorem have recently appeared in the literature; see [14] and [18]. However, these results depend heavily on the study of sections of certain fibrations. Since the category of proper maps provides an example of homotopy theory with good properties only on the cofibration-side, it seems to be interesting to look for alternative proofs of those results in the cofibration realm of homotopy theory. We shall deal with the category Pof locally compact σ-compact Hausdorff spaces and proper maps. Recall that a proper map (p-map) is a continuous map f:X→Y such that f−1(K) is compact for each compact subset K⊆Y. All maps and homotopies are assumed to be proper unless stated otherwise. We use the symbol ≃for proper homotopy and P/≃stands for the corresponding homotopy category. Furthermore, the symbol R+denotes the half-line [0,∞) and more generally Rn +denotes the upper n-dimensional half-space {(x1, x2, . . . , xn)∈ Rn;xn≥0}. 1 Proper L-S category This section contains some technical observations about the notion of proper L-S category which will be used later. Recall that, given a space Xin P, a system of ∞-neighbourhoods of Xis a decreasing sequence {Wj}of subsets of Xsuch that the closures Kj=X−Wjform an increasing sequence of compact subsets with Kj⊆intKj+1 and X=∪intKj. Remark 1.1. Given a locally finite sequence of pairwise disjoint compact subsets Ci⊆X(i≥1), it is possible to choose the compact sets Kjabove satisfying Ci∩Fr Kj=∅for all i, j ≥1. For this we consider the compact set ˆ L1= K1∪(∪{Ci;Ci∩K16=∅}). By using the normality of Xwe find a compact set L1⊆Xwith ˆ L1⊆int L1and L1∩Ci=∅whenever Ci∩K1=∅. Hence Ci∩ Fr L1=∅for all i≥1. Then we pick n1such that L1⊆int Kn1and we set ˆ L2=Kn1∪(∪{Ci;Ci∩Kn16=∅}). Similarly we find a compact set L2⊆Xwith ˆ L2⊆int L2and Ci∩Fr L2=∅for all i≥1. We proceed inductively to obtain an increasing sequence of compact sets Lj⊆int Lj+1 with the required properties. Minimal covers and the proper L-S category of product spaces 421 In the category Pthe constant map X→ {p}is not defined if Xis not compact. Notwithstanding, the role of the point is played partially in Pby the half-line R+ since for any space Xin Pthere always exists a proper map r:X→R+. Moreover the map ris unique up to p-homotopy. We shall briefly describe the construction of such a map r. If {Uj}j≥0is a system of ∞-neighbourhoods of Xwith U0=X, the Tietze Extension Theorem yields continuous maps rj=Uj−Uj+1 →[j, j + 1] with rj(FrUj+1) = j+ 1, rj(F rUj) = j. It is now clear that the maps rjdefine a proper map r:X→R+. Definition 1.2. A proper map r:R+→Xis called a ray in X. Moreover a proper map f:X→Yis properly inessential if there exists a commutative diagram in P/≃ X * Y ? ∗ HHH Hj α f β (1) where ∗is either R+or the one-point space {p}. Notice that ∗={p}only if Xis compact. Given a space Xin Pa closed subset A⊆Xis called inessential if the inclusion i:A→Xis an inessential map. A set A⊆Xis called properly categorical if A⊆Uwith Uan open set in Xand the closure ¯ Uis inessential. Moreover an open cover {Uα}of Xis said to be properly categorical if each ¯ Uαis an inessential set. The proper Lusternik-Schnirelmann category of X,p−cat(X), is the least number nsuch that Xadmits a properly categorical open cover V={U1, U2, . . . , Un}with nelements. In case Xis compact p−cat(X) = cat(X) is the ordinary L-S category of X. Remark 1.3. As in ordinary homotopy theory closed covers can also be used to define the proper L-S category of ANR-spaces. Furthermore for polyhedra in Pone can use covers