PROPER HOMOTOPY CLASSIFICATION OF GRAPHS
R. AYALA, E. DOMINGUEZ, A. MARQUEZ
AND
A. QUINTERO
ABSTRACT
This
wo k p esen s a classi ica ion o he p ope homo opy ypes o locally ini e 1-dimensional CW-
complexes.
0. In oduc ion
A p ope map (p-map) is a con inuous map/: X-> Y such ha
~ K)
is compac
o any compac K. P ope homo opy (p-homo opy), p ope homo opy equi alence,
e c.,
a e de ined in he na u al way. A g aph is he unde lying space o a connec ed
locally ini e 1-dimensional CW-complex. We conside on he hal -line
IR
+
he na u al
CW-s uc u e whose 0-cells a e he se o posi i e in ege s. Any cellula embedding
:U
+
-*G is called an in ini e b anch o he g aph
G.
The e ex /[0) is called he oo
e ex o he b anch.
I is a well-known ac ha any compac g aph has he same o dina y homo opy
ype as a wedge o ini ely many copies o he
1-sphe e
S1. We call ha numbe he
genus o he g aph. So he genus classi ies he o dina y homo opy ype o compac
g aphs and ac ually gi es us an equi alence wi h he ca ego y o
ini ely
gene a ed ee
g oups. Ne e heless, he e a e easy examples o non-compac g aphs wi h he same
o dina y homo opy ype bu di e en p-homo opy ypes. The aim o his wo k is o
s a e he co esponding p ope homo opy classi ica ion o non-compac g aphs. This
p oblem was posed o us by P o esso H. J. Baues as he s a ing poin o he s udy
o p ope homo opy om a combina o ial poin o iew.
As in he case o open su aces (see [6]), he no ion o F euden hal end is he main
ool used o ob aining such a classi ica ion. A F euden hal end o a space X is an
elemen o he in e se limi ^(X) =
imnQ(X—
K),
whe e G anges o e he amily o
compac se s o X and
no(X—
K) s ands o he se o connec ed componen s. When
A'
is
a 7^-locally compac c -compac space, we can use a coun able sequence Kx c K2
c ... o compac subse s o ob ain !F{X). The opology o X can be enla ged o a
opology on X = X
U
^(X) in such a way ha ^(X) u ns ou o be homeomo phic
o a closed se o he Can o se (see [3] and [1]), as ollows. The opology on X is
gene a ed by he opology o X and he se s U = U J U*, whe e Ueno(X—Kn) o
some
n
and
U*
is he se o ends gi en by he sequences {Un} e
lim
no(X—
Kn)
such ha
he e is a posi i e in ege «0 wi h Un a U.
Ano he use ul p ope in a ian is he no ion o p ope end. A p ope end is a p-
homo opy class o
p-maps/:
U+
-*•
X. The se o p ope ends o Xis deno ed by F(X)
and he e is a map 0 om F(X) on o
Recei ed
11 Sep embe 1989; e ised 8 Decembe 1989.
1980
Ma hema ics Subjec Classi ica ion 05CXX,
54C10,
55P15.
Bull. London
Ma h.
Soc. 22 (1990) 417-421
15 BLM 22
418 R. AYALA, E. DOMINGUEZ, A. MARQUEZ AND A. QUINTERO
A F euden hal end a is said o be s able i ^-1(a)
is
jus one poin . O he wise, a is
non-s able o uns able. We deno e he se o uns able ends o X by
^
ns
(X),
which
u ns o be a closed subse o ^{X). In o de o know whe he a F euden hal end a
is s able o no , we can use he bijec ion Vax n-^X—K^ = 6~ a) (see [5]), whe e *j
is a poin in he componen o X— K} belonging o a (see [1] o mo e de ails).
I is easy o check ha any p-map
:X-*Y
induces a con inuous map
m
{P{X),
&ns{X))^{3?{X
&n%Y)) such ha i /is a p-homo opy equi alence,
+ is a homeomo phism. No ice ha he e exis p-maps/which a e no p-homo opy
equi alences bu hei induced /* a e homeomo phisms.
I
we
embed Can o 's se in [0,1] x
{0}
e U2, i is a well-known esul ha he e is
a bina y ee, called he Can o ee, embedded in
IR
2
in such a way ha he ollowing
condi ions hold. (1) The numbe o e ices on he i h le el is 2*. (2) The space o
bounda y poin s o he embedded ee is Can o 's se . (3) The e is a na u al 1-1
co espondence be ween he poin s o he Can o se and he se o in ini e b anches
o he ee s a ing om he oo e ex.
1.
Cha ac e is ic
pai o
a
g aph
DEFINITION 1.1. A g aph is oo-s able o s able a in ini y i any F euden hal end
o X is s able. A g aph is o ally uns able i any end o X is uns able.
