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

Minimal co e s o open mani olds wi h hal -spaces and he p ope L-S ca ego y o p oduc spaces M. C´a denas F.F. Lashe as A. Quin e o Abs ac Classical esul s abou he Lus e nik-Schni elmann ca ego y o p oduc spaces ha e hei analogues in he ca ego y o p ope maps. By compa ing he p ope Lus e nik-Schni elmann ca ego y o an open mani old Xwi h he smalles numbe o closed hal -spaces needed o co e X, we ob ain a p ope analogue o Singho ’s heo em on he ca ego y o X×S1. In oduc ion The Lus e nik-Schni elmann ca ego y (L-S ca ego y) ca (X) o a space Xis he smalles numbe ksuch ha he e exis s an open co e {X1,...Xk} o which each inclusion Xj⊆Xis nullhomo opic in X. The L-S ca ego y u ns o be a homo opy in a ian o he space X. See [9] o a su ey on L-S ca ego y. O dina y homo opy in a ian s do no ake ca e o he beha iou o spaces a in ini y. So, “p ope ” homo opy in a ian s a e needed o he s udy o non-compac spaces. P ope analogues o Lus e nik-Schni elmann nume ical in a ian s we e in- oduced in [2] and [3]. The c ucial poin in he de ini ion o he p ope L-S ca ego y is he ac ha he hal -line [0,∞) plays in p ope homo opy pa o he ole played by he poin in o dina y homo opy. The pa allelism be ween bo h oles b eaks down on he Recei ed by he edi o s Janua y 2001. Communica ed by Y. F´elix. 1991 Ma hema ics Subjec Classi ica ion : 55M30, 55P57, 57Q30(Seconda y). Key wo ds and ph ases : p ope map, Lus e nik-Schni elmann ca ego y, open mani old, p ope collapse. Bull. Belg. Ma h. Soc. 9 (2002), 419–431 420 M. C´a denas – F.F. Lashe as – A. Quin e o ib a ion-side o homo opy heo y since p oduc and ib a ion p ojec ions a e no always p ope maps. Howe e , he class o p ope maps s ill keeps he basic p op- e ies on he co ib a ion-side o homo opy heo y which lead o a “combina o ial” homo opy in he sense o J.H.C. Whi ehead [19]; see [1]. This pape con inues he s udy o he p ope L-S ca ego y o non-compac spaces. He e, we ocus ou in e es on he beha iou o he p ope L-S ca ego y on p oduc spaces. Mo e explici ly we wo k ou he p ope L-S ca ego y o spaces o he o m W×Skand M×Rkwi h Wand Mopen and closed mani olds espec i ely. These esul s can be ega ded as p ope analogues o a heo em due o Singho [17] in ela ion wi h a classical ques ion posed by Ganea; see Rema k 3.9. In his pape we ollow closely he combina o ial p oo o Singho ’s heo em gi en by Mon ejano [12] since p ope collapses [16] and a sui able Engul ing Theo em o open mani olds [10] a e a ailable; see Appendix A o de ails. This way we can compa e he p ope L-S ca ego y o an open mani old Xwi h he smalles numbe o p ope ly embedded hal -spaces needed o co e X. Se e al new imp o emen s o Singho ’s heo em ha e ecen ly appea ed in he li e a u e; see [14] and [18]. Howe e , hese esul s depend hea ily on he s udy o sec ions o ce ain