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 in oduce and s udy p ope homo opy in a ian s o he Lus e - nik-Schni elmann ype, p-ca (-), p-Ca (-), and ca e(-) in he ca - ego y o Γ2-locally compac spaces and p ope maps. As an applica- ion, Rn (n Φ 3) is cha ac e ized as (i) he unique open mani old X wi h p-Ca (ΛΓ) = 2, o (ii) he unique open mani old wi h one s ong end and p-ca ( c) = 2. In oduc ion. The ca ego y ca (JΓ) o a space X in he sense o Lus e nik and Schni elmann (L-S ca ego y) is he smalles numbe k such ha he e exis s an open co e ing {X , ... , Xk} o X o which each inclusion Xj c X is nullhomo opic in X. This concep was in oduced by he quo ed au ho s in hei s udies on calculus o a ia ions [16] and hey used i as a lowe bound o he numbe o c i ical poin s o a di e en iable eal unc ion on a mani old. The basic wo k on he homo opical signi icance o ca is due o Bo suk (see [5]). Bo suk's wo k was con inued by Fox [10]. He e we p esen he de ini ion and he basic p ope ies o a new nume ical opological in a ian o Γ2-locally compac spaces which ag ees wi h he no ion o L-S ca ego y o ^-compac spaces. This in- a ian , deno ed p-ca (X), is called he p ope L-S ca ego y o X and u ns ou o be a p ope homo opy in a ian o X. Hence, p-ca (X) is a ine in a ian han ca (X). In [10] se e al gene aliza ions o L-S ca ego y a e sugges ed. Mo e explici ly, a gene al no ion o L-S si -ca ego y wi h espec o a class si o spaces has been de eloped by Puppe and Clapp in [6]. Ou wo k sha es some common poin s wi h [6] bu does no i in o he no ion o L-S si -ca ego y since we en i ely deal wi h p ope maps ins ead o o dina y con inuous maps. Ano he gene aliza ion o L-S ca ego y has been gi en in [1], whe e L-S ca ego y o p o-objec s in p o-c%/* is de ined. This idea is e- la ed o p ope L-S ca ego y by he Edwa ds-Has ings embedding (see [8]) which p o ides a close link be ween p ope homo opy heo y and homo opy in 201 202 R. AYALA, E. DOMINGUEZ, A. MARQUEZ, AND A. QUINTERO We shall wo k en i ely in he ca ego y <Poo o non-compac T2- locally compac σ-compac spaces and p ope maps. No ice ha any ^-compac space X can be ega ded in ^oo as he wedge I /, whe e / will s and o he hal -line [0, oo). We ecall ha a p ope map (p-map) is a con inuous map /: X —• Y such ha ~ι(K) is compac o each compac K c Y. P ope ho- mo opy (p-homo opy), p ope de o ma ion (p-de o ma ion), e c. can be de ined in he na u al way. The symbols "~", "~p" and "=" s and o homo opy equi alence, p-homo opy equi alence and homeomo - phism espec i ely. Finally we also ecall he no ion o end in p ope homo opy. A F euden hal end o X e^oo is an elemen o he in e se limi ^(X) = lim πo(X-K) whe e K anges o e he amily o compac subse s o X and πo s ands o he se o connec ed componen s. The opology o X can be enla ged o a opology on IuF(I) in such a way ha ^(X) u ns ou o be homeomo phic o a closed se o he Can o se (see [11] o de ails). I is easy o check ha any p-map /: X —• Y induces a con inuous map *: ^(X) —• &~(Y) such ha /j= is a homeomo phism when / is a p-homo opy equi alence. 