consisting of subpolyhedra in the definition of proper L-S category; see [2] and [3] for details. AFreudenthal end of a space Xin Pis an element of the inverse limit F(X) = lim ←− U(Wj). Here {Wj}is a system of ∞-neighbourhoods of Xand U(−) stands for the family of unbounded connected components. A subset A⊆Xis termed unbounded if its closure ¯ Ais non-compact. If F(X) = {∗} then Xis said to be one-ended. Remark 1.4. Notice that a properly categorical set A⊆Xcannot contain sequences of points defining two different Freudenthal ends. Indeed, it is immediate to check that any ray r:R+→Xdefines a unique Freudenthal end. As a consequence, for any system of ∞-neighbourhoods {Wj}j≥1of X, there exists j0such that for each j≥j0there is at most one component Uj∈ U(Wj) with A∩Wj6=∅; moreover these components form a nested sequence Uj0+1 ⊇Uj0+2 ⊇... which defines a unique Freudenthal end of X. 422 M. C´ardenas – F.F. Lasheras – A. Quintero Example 1.5. A lower bound for the proper L-S category of spaces with cylindrical ends can be easily obtained as follows. Let Xbe a space in Pwith mcylindrical ends; that is, there exists a relatively compact open set A⊆Xsuch that X−Ahas munbounded components Wj(1 ≤j≤m) homeomorphic to cylinders Zj×[0,∞) where each Zjis compact. Then the inequality p−cat(X)≥k= m X j=1 cat(Zj) holds. Indeed, given a properly categorical open cover {Us}1≤s≤nof X, by Remark 1.4, there is a compact subset B⊆Xsuch that A⊆B, and each non-empty difference Us−Bis contained in exactly one component of X−B. Therefore, if n≤k−1 then at least one component Ωj0⊆X−Bis covered by less than cat(Zj0) differences Usi−B(1 ≤i≤q < cat(Zj0)). Moreover, we can assume without loss of generality that Ωj0is of the form Zj0×[t0,∞). As each Usiis properly categorical there exists t≥t0such that the deformation of Usi−(Zj0×[t, ∞)) occurs inside Ωj0. From this fact, we easily derive that the intersections Usi∩(Zj0× {t+ 1}) (1 ≤i≤q < cat(Zj0)) provide an ordinary categorical cover of Zj0× {t+ 1}which is a contradiction. The proper homotopy class of the map βin diagram (1) is unique. However for ∗=R+the proper homotopy class of the map αin diagram (1) depends on the set of proper homotopy classes [R+, Y ]. Each class [α]∈[R+, Y ] is called a strong end of Y. When [R+, Y ] consists of only one element we say that Yis strongly one-ended. Clearly each strong end defines a Freudenthal end. More precisely there exists an onto map [R+, Y ]→ F(Y). It is obvious that for strongly one-ended spaces all rays αican be chosen to be the same. Next proposition shows that the same holds for one-ended polyhedra. Proposition 1.6. Let Xbe a connected one-ended polyhedron in Pwith p−cat(X) = n. Then there exists a properly categorical (polyhedral) cover {V1, V2, . . . , Vn}of X such that in the diagrams Vi  * X ? R+ HHH Hj αi ki r (2) all rays αi(i≤n) define the same strong end. Proof. Let {U1, U2,...,Un}be a properly categorical cover of Xconsisting of subpolyhedra; see Remark 1.3. Moreover, it is well known that any ray α:R+→X is properly homotopic to a ray embedded in the 1-skeleton of X. In case all connected components of some Uiare compact, we next show that αiin diagram (2) can be chosen to be arbitrary. Indeed, since all components C⊆Uiare compact, one readily checks that there is a proper deformation Hwhich contracts each Cto a point in X. Furthermore, by using the proper homotopy extension property we can replace Hby a new proper deformation H0which shrinks each C to a point xC∈Crelative xC. Finally, given