ib a ions. Since he ca ego y o p ope maps p o ides an example o homo opy heo y wi h good p ope ies only on he co ib a ion-side, i seems o be in e es ing o look o al e na i e p oo s o hose esul s in he co ib a ion ealm o homo opy heo y. We shall deal wi h he ca ego y Po locally compac σ-compac Hausdo spaces and p ope maps. Recall ha a p ope map (p-map) is a con inuous map :X→Y such ha −1(K) is compac o each compac subse K⊆Y. All maps and homo opies a e assumed o be p ope unless s a ed o he wise. We use he symbol ≃ o p ope homo opy and P/≃s ands o he co esponding homo opy ca ego y. Fu he mo e, he symbol R+deno es he hal -line [0,∞) and mo e gene ally Rn +deno es he uppe n-dimensional hal -space {(x1, x2, . . . , xn)∈ Rn;xn≥0}. 1 P ope L-S ca ego y This sec ion con ains some echnical obse a ions abou he no ion o p ope L-S ca ego y which will be used la e . Recall ha , gi en a space Xin P, a sys em o ∞-neighbou hoods o Xis a dec easing sequence {Wj}o subse s o Xsuch ha he closu es Kj=X−Wj o m an inc easing sequence o compac subse s wi h Kj⊆in Kj+1 and X=∪in Kj. Rema k 1.1. Gi en a locally ini e sequence o pai wise disjoin compac subse s Ci⊆X(i≥1), i is possible o choose he compac se s Kjabo e sa is ying Ci∩F Kj=∅ o all i, j ≥1. Fo his we conside he compac se ˆ L1= K1∪(∪{Ci;Ci∩K16=∅}). By using he no mali y o Xwe ind a compac se L1⊆Xwi h ˆ L1⊆in L1and L1∩Ci=∅whene e Ci∩K1=∅. Hence Ci∩ F L1=∅ o all i≥1. Then we pick n1such ha L1⊆in Kn1and we se ˆ L2=Kn1∪(∪{Ci;Ci∩Kn16=∅}). Simila ly we ind a compac se L2⊆Xwi h ˆ L2⊆in L2and Ci∩F L2=∅ o all i≥1. We p oceed induc i ely o ob ain an inc easing sequence o compac se s Lj⊆in Lj+1 wi h he equi ed p ope ies. Minimal co e s and he p ope L-S ca ego y o p oduc spaces 421 In he ca ego y P he cons an map X→ {p}is no de ined i Xis no compac . No wi hs anding, he ole o he poin is played pa ially in Pby he hal -line R+ since o any space Xin P he e always exis s a p ope map :X→R+. Mo eo e he map is unique up o p-homo opy. We shall b ie ly desc ibe he cons uc ion o such a map . I {Uj}j≥0is a sys em o ∞-neighbou hoods o Xwi h U0=X, he Tie ze Ex ension Theo em yields con inuous maps j=Uj−Uj+1 →[j, j + 1] wi h j(F Uj+1) = j+ 1, j(F Uj) = j. I is now clea ha he maps jde ine a p ope map :X→R+. De ini ion 1.2. A p ope map :R+→Xis called a ay in X. Mo eo e a p ope map :X→Yis p ope ly inessen ial i he e exis s a commu a i e diag am in P/≃ X * Y ? ∗ HHH Hj α β (1) whe e ∗is ei he R+o he one-poin space {p}. No ice ha ∗={p}only i Xis compac . Gi en a space Xin Pa closed subse A⊆Xis called inessen ial i he inclusion i:A→Xis an inessen ial map. A se A⊆Xis called p ope ly ca ego ical i A⊆Uwi h Uan open se in Xand he closu e ¯ Uis inessen ial. Mo eo e an open co e {Uα}o Xis said o be p ope ly ca ego ical i each ¯ Uαis an inessen ial se . The p ope Lus e nik-Schni elmann ca ego y