1. Basic p ope ies. In [8] he ollowing lemma is p o en, 1.1. LEMMA ([8; 6.3.5]). Any space X in φoo admi s a p-map : X —• / unique up o p-homo opy. 1.2. DEFINITION. A closed subse C C X is said o be p ope ly de o mable o / in X i he e exis s a diag am in φoo C ^X commu a i e up o p-homo opy. No ice ha we may use as he es ic ion o a p-map X —• / gi en by Lemma 1.1. 1.3. DEFINITION. Gi en a space X in φoo, A c X is said o be p ope ly ca ego ical (p-ca ego ical) in X i he e is a closed neigh- bou hood o A p ope ly de o mable o / in X. An open co e ing {Ua} o X is said o be p-ca ego ical i each Ua is p-ca ego ical in X (i.e. Ua is p ope ly de o mable o / in X). The p-ca ego y o X, p-ca (X), is he leas numbe n such ha X admi s a p-ca ego ical open co e ing wi h n elemen s. I no ini e p-ca ego ical co e ing exis s hen ρ-ca (Z) = oo. LUSTERNIK-SCHNIRELMANN INVARIANTS 203 LEMMA 1.4 [4; 3.4]. Le P be a locally compac me izable space and Q a locally compac ANR. Suppose X is a closed subse o P and ', g: X —• Q a e p-homo opic maps. I ,~g a e ex ensions o and g espec i ely, he e exis s a closed neighbou hood U o X such ha U and ~g U a e p-homo opic. Hence, i we deal wi h locally ini e polyhed a we ha e 1.5. PROPOSITION. Gi en a locally ini e polyhed on P, ρ-ca (P) is he smalles numbe n such ha P can be co e ed by n p-ca ego ical subpolyhed a. P oo . Gi en a p-ca ego ical open co e ing {Wi} le Hι: TFZ x / -* P be a p-de o ma ion o Wi. Since P is ANR by Lemma 1.4 he e is a closed neighbou hood Ω; o Wι and a p-ex ension Hι: Ω; xl —• P. Now by [23; 3.5] we may ake a closed subpolyhed on P/ such ha Wi C Pi C in Ω/ and hus {Pi} is a co e ing o P wi h P/ p- ca ego ical in P. Con e sely, i P is co e ed by m p-ca ego ical sub- polyhed a we may use egula neighbou hoods o ob ain a p-ca ego ical open co e ing wi h m elemen s. The ollowing p ope ies o p-ca (-) a e s aigh o wa dly checked: 1.6. PROPOSITION, (i) ca (X) < ρ-ca (X). I X is compac he equali y holds. (ii) p-ca (X) = 1 i and only i X ~p J. (iii) Le : X —>Y and g Y-*X bep-mapssuch ha g ~p idy. Then ρ-ca (Γ) < ρ-ca (X). In pa icula , ρ-ca (-) is a p-homo opy in a ian . (i ) I A is a p- e ac o X hen p-ca (^) < p-ca (X). ( ) I A and B a e closed and X = in ^Uin i?, hen p-ca (X) < p-ca (Λ) + p-ca (£). The nex p oposi ion gi es us an elemen a y ela ion be ween he se o F euden hal ends o X and ρ-ca (X). 1.7. PROPOSITION. Gi en a space X in φoo we ha e ca d(^(X)) < ρ-ca (X). P oo . I p-ca (X) = oo he e is no hing o p o e. O he wise, i C , Cι, ... , Cm is a p-ca ego ical closed co e ing o X, he na u- al inclusions Kj Cj —• X induce con inuous maps kj 204 R. AYALA, E. DOMINGUEZ, A. MARQUEZ, AND A. QUINTERO . Mo eo e , i is easy o check ha ^(X) = U Im ^/* and since Cj is p-de o mable o / we ge ha each Im c,* is an one-poin se . Hence ca d(^(X)) < p-ca (X). 