any ray R⊆X, we use that Xis Minimal covers and the proper L-S category of product spaces 423 one-ended to move each point xCto some point yC∈Rvia a proper homotopy {xC} × I→X. By using the previous arguments, we can assume without loss of generality that some element of the cover {U1, U2,...,Un}has at least one unbounded component. Let U1be such an element. We assume inductively that Uican be properly deformed to a ray R⊆Xfor i≤k. Now we consider Uk+1. By the arguments above we can assume that Uk+1 is not compact. Moreover we can also assume that U1∩Uk+1 6=∅ is non-compact as well; otherwise we use the fact that Xis one-ended to join U1to Uk+1 with a locally finite sequence of pairwise disjoint arcs which we add to U1. In addition, if the intersection U1∩Uk+1 contains a ray R0then the proper deformation of U1to Ryields that both rays Rand R0represent the same strong end of X, and so Uk+1 can be properly deformed to the ray R. It only remains to consider the case when U1∩Uk+1 consists of a locally finite sequence {K1, K2,...}of compact components. In such a case one finds two pairwise disjoint families {A1, A2,...}and {B1, B2,...}of compact subpolyhedra1of Uk+1 with Uk+1 = (∪∞ i=1Ai)∪(∪∞ t=1Bt) and Ki⊆intUk+1 Aifor i≥1. Also one chooses a pairwise disjoint sequence {L1, L2,...}of compact subpolyhedra of U1with Ki⊆ intU1Lifor all i≥1. Finally we take a locally finite sequence of pairwise disjoint arcs γi⊆U1joining Kito Z=U1− ∪∞ i=1Liwith intγi⊆intU1Li. Then we replace U1and Uk+1 by e U1= (∪∞ i=1Ai)∪Z∪(∪∞ j=1γj) and e Uk+1 = (∪∞ t=1Bt)∪(∪∞ j=1Lj) respectively. Now by the arguments above, we can assume in addition that the disjoint union ∪∞ s=1Asis properly deformed to the discrete set D=∪∞ j=1(γj∩Kj) relative to D. From this one easily shows that e U1can be properly deformed to the ray R. The set e Uk+1 is a locally finite disjoint union of compact subpolyhedra, and so it can be properly deformed to the ray Ras well. Since U1∪Uk+1 =e U1∪e Uk+1 we can replace the cover {U1, U1,...,Un}by {e U1, U2,...,Uk,e Uk+1, Uk+2, . . . , Un}. After a finite number of steps we get a properly categorical (polyhedral) cover of Xsuch that all elements in it can be properly deformed to the ray R. 2 Proper L-S category of product spaces For ordinary L-S category the following formula is well known for product spaces; see [5] max{cat(X), cat(Y)} ≤ cat(X×Y)≤cat(X) + cat(Y)−1 (∗) In this section we study the proper analogue for this formula. Recall that the proper L-S category p−cat(−) is a proper homotopy invariant; in fact if f:X→Yand g:Y→Xare proper maps with gf ≃idXone has p−cat(X)≤p−cat(Y). In particular, if Yis compact the projection p1:X×Y→Xyields 1One finds these families as follows. Let {Lj}be an increasing sequence of compact subpolyhedra in Uk+1 with Ki∩F r Lj=∅for i, j ≥1; see Remark 1.1. Then we pick n1< n2< . . . and we choose any locally finite family of pairwise disjoint compact subpolyhedra Aiwith Ki⊆intUk+1 Ai for i6=njand Knj∪F r Lj⊆intUk+1 Anjfor all j≥1. It is immediate to check that the closure Uk+1 − ∪∞ i=1Ai=∪∞ t=1Btis a locally finite union of pairwise disjoint compact subpolyhedra. 