o X,p−ca (X), is he leas numbe nsuch ha Xadmi s a p ope ly ca ego ical open co e V={U1, U2, . . . , Un}wi h nelemen s. In case Xis compac p−ca (X) = ca (X) is he o dina y L-S ca ego y o X. Rema k 1.3. As in o dina y homo opy heo y closed co e s can also be used o de ine he p ope L-S ca ego y o ANR-spaces. Fu he mo e o polyhed a in Pone can use co e s consis ing o subpolyhed a in he de ini ion o p ope L-S ca ego y; see [2] and [3] o de ails. AF euden hal end o a space Xin Pis an elemen o he in e se limi F(X) = lim ←− U(Wj). He e {Wj}is a sys em o ∞-neighbou hoods o Xand U(−) s ands o he amily o unbounded connec ed componen s. A subse A⊆Xis e med unbounded i i s closu e ¯ Ais non-compac . I F(X) = {∗} hen Xis said o be one-ended. Rema k 1.4. No ice ha a p ope ly ca ego ical se A⊆Xcanno con ain se- quences o poin s de ining wo di e en F euden hal ends. Indeed, i is immedia e o check ha any ay :R+→Xde ines a unique F euden hal end. As a conse- quence, o any sys em o ∞-neighbou hoods {Wj}j≥1o X, he e exis s j0such ha o each j≥j0 he e is a mos one componen Uj∈ U(Wj) wi h A∩Wj6=∅; mo e- o e hese componen s o m a nes ed sequence Uj0+1 ⊇Uj0+2 ⊇... which de ines a unique F euden hal end o X. 422 M. C´a denas – F.F. Lashe as – A. Quin e o Example 1.5. A lowe bound o he p ope L-S ca ego y o spaces wi h cylind ical ends can be easily ob ained as ollows. Le Xbe a space in Pwi h mcylind ical ends; ha is, he e exis s a ela i ely compac open se A⊆Xsuch ha X−Ahas munbounded componen s Wj(1 ≤j≤m) homeomo phic o cylinde s Zj×[0,∞) whe e each Zjis compac . Then he inequali y p−ca (X)≥k= m X j=1 ca (Zj) holds. Indeed, gi en a p ope ly ca ego ical open co e {Us}1≤s≤no X, by Rema k 1.4, he e is a compac subse B⊆Xsuch ha A⊆B, and each non-emp y di e ence Us−Bis con ained in exac ly one componen o X−B. The e o e, i n≤k−1 hen a leas one componen Ωj0⊆X−Bis co e ed by less han ca (Zj0) di e ences Usi−B(1 ≤i≤q < ca (Zj0)). Mo eo e , we can assume wi hou loss o gene ali y ha Ωj0is o he o m Zj0×[ 0,∞). As each Usiis p ope ly ca ego ical he e exis s ≥ 0such ha he de o ma ion o Usi−(Zj0×[ , ∞)) occu s inside Ωj0. F om his ac , we easily de i e ha he in e sec ions Usi∩(Zj0× { + 1}) (1 ≤i≤q < ca (Zj0)) p o ide an o dina y ca ego ical co e o Zj0× { + 1}which is a con adic ion. The p ope homo opy class o he map βin diag am (1) is unique. Howe e o ∗=R+ he p ope homo opy class o he map αin diag am (1) depends on he se o p ope homo opy classes [R+, Y ]. Each class [α]∈[R+, Y ] is called a s ong end o Y. When [R+, Y ] consis s o only one elemen we say ha Yis s ongly one-ended. Clea ly each s ong end de ines a F euden hal end. Mo e p ecisely he e exis s an on o map [R+, Y ]→ F(Y). I is ob ious ha o s ongly one-ended spaces all ays αican be chosen o be he same. Nex p oposi ion shows ha he same holds o one-ended polyhed a. P oposi