1.8. EXAMPLES, (a) p-ca (Rπ) = 2 > 1 = ca (RΛ). (b) Fo any Γ2-compac space X, ca (X) = p-ca (X x /) = ρ-ca (X V /). (c) I Sn is he space ob ained om / by a aching one copy o Sn a each na u al numbe n e / we ha e p-ca (*SΛ) = 2. (d) I X is a space in <Poo and : X -> / is a p-map we can de ine (up o p-homo opy) he p ope cone CPX oϊ X as he push-ou o he diag am X x I <— X -^ J and he p ope mapping cone Cp o a p-map / is de ined in he na u al way (see [2] o de ails). I /= , Cp u ns o be he p ope suspension Σp X o X. As in he o dina y case i ollows om 1.6(iii) and (i ) p-ca (Q/) < p-ca (Γ) + 1. In pa icula p-ca (^ X) < 2. (e) As a consequence o (d), p-ca (X) < dimX+1, i X is a locally ini e CW-complex wi h only one F euden hal end. ( ) The no ion o ca ego ical sequence due o Fox (see [10]) can be ansla ed in o p ope e ms. Namely, gi en a space X in φ^ a sequence o open se s V c c Vn — X o which each di e ence Vι - Vι_ is p-ca ego ical (Vo = 0), is called a p-ca ego ical sequence. I is easy o check ha p-ca X < n i and only i X admi s a p- ca ego ical sequence o leng h n . By using his esul , one shows he inequali y p-ca (X xF)< p-ca (ΛT) + p-ca (Γ) - 1. (g) The inequali y max{ca X, ca Y} < ca (X x Y) holds in o - dina y L-S ca ego y bu no in p ope L-S ca ego y as he ollowing example shows. Le X = R2 and Y = J hen ρ-ca (X x Y) = 1 < p-ca (R2). The ollowing de ini ion p o ides a new p ope L-S in a ian which is he ansla ion o Ganea's s ong L-S ca ego y in o p ope homo opy (see [15]). DEFINITION 1.9. gi en a space X in φ^, he s ong L-S ca ego y o X is he smalles in ege p-Ca (X) such ha he e exis s a space Y in *Poo p-homo opically equi alen o X which can be co e ed LUSTERNIK-SCHNIRELMANN INVARIANTS 205 by p-Ca (X) closed se s each wi h he same p-homo opy ype as /. Ob iously p-ca (X) < p-Ca (X). The ela ion be ween p-Ca (X) and he 1-LC a oo condi ion is gi en in he ollowing heo em. We i s ly ecall ha a space X in φoo is 1-LC a oo i gi en a sequence o compac subse s {Lj} wi h Lj C in Lj+ι and X = JLj, he in e se sequence (1.9.1) πι(X-Lι,a{ x))<^-πι{X-L2,a{ 2))X is i ial, whe e a: J —• X is a p ope map wi h a{[ ι?, oo)) C X - Li (/ > 1) and θj is he inclusion induced homomo phism ollowed by he change o basepoin isomo phism gi en by α|[ί, , ί, +i]. THEOREM 1.10. Le X be a space in y^ wi h one F euden hal end. I p-Ca (X) = 2, X is l-LC a oo i and only i he in e se sequence (1.10.1) H^X - Lό+-Hi(X - I*) < *-Hι(X-Ln)< is i ial. This heo em is an immedia e co olla y o PROPOSITION 1.11. Le X be as in Theo em 1.10. Assume ha X can be co e ed by wo one-ended p-ca ego ical closed subse s U and V. Then X is 1 - LC a 00 i and only i he in e se sequence (1.10.1) is i ial. P oo . Le {£/, V) be a p-ca ego ical open co e ing wi h U and V one-ended. We claim ha UnV also is one-ended o equi alen ly ha he in e se sequence (1.11.1) HoiUnV-Li)* < -HQ(UnV-Ln) is i ial. In he commu a i e diag am Hχ{X-Ln) « Hχ{X-Lnχ) H0(U nV - Ln) H0(U-Lnι)QHQ(V-Lnι) « H0(U-Ln whe e he e ical a ows a e p o ided by he espec i e Maye - Vie o is sequences. We may choose nι > n > n such ha he uppe 206 R. AYALA, E. DOMINGUEZ, A. MARQUEZ, AND A. QUINTERO and lowe ho izon al a ows a e i ial. This implies ha he com- posi ion o he mo phisms (1) and (2) is i ial and so he in e se sequence (1.11.1) is i ial. We ha e he ollowing ac s: (i) Since U and V a e p-ca ego ical, we may choose n?