HOUSTON JOURNAL OF MATHEMATICS, Volume 1, No. 1, 1975.
THE HOPF EXTENSION
THEOREM
FOR TOPOLOGICAL
SPACES
Kiiti Morita
In celebration of the start of
Houston Journal of Mathematics
1. Introduction. Let X be a topological space. Then by the coveringdimension
of X, denoted by dim X, we mean, as in our previous paper [5], the least integer n
such that every finite normal open cover of X is refined by a finite normal open cover
of X of order •< n+l; in casethere is no such an integer n, we define dim X = o•.
Let A be a subset
of X. Foreachinteger
n •>0, let Hn(X,A)bethen-th•ech
cohomologygroup of (X,A) with coefficientsin the additive group of integerswhich is
defined by usinglocally finite normal open coversof X.
Let I be the closed
unitinterval[0,1] in thereallineandi n theboundary
of In.
A continuous
map f: (X,A)->(I
n,in) is calledessential
if any continuous
map
g:(X,A)->(I
n,in) with glA=flA satisfies
g(X)=In; otherwise
f is saidto be
inessential.
Hencef is inessential
iff thereisa continuous
mapg:X->•n whichis an
extension of flA.
Thus, the following theorem may be viewed as a generalization of the Hopf
extension
theorem.
THEOREM 1. Let (X,A) be a pair of topologicalspacessuchthat dim X/A •< n.
Thena continuous
mapf: (X,A)->(In,;)n)isinessential
ifff*.' Hn(In,i n)-->Hn(X,A)is
zero, where n •> 2.
Now, let us considerthe casewhere A is C-embeddedin X. Then, by Shapiro [8],
for every locally finite, countable, normal open cover U of A there is a locally finite,
countable, normal open cover V of X such that V C3A refines U. On the other hand,
¾
the Cech cohomology groups with coefficients in the additive group of integers are
¾
naturally isomorphicto the correspondingCech cohomologygroupsdefined by using
locally finite, countable, normal open covers (cf. [5, Theorem 6.8]). Hence we can
definethe coboundary
operator6: Hn-I(A)->Hn(X,A),
and the cohomology
sequenceof (X,A).
121
122
KIITI
MORITA
ß" ->Hn-1(X)• Hn-1(A)-• Hn(X,A)->"is exact, where i: A --* X is the inclusion map.
In the commutative diagram
Hn-1
(X)
i*
>Hn-!(A)
6
, Hn(X,A)
l(flA)*
0
, Hn-1
(in)
>Hn(In, i n)
)0
the upper and lower sequencesare exact.
Therefore from Theorem 1 we obtain the following theorem which has the same
form as the usualHopf extensiontheorem.
THEOREM 2. Let X be a topological space and A a subset of X which is
C-embedded
in X. Let dimX/A •<n andn •>2. Thena continuous
mapg:A ->)n can
beextended
toa continuous
mapfromX to)n iffg,Hn-l(]n)Ci*Hn-l(x).
The Hopf extension theorem has been proved hitherto for the following cases.
I.
(X,A) is a relative CW complex and dim(X-A) •< n (Spanier [9] ).
II. X is a paracompactnormal spacewith A closedand dim X •< n (Dowker [ 1] ).
In case! the condition "dim(X-A) •< n" is equivalentto "dim X/A •< n", and, as
wasproved in [5 ], dim X •< n implies dim X/A •< n for any pair (X,A) of spaces.
In both cases A is C-embedded
in X.
In case I the singular cohomology groups are used. But, if (X,A) is a relative CW
complex,
thenthe •'echcohomology
groupHn(X,A)is naturally
isomorphic
to the
correspondingsingular cohomologygroup of (X,A) by virtue of [ 5, Theorem 6.1 ] and
[9, P. 428], and the cohomology sequenceof (X,A) is exact for both cohomology
groups,
andhence
thecondition
"g*Hn-l(i
n)C i*Hn'l(x)"remains
to beequivalent
if Hn-I(A) andHn-I(x) arereplaced
by the corresponding
singular
cohomology
groups.
Thus, our Theorem 2 contains the Hopf extensiontheorem for the casesI and II
above.
In õ4, Theorem 1 and the arguments in its proof will be applied to covering
dimension.