ion 1.6. Le Xbe a connec ed one-ended polyhed on in Pwi h p−ca (X) = n. Then he e exis s a p ope ly ca ego ical (polyhed al) co e {V1, V2, . . . , Vn}o X such ha in he diag ams Vi  * X ? R+ HHH Hj αi ki (2) all ays αi(i≤n) de ine he same s ong end. P oo . Le {U1, U2,...,Un}be a p ope ly ca ego ical co e o Xconsis ing o subpolyhed a; see Rema k 1.3. Mo eo e , i is well known ha any ay α:R+→X is p ope ly homo opic o a ay embedded in he 1-skele on o X. In case all connec ed componen s o some Uia e compac , we nex show ha αiin diag am (2) can be chosen o be a bi a y. Indeed, since all componen s C⊆Uia e compac , one eadily checks ha he e is a p ope de o ma ion Hwhich con ac s each C o a poin in X. Fu he mo e, by using he p ope homo opy ex ension p ope y we can eplace Hby a new p ope de o ma ion H0which sh inks each C o a poin xC∈C ela i e xC. Finally, gi en any ay R⊆X, we use ha Xis Minimal co e s and he p ope L-S ca ego y o p oduc spaces 423 one-ended o mo e each poin xC o some poin yC∈R ia a p ope homo opy {xC} × I→X. By using he p e ious a gumen s, we can assume wi hou loss o gene ali y ha some elemen o he co e {U1, U2,...,Un}has a leas one unbounded componen . Le U1be such an elemen . We assume induc i ely ha Uican be p ope ly de o med o a ay R⊆X o i≤k. Now we conside Uk+1. By he a gumen s abo e we can assume ha Uk+1 is no compac . Mo eo e we can also assume ha U1∩Uk+1 6=∅ is non-compac as well; o he wise we use he ac ha Xis one-ended o join U1 o Uk+1 wi h a locally ini e sequence o pai wise disjoin a cs which we add o U1. In addi ion, i he in e sec ion U1∩Uk+1 con ains a ay R0 hen he p ope de o ma ion o U1 o Ryields ha bo h ays Rand R0 ep esen he same s ong end o X, and so Uk+1 can be p ope ly de o med o he ay R. I only emains o conside he case when U1∩Uk+1 consis s o a locally ini e sequence {K1, K2,...}o compac componen s. In such a case one inds wo pai wise disjoin amilies {A1, A2,...}and {B1, B2,...}o compac subpolyhed a1o Uk+1 wi h Uk+1 = (∪∞ i=1Ai)∪(∪∞ =1B ) and Ki⊆in Uk+1 Ai o i≥1. Also one chooses a pai wise disjoin sequence {L1, L2,...}o compac subpolyhed a o U1wi h Ki⊆ in U1Li o all i≥1. Finally we ake a locally ini e sequence o pai wise disjoin a cs γi⊆U1joining Ki o Z=U1− ∪∞ i=1Liwi h in γi⊆in U1Li. Then we eplace U1and Uk+1 by e U1= (∪∞ i=1Ai)∪Z∪(∪∞ j=1γj) and e Uk+1 = (∪∞ =1B )∪(∪∞ j=1Lj) espec i ely. Now by he a gumen s abo e, we can assume in addi ion ha he disjoin union ∪∞ s=1Asis p ope ly de o med o he disc e e se D=∪∞ j=1(γj∩Kj) ela i e o D. F om his one easily shows ha e U1can be p ope ly de o med o he ay R. The se e Uk+1 is a locally ini e disjoin union o compac subpolyhed a, and so i can be p ope ly de o med o he ay Ras well. Since U1∪Uk+1 =e U1∪e Uk+1 we can eplace he co e {U1, U1,...,Un}by {e U1, U2,...,Uk,e Uk+1, Uk+2, . . . , Un}. A e a ini e numbe o s eps we ge