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 .