424 M. C´ardenas – F.F. Lasheras – A. Quintero p−cat(X)≤p−cat(X×Y) (3) However, Example 3.10 below shows that inequality (3) does not hold if Yis not compact. Concerning the right-hand side inequality in (*) we can prove the following proposition; compare [5] Proposition 2.1. Let Xand Ybe two connected polyhedra in P. Assume Xhas at most one end in case Yis compact. Then p−cat(X×Y)≤p−cat(X)+p−cat(Y)−1. Proof. Assume Xand Yare not compact. Then by ([11];2.2) X×Yis strongly one-ended. Let p−cat(X) = nand p−cat(Y) = mand let {U1, . . . , Un}and {W1,...,Wm}be families of inessential subpolyhedra whose interiors cover Xand Yrespectively. By using regular neighbourhoods we find new properly categorical covers {e Ui}and {f Wj}consisting of closed subpolyhedra with Ui⊆int e Uiand Wj⊆ int f Wj. Since R+×R+has the proper homotopy type of R+it is clear that each product e Ui×f Wjis inessential. Moreover, since X×Yis strongly one-ended we have a commutative diagram in P/≃ e Ui×f Wj  * X×Y ? R+ HHH Hj α r (4) with the same αfor all i, j. We now consider the unions Ai=∪i k=1Uk,i≤n, and Bj=∪j h=1Wh,j≤m. Moreover let C1⊆C2⊆ · · · ⊆ Cn+m−1=X×Ybe the increasing sequence of closed sets Cs=∪i+j=s+1Ai×Bj,1≤s≤n+m−1. From this we can write X×Y=∪n+m−1 s=1 Dsfor the sets D1=C1and Ds=Cs−Cs−1, s≥2. Moreover we have Ds=∪{Ei×Fj;i+j=s+ 1}where Ei=Ai−Ai−1and Fj=Bj−Bj−1. We set A0=B0=∅. Clearly the sets Ei×Fj⊆Dsare pairwise disjoint. Moreover they are pairwise separated; that is, we have (Ei×Fj)∩(Ei0×Fj0) = (Ei×Fj)∩(Ei0×Fj0) = ∅ if i+j=i0+j0. Since X×Yis hereditarily normal we find for each s≤n+m−1 a pairwise disjoint family of open sets Vs={Vj i}i+j=s+1 with Ei×Fj⊆Vj i⊆ int e Ui×intf Wj; see ([4];2.1.7). In particular the union V=∪m+n−1 s=1 Vsis an open cover of X×Y, and the paracompactness of X×Yprovides us with a finite open refinement of V,Z=∪n+m−1 s=1 {Zj i}i+j=s+1 with Zj i⊆Vj i; see ([4]; 5.1.7). Moreover for each s the closure of the open set Ωs=∪i+j=s+1Zj iis the disjoint finite union of closed sets Ωs=∪i+j=s+1Zj i. Finally diagram (4) shows that {Ωs}1≤s≤n+m−1is a properly categorical open cover of X×Yand hence p−cat(X×Y)≤p−cat(X)+p−cat(Y)−1. Here we use the crucial fact that the ray αis the same for all i, j. In case Yis compact the subpolyhedra Wjare contractible to a point in Y. Moreover, since Xis supposed to be at most one-ended, we can use Proposition 1.6 to assume that for the subpolyhedra Uione has a commutative diagram Minimal covers and the proper L-S category of product spaces 425 Ui  * X ? R+ HHH Hj α ki r in P/≃for all i≥1. Next we argue as above to obtain the open sets Zj i. Then by using the previous diagram and the fact that each Wjis contractible in Yone easily shows that the disjoint union ∪{ ¯ Zj i;i+j=s+ 1}is an inessential set in X×Yand the result follows. Remark 2.2. It is clear that Proposition 2.1 does not hold if Xis not one-ended in case Yis compact. Indeed, it is clear that p−cat(Sn×R)≤4, and so from Example 1.5 it follows p−cat(Sn×R) = 4 >3 = cat(Sn) + p−cat(R)−1. 3 Minimal covers with half-spaces and proper L-S category If Mis an open n-manifold one can consider covers consisting of closed subspaces homeomorphic to the half-space Rn +, and it is natural to ask for the comparison of p−cat(M) with the smallest number, h(M), of half-spaces needed to cover M. By using the proper Engulfing Theorem (A.6) we have the following result; compare ([20]; Ch. VII). Theorem 3.1. Let Mbe a one-ended open PL n-manifold. Then p−cat(M)≤ h(M)≤n+ 1. Proof. Let Kbe a triangulation of M. If b(σ) denotes the barycentre of σ∈Kwe consider the discrete sets Γi={b(σ); dimσ =i}. Since Mis one-ended it is easily checked that all sets Γi, 0 ≤i≤n, are inessential in Mand so by (A.6) there exist half-spaces Hiwith Γi⊆Hi. Next we consider the regular neighbourhoods Niof Γiin the second barycentric subdivision of K. Then M=∪n i=0Niand moreover by the uniqueness of regular neighbourhoods (A.3) there