The ollowing esul s a e easy o p o e.
PROPOSITION
1.2. (a) The s able ends and he uns able ends a e p-homo opy
in a ian s.
(b) A g aph is oo-s able i and only i i s undamen al g oup is ini ely gene a ed.
(c)
Any
non-compac g aph
G ^
U+
is
p-homo opy
equi alen
o a g aph
wi h no
end-
e ices.
DEFINITION 1.3. The genus o an oo-s able g aph G is he ank o he
undamen al g oup n^G). The cha ac e is ic pai o an oo-s able g aph
is
(J
5
",
g),
whe e
!F is he space o F euden hal ends o G and g is he genus o
G.
The cha ac e is ic
pai o a non-oo-s able g aph is (& ,&'DS).
Two cha ac e is ic pai s (^gj and (&2ig2) a e isomo phic i gx = g2 and
3F
X
and
^2 a e homeomo phic. Two cha ac e is ic pai s (&'x,!F ) and ($F2,!F ) a e
isomo phic i he e exis s a homeomo phism o pai s h.iJF^SF' )
-»•
(^2,^ls).
PROPOSITION
1.4. (a)
Gi en
a pai {JF,
g),
whe e
^
is
a closed subse o he Can o
se and g is a na u al
numbe ,
he e exis s an oo
-s able g aph whose cha ac e is ic
pai
is
( ,g).
(b) Gi en a pai o
closed
subse s
(#",
!F') o he Can o se he e exis s a g aph
whose
cha ac e is ic
pai is he gi en one.
P oo .
Gi en a pai
(i^,
g),
we may conside in a na u al way he Can o sub ee
de ined by he subspace J5", and hen we glue a wedge o g copies o
1-sphe es
a he
oo e ex. We ob ain in his way an oo-s able g aph whose cha ac e is ic pai is
{^,g). A g aph associa ed o (&',&'') is ob ained by gluing exac ly one
1-sphe e
a
each e ex belonging o an in ini e b anch de e mined by a poin o $F'.
PROPER HOMOTOPY CLASSIFICATION
OF
GRAPHS
419
DEFINITION
1.5. The
g aphs gi en
in
P oposi ion
1.4 a e
called canonical g aphs.
An uns able b anch
o a
canonical g aph
is an
in ini e b anch whe e he e
is an
1-sphe e
a ached
a
each e ex.
2.
P ope classi ica ion o g aphs
F om P oposi ion
1.2(c) we
shall deal only wi h g aphs wi h
no
end- e ices.
Gi en
a
non-compac connec ed g aph
G
wi h
no
end- e ices,
we se up a
sequence
o compac connec ed g aphs
Kx c K2 c ...
wi h
[j K} = G and a
spanning ee
T
c G, as
ollows.
Le
Kx be any
connec ed compac subg aph
o
G.
Le K* be he
ex ension
o Kx
by adding
all he
inciden edges
o K Le {D)} be he
connec ed componen s
o
G
—
K The
e ices
in
(K*—Kx)
n D] a e
joined
by a
ini e ee
T) a D). We
de ine
K2 by
adding
o K'2 =
K *
U
{[jj T)} all he
edges wi h bo h e ices
in K2. We
i e a e
ha p ocess
o
ob ain
he
abo e sequence. No ice ha i
Z)"
is a
connec ed componen
o
G —
Kn,
we ha e ha cl
(Kn
—
Kn_x)
D
cl
(Z>")
is a
compac connec ed g aph L" joined
o
Kn_x
wi h
a se o
edges
A".
We
now
choose
a
spanning ee
T
c:
G as
ollows.
We
ake
a
maximal ee
in Kx
and
all he
ees
TJ(n ^ 1). We
join hem wi h edges
o A".
No ice ha hose pa hs
in
T
going be ween
wo
e ices
o Kn
—
Kn_2 miss
he
subg aph
Kn_2.
PROPOSITION
2.1.
Gi en
G and T as
abo e, he e
is a
g aph
G'
p-homo opically
equi alen
o G
wi h
he
same ee
T and
such ha
all he
cycles
o
G'
a e
loops.
P oo .
Gi en
an
edge eeG—T which de e mines
a
cycle
o
G,
le e and we be he
e ices
o
e
and Le be he
unique pa h
in T
unning om
e o we. We
subdi ide
e by
adding
a new
e ex
ce and we
a ach
a new
edge
ye
wi h e ices
e and ce o G. We
can
now
glue
a
2-disk
D
iden i ying
dDe
wi h
Le
U
c~^W~e
U
ye,
whe e cjwe
is he
1 -cell
de ined
by ce and we.