a p ope ly ca ego ical (polyhed al) co e o Xsuch ha all elemen s in i can be p ope ly de o med o he ay R. 2 P ope L-S ca ego y o p oduc spaces Fo o dina y L-S ca ego y he ollowing o mula is well known o p oduc spaces; see [5] max{ca (X), ca (Y)} ≤ ca (X×Y)≤ca (X) + ca (Y)−1 (∗) In his sec ion we s udy he p ope analogue o his o mula. Recall ha he p ope L-S ca ego y p−ca (−) is a p ope homo opy in a ian ; in ac i :X→Yand g:Y→Xa e p ope maps wi h g ≃idXone has p−ca (X)≤p−ca (Y). In pa icula , i Yis compac he p ojec ion p1:X×Y→Xyields 1One inds hese amilies as ollows. Le {Lj}be an inc easing sequence o compac subpolyhe- d a in Uk+1 wi h Ki∩F Lj=∅ o i, j ≥1; see Rema k 1.1. Then we pick n1< n2< . . . and we choose any locally ini e amily o pai wise disjoin compac subpolyhed a Aiwi h Ki⊆in Uk+1 Ai o i6=njand Knj∪F Lj⊆in Uk+1 Anj o all j≥1. I is immedia e o check ha he closu e Uk+1 − ∪∞ i=1Ai=∪∞ =1B is a locally ini e union o pai wise disjoin compac subpolyhed a. 424 M. C´a denas – F.F. Lashe as – A. Quin e o p−ca (X)≤p−ca (X×Y) (3) Howe e , Example 3.10 below shows ha inequali y (3) does no hold i Yis no compac . Conce ning he igh -hand side inequali y in (*) we can p o e he ollowing p opo- si ion; compa e [5] P oposi ion 2.1. Le Xand Ybe wo connec ed polyhed a in P. Assume Xhas a mos one end in case Yis compac . Then p−ca (X×Y)≤p−ca (X)+p−ca (Y)−1. P oo . Assume Xand Ya e no compac . Then by ([11];2.2) X×Yis s ongly one-ended. Le p−ca (X) = nand p−ca (Y) = mand le {U1, . . . , Un}and {W1,...,Wm}be amilies o inessen ial subpolyhed a whose in e io s co e Xand Y espec i ely. By using egula neighbou hoods we ind new p ope ly ca ego ical co e s {e Ui}and { Wj}consis ing o closed subpolyhed a wi h Ui⊆in e Uiand Wj⊆ in Wj. Since R+×R+has he p ope homo opy ype o R+i is clea ha each p oduc e Ui× Wjis inessen ial. Mo eo e , since X×Yis s ongly one-ended we ha e a commu a i e diag am in P/≃ e Ui× Wj  * X×Y ? R+ HHH Hj α (4) wi h he same α o all i, j. We now conside he unions Ai=∪i k=1Uk,i≤n, and Bj=∪j h=1Wh,j≤m. Mo eo e le C1⊆C2⊆ · · · ⊆ Cn+m−1=X×Ybe he inc easing sequence o closed se s Cs=∪i+j=s+1Ai×Bj,1≤s≤n+m−1. F om his we can w i e X×Y=∪n+m−1 s=1 Ds o he se s D1=C1and Ds=Cs−Cs−1, s≥2. Mo eo e we ha e Ds=∪{Ei×Fj;i+j=s+ 1}whe e Ei=Ai−Ai−1and Fj=Bj−Bj−1. We se A0=B0=∅. Clea ly he se s Ei×Fj⊆Dsa e pai wise disjoin . Mo eo e hey a e pai wise sepa a ed; ha is, we ha e (Ei×Fj)∩(Ei0×Fj0) = (Ei×Fj)∩(Ei0×Fj0) = ∅ i i+j=i0+j0. Since X×Yis he edi a ily no mal we ind o each s≤n+m−1 a pai wise disjoin amily o open se s Vs={Vj i}i+j=s+1 wi h Ei×Fj⊆Vj i⊆ in e Ui×in Wj; see ([4];2.1.7). In pa icula he union V=∪m+n−1 s=1 Vsis an open co e o X×Y, and he pa acompac ness o X×Yp o ides us wi h a ini e open e inemen o V,Z=∪n+m−1 s=1 {Zj i}i+j=s+1 wi h Zj i⊆Vj i; see ([4]; 5.1.7). Mo eo e o each s he closu e o he open se