>> n such ha any loop ei he in U - Ln o in F- Ln^ is null-homo opic in X-Ln. (ii) On he o he hand, since U n V is one-ended he e is n4 > n-$ such ha all he connec ed componen s o U Π V - Ln^ a e included in he same connec ed componen o U Π V - Ln^. Gi en any loop /: (/, {0, 1}) —> (X-Ln^, *), by Lebesgue's Lemma we may ind a pa i ion O = ίo<ίi <•••<*„ = 1 such Λa [ i> U~ ] is included ei he in U-Ln^ o in V -Ln^ The e o e, by (ii), / is a loop in X-Ln^ ha can be exp essed as a p oduc o loops ei he in U - Ln^ o in V - Ln^ and each ac o is null-homo opic in X - Ln by (i). Thus he mo phism LnJ ^ πx(X - Ln) is i ial and X is 1-LC a oo. COROLLARY 1.12. Le X be a homologically i ial open n-mani old (n > 3) wi h p-Ca (JΓ) = 2. Then X is -LCa oo. P oo . As n > 3, Poinca e-Le sche z Duali y a gumen s (see [7; 3.2]) show ha he in e se sequence (1.10.1) is semis able (i.e. sa is ies he Mi ag-Le le condi ion) and i s in e se limi is i ial. Then by [18; Π.6.2.2] he in e se sequence (1.10.1) is i ial. Now he esul ollows om Theo em 1.11. 2. P ope L-S ca ego y and L-S ca ego y in p o-Top. We ecall ha ca (/) < n o a con inuous map / om X o Y, i he e is an open co e ing {V , ... , Vn} o X such ha Vj is homo opically i ial o 1 < j < n . In [1] a no ion o L-S ca ego y o in e se sys ems is de ined. Name- ly, gi en an in e se sys ems χ = {Xa Paβ} > he L-S ca ego y o χ, ca (χ), is < n i o each a he e exis s β > a such ha C3 (paβ) < n. I p o S0/1, is he ca ego y o opological in e se sys ems and p o- mo phisms, Edwa ds and Has ings (see [8]) ha e p o en ha he e exis s a na u al unc o ε: φoo —> p o-<5ζ^, ε(X) being he in e se LUSTERNIK-SCHNIRELMANN INVARIANTS 207 sys em {U <— U2 *— ••} whe e £4 = X - Q and Q is an in- c easing sequence o compac s wi h Q C in Q+1 and X = |J Q . A ela ion be ween ρ-ca (X) and ca (ε(X)) is gi en by 2.1. THEOREM. 7 X is a space in φ^, ρ-ca (X) > ca (ε(X)). P oo . I {Wγ, ... ,Wn} is a p-ca ego ical open co e ing o X, we conside he co e ing {WjΠUk} o U^. Le 7: PF7 —• / and α;:/->Ibe p-maps such ha he e a e p-homo opies W: WjXl —• X be ween α7 o 7 and he inclusion W7 c X. Fo each k he e exis s s(k) > k such ha Hj(Wj Π C/J( c) x /) Q Uk. Bu since / is con ac ible i ollows ha Us^ is nuU-homo opic in each Wj n J75 So ca (β(JΓ)) < p-ca (X). 2.2. REMARKS, (a) We now conside he ca ego y (p o-,%^, whose objec s a e a ows :χ—>A whe e χ is an objec in and A is a space ega ded as he cons an in e se sys em. Mo phisms a e pai s (φ, h): whe e φ: χ -^ ξ is a p o-mo phism and h: A -+ 5 is a con inuous map compa ible wi h 0 ia he bonding maps. Edwa ds and Has ings p o ed ha ε: <Poo —• (ρ o-,%^, 3 c/ι) gi en by e(X) = X «— C/i *- C/2 is a ull embedding (see [8]). We can also p o e ha ca (ε(X)) < p-ca (X) in his case. (b) The e exis spaces wi h ca (ε(X)) < p-ca (X). Indeed, le X be he punc u ed o us. Then p-ca (X) = 3 acco ding o Co olla y 3.3 bu ob iously ca (ε(X)) = ca ^S1) = 2. 