A e doing ha
o
each e ex
e
which does
no
belong
o T
(no ice ha
all
hose gluings
a e
locally ini e
by he
cons uc ion
o T), we
ob ain
a
2-dimensional CW-complex
G
p-homo opically equi alen
o G
ela i ely
o T.
Pe o ming
now he
ob ious collapses,
we
ob ain
a
g aph
G'
p-homo opically
equi alen
o G and
such ha
all he
cycles
a e
loops.
PROPOSITION 2.2.
Any
g aph G is
p-homo opically
equi alen
o a
canonical
g aph.
P oo
By
P oposi ion
2.1 we may
assume ha
G is a
di ec ed ee Twi h some
loops a ached
a he
e ices.
By an
ob ious p-homo opy equi alence
we may
conside ha
he
numbe
o
inciden edges om
a
e ex
is
1
o 2 and he
numbe
o
inciden edges
o a
e ex
is 1
(excep
he
i s e ex, which
has no
inciden edge
o
i ).
The e o e
he
ee
o G is now a
sub ee
o he
Can o ee.
I
G is
oo-s able, a e
a
sui able p-homo opy equi alence
we may
ega d
all he
loops
a he
oo e ex. O he wise
we can
a gue
in he
same
way o
each maximal
s able in ini e b anch whose oo e ex belongs
o an
uns able b anch. Only
he
p oblem
o
dis ibu ing
he
loops emains.
We now
ake
any
e ex
wi h
n >
1
loops
and subdi ide
any
inciden edge om
adding (n— 1)
new
e ices.
We may
emo e
(«—
1) loops om
and pu
each o hem
a one new
e ex. This comple es
he p oo .
15-2
420 R.
AYALA,
E.
DOMINGUEZ,
A.
MARQUEZ
AND A.
QUINTERO
LEMMA
2.3. Le L be a sub ee o he Can o ee. Gi en a ini e disjoin open
co e ing {Vx,
V
2
,...,
Vn) o ^(L), we can choose a compac sub ee Locz L such ha
each se U* o ends de ined by a componen U c L —
L
o
is included in some V .
P oo .
We ha e ha each
V
is closed in ^(L) and so compac . Since he amily
o se s U*, whe e U is a componen o L — K, K unning o e he compac se s o L,
is an open basis o he opology o i^(L), we can co e each V wi h only ini ely
many U* and hen we can easily choose he ee Lo we a e looking o .
PROPOSITION
2.4. Le G and G' be wo
canonical
g aphs wi h ees L and L'.
Then any con inuous map h:{& {G),P(Gns))->{&{G'),&{G'ns)) induces a
p-map
h':
(L,
Lns) -»(Z/, L'ns) unique up o p-homo opy o pai s such ha
h'
i
,
=
P oo .
We ake an inc easing sequence o compac sub ees L[ c
L'
2
c ... o L.
Le nx be he numbe o connec ed componen s o L'
—
L[
and } be he i s e ex in
he componen K^(l ^y'^ nx). Applying he abo e lemma o
we may choose a ini e sub ee Lx o L such ha he ends de ined by each componen
Vk o L
—
L1 a e included in some
h~xV'*k).
Le k be he i s e ex in Vk. We de ine
h'{ k) = 'm. All he e ices in L1 a e mapped o he oo e ex
'
o
o
L'.
Now we ake
L'
— L[ and we a gue eplacing L and L by he maximal ees in each V and he
maximal ee in each Vk. So we may choose a ini e sub ee L2 a L such ha he se
o ends de ined by each componen o L
—
L2 is mapped by h in he se o ends de ined
by some componen o L'
—
L'
2
.
We de ine h' as abo e o he i s e ices and h )
= 'm i is a e ex in L2 —
L
x
and
V
k
is he componen o in L —
L
We may i e a e
his p ocedu e and de ine induc i ely a p-map h'.L^L' such ha h' = h , (L).
No ice ha h' ac ually maps Lns in o L'ns.
Le h be ano he p-map wi h h# = h. By using a p ope simplicial app oxima ion
[2],
we may assume ha h maps e ices in o e ices and h( 0) =
h'(
Q
),
whe e
Q
is
he oo e ex o L. F om he p ope ness o h we can choose inc easing sequences
^c^c.cL and IJcLjC.cL'o compac sub ees such ha any com-
ponen o L — K} is mapped by h in o a componen o L'
—
L' We can easily de ine
a sequence Ln a Kn a Ln
c=
AT
n
c: ...,whe e Ln is gi en in he de ini ion o h'.