Ωs=∪i+j=s+1Zj iis he disjoin ini e union o closed se s Ωs=∪i+j=s+1Zj i. Finally diag am (4) shows ha {Ωs}1≤s≤n+m−1is a p ope ly ca ego ical open co e o X×Yand hence p−ca (X×Y)≤p−ca (X)+p−ca (Y)−1. He e we use he c ucial ac ha he ay αis he same o all i, j. In case Yis compac he subpolyhed a Wja e con ac ible o a poin in Y. Mo eo e , since Xis supposed o be a mos one-ended, we can use P oposi ion 1.6 o assume ha o he subpolyhed a Uione has a commu a i e diag am Minimal co e s and he p ope L-S ca ego y o p oduc spaces 425 Ui  * X ? R+ HHH Hj α ki in P/≃ o all i≥1. Nex we a gue as abo e o ob ain he open se s Zj i. Then by using he p e ious diag am and he ac ha each Wjis con ac ible in Yone easily shows ha he disjoin union ∪{ ¯ Zj i;i+j=s+ 1}is an inessen ial se in X×Yand he esul ollows. Rema k 2.2. I is clea ha P oposi ion 2.1 does no hold i Xis no one-ended in case Yis compac . Indeed, i is clea ha p−ca (Sn×R)≤4, and so om Example 1.5 i ollows p−ca (Sn×R) = 4 >3 = ca (Sn) + p−ca (R)−1. 3 Minimal co e s wi h hal -spaces and p ope L-S ca ego y I Mis an open n-mani old one can conside co e s consis ing o closed subspaces homeomo phic o he hal -space Rn +, and i is na u al o ask o he compa ison o p−ca (M) wi h he smalles numbe , h(M), o hal -spaces needed o co e M. By using he p ope Engul ing Theo em (A.6) we ha e he ollowing esul ; compa e ([20]; Ch. VII). Theo em 3.1. Le Mbe a one-ended open PL n-mani old. Then p−ca (M)≤ h(M)≤n+ 1. P oo . Le Kbe a iangula ion o M. I b(σ) deno es he ba ycen e o σ∈Kwe conside he disc e e se s Γi={b(σ); dimσ =i}. Since Mis one-ended i is easily checked ha all se s Γi, 0 ≤i≤n, a e inessen ial in Mand so by (A.6) he e exis hal -spaces Hiwi h Γi⊆Hi. Nex we conside he egula neighbou hoods Nio Γiin he second ba ycen ic subdi ision o K. Then M=∪n i=0Niand mo eo e by he uniqueness o egula neighbou hoods (A.3) he e exis ambien iso opies φiin Mca ying Niinside Hi. Hence M=∪n i=0φ−1 i(Hi) and he esul ollows. We now p oceed o p o e a p ope analogue o a heo em due o Singho [17] which p o ides su icien condi ions o he equali y p−ca (M) = h(M). Recall ha a space Xin Pis said o be p ope ly k-connec ed i o any q≤kany p ope map :K→X om a q-dimensional polyhed on Kin Pis p ope ly inessen ial. No ice ha Xis p ope ly 0-connec ed i and only i Xis one-ended. Lemma 3.2. Le Pbe a p ope ly k-connec ed polyhed on in Pand le (K, L)be a polyhed al pai in Pwi h dim(K−L)≤k+ 1. Then any p ope map :L→P admi s a p ope ex ension e :K→P. In pa icula , i R⊂Q⊂P,dim(Q−R)≤k, and Ris p ope ly inessen ial hen so is Q. P oo . Assume ha ˜ :K ∪L→Pexis s. Then he es ic ion o ˜ o he union ∆ = ∪{∂σ;σ +1 ∈K}is p ope ly inessen ial and hence ˜ |∆ admi s a p ope ex ension o K +1 which yields a p ope ex ension o ˜ o K +1 ∪L. Fo he second pa , le H:R×I→Pbe a p ope de o ma ion o Rwi h H1a composi e H1:R −→ R+ α −→ P. Then we apply he i s pa o he lemma o K=Q×I, L=R×I∪Q× {1}and =H∪˜ :L→Pwhe e ˜ :Q→R+is any p ope ex ension o . 