exist ambient isotopies φiin Mcarrying Niinside Hi. Hence M=∪n i=0φ−1 i(Hi) and the result follows. We now proceed to prove a proper analogue of a theorem due to Singhof [17] which provides sufficient conditions for the equality p−cat(M) = h(M). Recall that a space Xin Pis said to be properly k-connected if for any q≤kany proper map f:K→Xfrom a q-dimensional polyhedron Kin Pis properly inessential. Notice that Xis properly 0-connected if and only if Xis one-ended. Lemma 3.2. Let Pbe a properly k-connected polyhedron in Pand let (K, L)be a polyhedral pair in Pwith dim(K−L)≤k+ 1. Then any proper map f:L→P admits a proper extension e f:K→P. In particular, if R⊂Q⊂P,dim(Q−R)≤k, and Ris properly inessential then so is Q. Proof. Assume that ˜ f:Kr∪L→Pexists. Then the restriction of ˜ fto the union ∆ = ∪{∂σ;σr+1 ∈K}is properly inessential and hence ˜ f|∆ admits a proper extension to Kr+1 which yields a proper extension of ˜ fto Kr+1 ∪L. For the second part, let H:R×I→Pbe a proper deformation of Rwith H1a composite H1:Rr −→ R+ α −→ P. Then we apply the first part of the lemma to K=Q×I, L=R×I∪Q× {1}and f=H∪˜r:L→Pwhere ˜r:Q→R+is any proper extension of r. 426 M. C´ardenas – F.F. Lasheras – A. Quintero Theorem 3.3. Let Mbe a properly c-connected open PL n-manifold (c≥0,n≥4). If p−cat(M)≥n+c+4 2(c+1) then p−cat(M) = h(M). The proof of Theorem 3.3 follows closely the proof due to Montejano [12] for ordinary L-S category. More precisely we first prove the following proper analogue of ([13], Thm. 1). See Appendix A for the definition of a proper collapse Y&pX. Proposition 3.4. Let Pbe a properly c-connected n-dimensional polyhedron in P, and let {P1, ..., Pm}be a properly categorical (polyhedral) cover of P. Then, for each 0≤q≤c, there is a properly categorical cover {R1, ..., Rm}of Psuch that, for each 1≤i≤m, Ri&pNiwhere dim(Ni)≤max{n−(m−1)(q+ 1), q}. Proof. Let Tbe a triangulation of Psuch that T1, ..., Tmare subcomplexes which triangulate P1, ..., Pm. Let L1be the (n−(m−1)(q+ 1))-skeleton of T1 and let L0 1be its dual skeleton. By Lemma A.2, there exists a polyhedral cover {R1 2, ..., R1 m}of |L0 1| ∪ ( m [ i=2 Pi) such that R1 i&pPi∪Niand dim(Ni)≤q, 2≤i≤m. Let R1 1=|J|, where Jis a second derived neighbourhood of L1in Tsuch that P=|J| ∪ ( m [ i=2 R1 i). Notice that each R1 i(i≥2) is properly categorical by Lemma 3.2. Moreover, R1 1&p|L1| ⊆ P1with dim(L1)≤n−(m−1)(q+ 1), and hence R1 1 is also properly categorical. Next, let us suppose we have constructed a polyhedral cover {Rk 1, ..., Rk m}satisfying (a)Rk iis properly categorical, 1 ≤i≤m. (b)kRk i&pNi, with dim(Ni)≤max{n−(m−1)(q+ 1), q}, 1 ≤i≤k < m. By replacing R1 1with Rk k+1 and using the same argument as above one constructs a polyhedral cover {Rk+1 1, ..., Rk+1 m}such that Rk+1 i&pRk i∪N0 i,dim(N0 i)≤q(i6= k+1) and Rk+1 k+1 &p|Lk+1| ⊆ Rk k+1 with dim(Lk+1)≤n−(m−1)(q+1). Moreover, for each 1 ≤i≤k,Rk+1 i&pRk i∪N0 iand Rk i&pNi. Thus, by Lemma A.1, there exist polyhedra Miwith dim(Mi)≤dim(N0 i)≤qand such that Rk i∪N0 i&pNi∪Mi=N00 i, whence Rk+1 i&pN00 iand dim(N00 i)≤max{n−(m−1)(q+1), q}. By Lemma 3.2, each Rk+1 i(i6=k+1) is properly categorical. Moreover, Rk+1 k+1 &p|Lk+1| ⊆ Rk k+1 and hence Rk+1 k+1 is also properly categorical. Therefore, the polyhedral cover {Rk+1 1, ..., Rk+1 m} satisfies properties (a) and (b)k+1. Proof of 3.3. Let q=min{c, n −3}. According to Proposition 3.4 there exists a properly categorical cover M=R1∪ · · · ∪ Rm(m=p−cat(M)) such that Ri&pNiwith dimNi≤max{n−(m−1)(q+ 1), q}. If c≤n−3 then q=cand dimNi≤n+c−2 