Le Ai be he se o i s e ices o componen s o L
—
Kn . As
h'*
= h* = h, we
claim ha h'( ) and h( ) a e in he same componen o L'
—
L'
n
i
eA
Indeed, i
h( )eC[ and
h'( )eC2
hen h'^C* and
h'
l
C*
a e disjoin open se s in ^(L), and
aking an end a s a ing om we ob ain ha
h((x,)eC[*
n C*. So we may ake a pa h
e in L'
—
L'
n
going om h( ) o h ). In his way we can de ine he p-map
by Ho
|
L x
{0}
= h, Ho
|
L x
{1}
= h' and H0( , ) = e ( ). Gi en eA} and
weAJ+l,
i is
ob ious ha he loop H0( w x
{0,1} U
{ ,
w}
x /) is null-homo opic in L'
—
L'n. So Ho
ex ends o a p-homo opy H be ween h and h', which is easily checked o be a
p-homo opy o pai s.
COROLLARY
2.5. h {^{G),^{Gm))^{^{G'),^{G'm)) is a homeomo phism i
and only i
h'
is a p-homo opy equi alence.
PROPER HOMOTOPY CLASSIFICATION
OF
GRAPHS
421
LEMMA 2.6.
Le :
A-* Xbe
a
p-map
wi h he p ope homo opy ex ension p ope y
and
h:X->
Yap-homo opy equi alence. Then A J X isp-homo opically equi alen
o
A
V
h
X.
P oo .
I is
s aigh o wa d, ollowing wi h mino changes
he
p oo in o dina y
homo opy
(see
[7,1.3.10]).
THEOREM 2.7. Two g aphs G
and
G' ha e he same p-homo opy ype i and only
i
hei cha ac e is ic pai s
a e
isomo phic.
P oo .
By
P oposi ion
2.2 we can
conside hese g aphs canonical.
I
hey
a e
bo h oo-s able
he
esul ollows om P oposi ion
2.4.
Assume ha hey ha e
non-
S able ends
and le h:
C^(G), ^(Gns)) -> (&(&), &(G'ns))
be a
homeomo phism
and T
and
T' he
ees
o G and G'
espec i ely.
By
Co olla y
2.5
he e
is a
p-homo opy
equi alence
o
pai s
h!
:(T,T0)
-•
(T,T'O),
whe e
To and T'o a e he
co esponding
non-s able sub ees. Ac ually, gi en
any
inc easing sequence
o
compac sub ees
L[
<=
L'2
e ... in 7", we may
ind
a
sequence
o
compac sub ees
Lx c L2 c ... in T
such ha
i Ap
B}
a e he
se s
o
i s e ices
o
componen s
o
T—
L} and T'
—
L'}
espec i ely,
h'
maps
A}
in o
Bj and all he
e ices
o L}
—
L}_x in o
Bj_1
(see
he
p oo
o P oposi ion 2.4).
Now by
Lemma
2.6, i
we a ach
one
copy
o S1 o
h'{ )
o
each
copy
o S1
a ached
a we
ob ain
a new
g aph G*
in he
p-homo opy class
o
G
wi h
ee
7". We
inally apply P oposi ion
2.4 o
educe
G* and G' o he
same g aph
up
o p-homo opy equi alence.
ACKNOWLEDGEMENTS. This wo k
was
pa ially suppo ed
by he
p ojec s
CAICYT 0812-84
and
'Consolida ion
de
g upos
de
T abajo'-PAICYT (Andalucia).
Re e ences
1.
E.
DOMINGUEZ
and L. J.
HERNANDEZ,
'
Rema ks abou p ope ends',
Re .
Roumaine Ma h. Pu es Appi,
o appea .
2.
F. T.
FARREL,
L. R.
TAYLOR
and J. B.
WAGONER,
'The
Whi ehead Theo em
in
p ope ca ego y',
Composi io Ma h.
27
(1973)
1-23.
3.
H.
FREUDENTHAL,
'Ube
die
Enden opologische Raume
and
G uppen', Ma h. Zei .
33
(1931)
692-713.
4.
W.
MASSEY,
Algeb aic opology.
An
in oduc ion, G adua e Tex s
in
Ma h.
56
(Sp inge , Be lin, 1977).
5.
M.
MIHALIK,
'
Semis abili y
a
in ini y
and he end o a
g oup ex ension', T ans. Ame . Ma h.
Soc. 277
(1983)
307-321.
6.
I.
RICHARDS,
'On he
classi ica ion
o
noncompac su aces', T ans. Ame . Ma h.
Soc. 106
(1963)
259-269.
7.
V.
ROHLIN
and D.
FUCHS,
P emie cou se
de
opologie
(Mi
Publishe s, Moscow, 1981).
R. Ayala,
A.
Ma quez
and A.
Quin e o
E.
Dominguez
Facul ad
de
Ma ema icas Depa amen o
de
Ma ema icas
Uni e sidad
de
Se illa Facul ad
de
Ciencias
41080 Se illa Uni e sidad
de
Za azoga
Spain 50009 Za azoga
Spain