426 M. C´a denas – F.F. Lashe as – A. Quin e o Theo em 3.3. Le Mbe a p ope ly c-connec ed open PL n-mani old (c≥0,n≥4). I p−ca (M)≥n+c+4 2(c+1) hen p−ca (M) = h(M). The p oo o Theo em 3.3 ollows closely he p oo due o Mon ejano [12] o o dina y L-S ca ego y. Mo e p ecisely we i s p o e he ollowing p ope analogue o ([13], Thm. 1). See Appendix A o he de ini ion o a p ope collapse Y&pX. P oposi ion 3.4. Le Pbe a p ope ly c-connec ed n-dimensional polyhed on in P, and le {P1, ..., Pm}be a p ope ly ca ego ical (polyhed al) co e o P. Then, o each 0≤q≤c, he e is a p ope ly ca ego ical co e {R1, ..., Rm}o Psuch ha , o each 1≤i≤m, Ri&pNiwhe e dim(Ni)≤max{n−(m−1)(q+ 1), q}. P oo . Le Tbe a iangula ion o Psuch ha T1, ..., Tma e subcomplexes which iangula e P1, ..., Pm. Le L1be he (n−(m−1)(q+ 1))-skele on o T1 and le L0 1be i s dual skele on. By Lemma A.2, he e exis s a polyhed al co e {R1 2, ..., R1 m}o |L0 1| ∪ ( m [ i=2 Pi) such ha R1 i&pPi∪Niand dim(Ni)≤q, 2≤i≤m. Le R1 1=|J|, whe e Jis a second de i ed neighbou hood o L1in Tsuch ha P=|J| ∪ ( m [ i=2 R1 i). No ice ha each R1 i(i≥2) is p ope ly ca ego ical by Lemma 3.2. Mo eo e , R1 1&p|L1| ⊆ P1wi h dim(L1)≤n−(m−1)(q+ 1), and hence R1 1 is also p ope ly ca ego ical. Nex , le us suppose we ha e cons uc ed a polyhed al co e {Rk 1, ..., Rk m}sa is ying (a)Rk iis p ope ly ca ego ical, 1 ≤i≤m. (b)kRk i&pNi, wi h dim(Ni)≤max{n−(m−1)(q+ 1), q}, 1 ≤i≤k < m. By eplacing R1 1wi h Rk k+1 and using he same a gumen as abo e one cons uc s a polyhed al co e {Rk+1 1, ..., Rk+1 m}such ha 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 wi h dim(Lk+1)≤n−(m−1)(q+1). Mo eo e , o each 1 ≤i≤k,Rk+1 i&pRk i∪N0 iand Rk i&pNi. Thus, by Lemma A.1, he e exis polyhed a Miwi h dim(Mi)≤dim(N0 i)≤qand such ha 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 p ope ly ca ego ical. Mo eo e , Rk+1 k+1 &p|Lk+1| ⊆ Rk k+1 and hence Rk+1 k+1 is also p ope ly ca ego ical. The e o e, he polyhed al co e {Rk+1 1, ..., Rk+1 m} sa is ies p ope ies (a) and (b)k+1. P oo o 3.3. Le q=min{c, n −3}. Acco ding o P oposi ion 3.4 he e exis s a p ope ly ca ego ical co e M=R1∪ · · · ∪ Rm(m=p−ca (M)) such ha Ri&pNiwi h dimNi≤max{n−(m−1)(q+ 1), q}. I c≤n−3 hen q=cand dimNi≤n+c−2 2≤n−3. O he wise c≥n−2 and q=n−3 yield dimNi≤n−3≤ n+c−2 2. Hence by he p ope Engul ing Theo em (A.6) we can ind mhal -spaces H1,...,Hmwi h Ni⊆Hi. As Ri&pNiany egula neighbou hood Ωio Riis a egula neighbou hood o Niand by he uniqueness o egula neighbou hoods he e exis s an iso opy φica ying Ωiin o Hi. See (A.3). Hence M=∪m i=1φ−1 i(Hi) and he p oo is inished. Acco ding o 2.1 and (2) abo e, o any one-ended open mani old Mwe see ha p−ca (M×Sk) is ei he p−ca (M) o p−ca (M) + 1. As a consequence o Theo em 3.3 we can de