2.3. PROPOSITION. I M is a compac connec ed iangulable man- i old wi h bounda y and W = M - dM hen ca (<9M) < p-ca ( Γ) < ca (9Jl/)+ca (A ). Fu he mo e, i M is con ac ible hen p-ca (W^) = ca (<9M). P oo . The in e se sys em ε(W) in (p o-^/^, 9*/ι) is Now we ha e ca [/: dM x [0, 00) -• W] < ca (dM x [0? 00)) = ca (0Λ/) and ca (ε(X)) = max{ca (/), ca (9M)} = ca (ΘM). So, by Rema k 2.2(a) caX(ΘM) < p-ca (FF). On he o he hand i we ake a copy o M, M' C W hen p-ca ( W) < p-ca (M; V /) + -cdi {dM x J) = ca (M) + ca (ΘM) by 1.6( iii) and 1.8(b). 208R. AYALA, E. DOMINGUEZ, A. MARQUEZ, AND A. QUINTERO I we assume ha M is con ac ible, we ge ca (<9M) < p-ca (FΓ) < ca (dM) + 1. Bu ac ually p-ca (W) < caX(dM). Indeed, i {Wj}j<n is a ca ego ical co e ing o dM, we may assume ha W is a sub- polyhed on (see P oposi ion 1.5). By [22; 6.30] we ex end he p ope de o ma ion o Wx x J in o {*} x / o a p ope de o ma ion o Wι x J UM. Thus {W x J uM,W2x J, ... , Wn x J} is a p- ca ego ical co e ing o W. The in a ian ca ε(X) can be used o gi e some esul s on he beha iou o he in e se sequence (2.3.1) πx(X - Kx) <- π1 (X - K2) < <-πx(X - Kn) < whe e {Ki} is a sequence o compac subse s wi h K C m Ki+x, and 2.4. THEOREM. Le X be a locally ini e polyhed on wi h one F eud- en hal end and ca ε(X) = 2. Assume ha he in e se sequence o abelian g oups HX{X - Kx) - K2)Hx{X-Kn) is i ial. Then he in e se sequence (2.3.1) is p o-isomo phic in o an in e se sequence o ini ely gene a ed g oups. THEOREM 2.5. Unde he hypo heses o Theo em 2.4, X is ί-LC a oc i and only i he in e se sequence (2.3.1) is semis able, i.e. imι πx(X - Kj) = *. P oo o Theo em 2.4. We may choose he sequence {Kj} wi h Uj = X - Kj subpolyhed on o X. Up o p o-isomo phism we may eplace nx{X-Kj) by Gj = πλ(Uj) in (2.3.1). By using he 1-skele on o X, i is easy o check ha he e is a commu a i e diag am F(L2) (2.4.1) whe e F(Ln) deno es he co esponding ee g oup o basis Ln , he di e ences Ln - LΛ+1 a e ini e, and he bonding mo phisms a e he na u al inclusions. LUSTERNIK-SCHNIRELMANN INVARIANTS 209 Since he in e se sequence G ^G ^Q , <_ G < is i ial, by abelianizing (2.4.1) we eadily check ha G*b is ini ely gene a ed o n > 1. Now, since ca ε(X) = sup{ca [/« : Un - Un^]} = 2 we ge om [15; 7.3] ha o any n > 2, he e is μn such ha Un > Un^VUn^ is commu a i e up o homo opy. This diag am induces a commu a i e diag am Gn • Gn- * Gn- (2.4.2) j/.. [in_u Gn- • Gn- x ^Λ-I whe e /„* induces he na u al epimo phism a*b —• (α, δ). F om he Ku osh Subg oup Theo em (see [17]) he g oup l~ uΔim{Gn) is a ee g oup. The e o e, μm : (?„ -• Im//^* is an epimo phism on o a ee g oup and so (Imμn*)ab is a ini ely gen- e a ed ee abelian g oup. Hence, Im/^* is a ini ely gene a ed ee g oup. On he o he hand, Δ is injec i e and so he commu a i i y o (2.4.2) yields a na u al epimo phism Imμ * -» Im/W*. Thus, s im is a ini ely gene a ed g oup. I is a well-known ac ha (2.3.1) is p o-isomo phic o he in e se sequence (2.4.3) Imiu ^-Im/2*^ ^- Im/„*<—••-. P oo o Theo em 2.5. I in addi ion lim1 G}• = *, we know ha he in e se sequence (2.3.1) sa is ies he Mi ag-Le le condi ion (see [18; p. 174]) and we may assume ha he bonding mo phisms in (2.4.3) a e on o. We shall deno e Im im by Hn .