2≤n−3. Otherwise c≥n−2 and q=n−3 yield dimNi≤n−3≤ n+c−2 2. Hence by the proper Engulfing Theorem (A.6) we can find mhalf-spaces H1,...,Hmwith Ni⊆Hi. As Ri&pNiany regular neighbourhood Ωiof Riis a regular neighbourhood of Niand by the uniqueness of regular neighbourhoods there exists an isotopy φicarrying Ωiinto Hi. See (A.3). Hence M=∪m i=1φ−1 i(Hi) and the proof is finished. According to 2.1 and (2) above, for any one-ended open manifold Mwe see that p−cat(M×Sk) is either p−cat(M) or p−cat(M) + 1. As a consequence of Theorem 3.3 we can determine the proper L-S category of M×Skin some cases. More explicitly, Minimal covers and the proper L-S category of product spaces 427 Theorem 3.5. Let Mbe a one-ended open PL n-manifold, n≥3. If p−cat(M)≥ n+k 2+ 2 then p−cat(M×Sk) = p−cat(M) + 1. In particular if p−cat(M)≥n+5 2 we have p−cat(M×S1) = p−cat(M) + 1. Example 3.6. It is clear that Theorem 3.5 does not hold if Mis not one-ended. Indeed, for M=S2×Rwe have p−cat(M×S1)≤6 since cat(S2×S1) = 3. Hence from Example 1.5 we get p−cat(M×S1) = 6 >5 = p−cat(M) + 1. In the proof of Theorem 3.5 we need the following Lemma 3.7. Let Xand Ybe path connected spaces in P. If Yis compact then the projection p1:X×Y→Xinduces a bijection p1∗: [R+, X ×Y]∼ =[R+, X]between strong ends. Proof. Given y0∈Ylet j:X→X×Ybe the inclusion j(x) = (x, y0). It is clear that pj =idXand hence p1∗is onto and j∗is injective. Moreover, given any ray r:R+→X×Ylet H:p2r≃cy0be a homotopy where cy0is the constant map cy0(t) = y0. Although His not a proper map the map ˜ H(t, s) = (p1r(t), H(t, s)) is a proper homotopy such that ˜ H(t, 0) = r(t) and ˜ H(t, 1) is a ray in X× {y0}. We have shown that j∗is onto and hence j∗as well as p1∗are bijections. Proof of 3.5 We have p−cat(M)≤p−cat(M×Sk)≤p−cat(M)+1 by Proposition 2.1. Assume for a moment s=p−cat(M×Sk) = p−cat(M)≥n+k 2+2. By Theorem 3.3 with c= 0 applied to M×Skthere are closed half-spaces H1, H2, . . . Hswith M×Sk=∪s i=1Hi. Let R⊆H1be an embedded ray with H1collapsing properly to Rand hence H1is a regular neighbourhood of R; see (A.3). Let x0∈Skbe any point. By Lemma 3.7 we can find an embedded ray R0⊆M× {x0}such that both Rand R0define the same strong end of M×Sk. Hence there exists a proper homotopy G:R+×I→M×Skwith G(R+× {0}) = Rand G(R+× {1}) = R0. As dimM ×Sk≥4 there exists an ambient isotopy of M×Skwhich carries Rto R0; see (A.5). Now we use the uniqueness of regular neighbourhoods (A.3) to find an isotopy carrying H1to a small regular neighbourhood Nof R0with N∩(M× {x}) = ∅ for some x6=x0. Hence M× {x} ⊆ Z=H2∪ · · · ∪ Hsand so the restriction r=p|Z:Z→M× {x}of the obvious projection is a proper retraction. Hence p−cat(M× {x})≤p−cat(Z)≤s−1 which is a contradiction. Recently Rudyak ([14]; 3.8) has proved that Singhof’s Theorem implies the stronger result cat(M×Sk) = cat(M) + 1. Rudyak’s arguments can be repeated here to derive from Theorem 3.5 the following Theorem 3.8. Let Mbe a one-ended open PL n-manifold, n≥3. If p−cat(M)≥ n+5 2we have p−cat(M×Sk) = p−cat(M) + 1 and p−cat(M×Sm1× · · ·×Smk) = p−cat(M) + kfor all k≥1. Remark 3.9. It has been a long standing conjecture due to Ganea that the equality cat(X×Sk) = cat(X)+1 always holds for any finite CW-complex X. In 1998, Iwase [7] gave counterexamples to this conjecture. In addition, Iwase [8] has obtained recently a closed manifold Mfor which cat(M×Sk) = cat(M). At present the authors do not know whether the corresponding version of Ganea’s conjecture is true for the proper L-S category of one-ended open manifolds.