e mine he p ope L-S ca ego y o M×Skin some cases. Mo e explici ly, Minimal co e s and he p ope L-S ca ego y o p oduc spaces 427 Theo em 3.5. Le Mbe a one-ended open PL n-mani old, n≥3. I p−ca (M)≥ n+k 2+ 2 hen p−ca (M×Sk) = p−ca (M) + 1. In pa icula i p−ca (M)≥n+5 2 we ha e p−ca (M×S1) = p−ca (M) + 1. Example 3.6. I is clea ha Theo em 3.5 does no hold i Mis no one-ended. Indeed, o M=S2×Rwe ha e p−ca (M×S1)≤6 since ca (S2×S1) = 3. Hence om Example 1.5 we ge p−ca (M×S1) = 6 >5 = p−ca (M) + 1. In he p oo o Theo em 3.5 we need he ollowing Lemma 3.7. Le Xand Ybe pa h connec ed spaces in P. I Yis compac hen he p ojec ion p1:X×Y→Xinduces a bijec ion p1∗: [R+, X ×Y]∼ =[R+, X]be ween s ong ends. P oo . Gi en y0∈Yle j:X→X×Ybe he inclusion j(x) = (x, y0). I is clea ha pj =idXand hence p1∗is on o and j∗is injec i e. Mo eo e , gi en any ay :R+→X×Yle H:p2 ≃cy0be a homo opy whe e cy0is he cons an map cy0( ) = y0. Al hough His no a p ope map he map ˜ H( , s) = (p1 ( ), H( , s)) is a p ope homo opy such ha ˜ H( , 0) = ( ) and ˜ H( , 1) is a ay in X× {y0}. We ha e shown ha j∗is on o and hence j∗as well as p1∗a e bijec ions. P oo o 3.5 We ha e p−ca (M)≤p−ca (M×Sk)≤p−ca (M)+1 by P oposi ion 2.1. Assume o a momen s=p−ca (M×Sk) = p−ca (M)≥n+k 2+2. By Theo em 3.3 wi h c= 0 applied o M×Sk he e a e closed hal -spaces H1, H2, . . . Hswi h M×Sk=∪s i=1Hi. Le R⊆H1be an embedded ay wi h H1collapsing p ope ly o Rand hence H1is a egula neighbou hood o R; see (A.3). Le x0∈Skbe any poin . By Lemma 3.7 we can ind an embedded ay R0⊆M× {x0}such ha bo h Rand R0de ine he same s ong end o M×Sk. Hence he e exis s a p ope homo opy G:R+×I→M×Skwi h G(R+× {0}) = Rand G(R+× {1}) = R0. As dimM ×Sk≥4 he e exis s an ambien iso opy o M×Skwhich ca ies R o R0; see (A.5). Now we use he uniqueness o egula neighbou hoods (A.3) o ind an iso opy ca ying H1 o a small egula neighbou hood No R0wi h N∩(M× {x}) = ∅ o some x6=x0. Hence M× {x} ⊆ Z=H2∪ · · · ∪ Hsand so he es ic ion =p|Z:Z→M× {x}o he ob ious p ojec ion is a p ope e ac ion. Hence p−ca (M× {x})≤p−ca (Z)≤s−1 which is a con adic ion. Recen ly Rudyak ([14]; 3.8) has p o ed ha Singho ’s Theo em implies he s onge esul ca (M×Sk) = ca (M) + 1. Rudyak’s a gumen s can be epea ed he e o de i e om Theo em 3.5 he ollowing Theo em 3.8. Le Mbe a one-ended open PL n-mani old, n≥3. I p−ca (M)≥ n+5 2we ha e p−ca (M×Sk) = p−ca (M) + 1 and p−ca (M×Sm1× · · ·×Smk) = p−ca (M) + k o all k≥1. Rema k 3.9. I has been a long s anding conjec u e due o Ganea ha he equali y ca (X×Sk) = ca (X)+1 always holds o any ini e CW-complex X. In 1998, Iwase [7] ga e coun e examples o his conjec u e. In addi ion, Iwase [8] has ob ained ecen ly a closed mani old M o which ca (M×Sk) = ca (M). A p esen he au ho s do no know whe he he co esponding e sion o Ganea’s conjec u e is ue o he p ope L-S ca ego y o one-ended open mani olds.