Throughout the paper, N denotesthe set of positive integers.
2. Proof of Theorem 1. Since the "only if" part is obvious, we have only to
HOPF EXTENSION
THEOREM
123
prove the "if" part.
Sincedim X/A •< n, for any normalopen coverG of X there existsa locally finite
normal open cover H of X such that H is a refinement of G and the order of
{ H, St(A,H,I H C H, H ClA = {b} doesnot exceedn+l.
Toprove
this,letG= { Gx I XCA} bea locally
fifiitecozero-set
cover
ofX and
letAo= {Xc A I Gx C3
Av• ½}.Let{Fx I XCA } bea locally
finitezero-set
cover
of
X suchthatFx C Gx for eachX C A. Letusput
Fo=kJ(Fx [ XGAo}, Go=kJ(Gx[XGAo}.
Thenby [6, Lemma2.3] Fo is a zero-set
and GO is a cozero-set.
Hencethereis a
continuous
maph:X-+I suchthath'l(0)=Fo
, h-I(1)=X-G
o. Let usput
L= (xGXIh(x)•<«}.
ThenACF oCIntL. Hence(Gx-L,G OlxGA-A o} is
the inverseimageof a locally finite cozero-setcoverof X/A underthe quotient map of
X ontoX/A. Hence
thereisa locallyfinitecozero-set
cover( HX,Ho I XGA - Ao} of
X such that its order does not exceed n+l
and
Hx C Gx- L for eachX GA- Ao;A C Ho C Go-
Then
H= ( HX,HoC3
G/aI XGA- Ao,/a
GAo} isa locally
finite
cozero-set
cover
of
X andHo = St(A,H).ThusH isa desired
coverof X.
Let ( Wi I i GN } bea normalsequence
of opencovers
of In suchthateachset
belonging
to Wi hasdiameter
•< 1/i.Thenthereisa normalsequence
• = ( Ui I i G N }
of open coversof X satisfyingthe following conditions:
(1) Ui isarefinement
off-1(Wi)fori GN,
(2) order(U,St(A,Ui)
I U GUi,U chA = • } •<n+l.
Let(X,•) bea topological
space
obtained
fromX by taking{ St(x,Ui)I i GN }
as a local base at each point x of X, and X/• the quotient spaceobtained from (X,•)
by identifyingtwo pointsx andy suchthaty G St(x,Ui)for eachi GN. Let usdenote
by ß the composite of the identity map from X to (X,•) and the quotient map from
(X,•) to X/•. Then ½:X -+ X/• is a continuousmap and the spaceX/• is metrizable.
This fact is proved in [4].
For any subset K of X let us put
Int(K;•)= {xGX[ EIiGN: St(x,Ui)cK} .
ThenInt(K;•) is an opensetof (X,•) and;b-lqb(Int(K;•))=Int(K;•).Let
usput
further
124
KIITI MORITA
(3) Bi = Int(St(A,Ui);•),
(4) Vi= (Int(U,•)lUcU i).
ThenVi is an opencoverof X suchthatVi+1> Ui+1> Vi where"•" means
"isa
refinement
of". Moreover,
( ½(Vi) [i c N) is a normalsequence
of opencovers
of
X/• suchthat (St(y,½(Vi))I i 6 N ) is a localbase
at anypointy of X/•. Thisfactis
proved in[4].
Since
(5) St(A,Ui+1) C Bi C St(A,Ui)
we have
C1½(A)C St(½(A),½(Vi+1)) C ½(Bi)C St(½(A),
½(Vi_
1))
and hence
(6) C1 ½(A)= c•(½(Bi){i • N).
On the other hand, forj •> i+l we have by (5)
(7)(½(X)
- ½(Bi))
t•½(U)
=½forU• Ujwith
U
Since
Vj isacover
ofX, wehave
(8) ½(X)-½(B
i) CL•{ ½(Int(U;
•) [U• Uj,U C•A=•) .
From(2), (7) and(8) it follows
thattheorderof ½(Vj)on•b(X)-½(B
i) does
not
exceed n+l forj > i. Hence by Nagata [7]
(9) dim(½(X)- •b(Bi))•<n for i C N.
Therefore,by (6) andby the sumtheoremon dimensionwe have
(10) dim(X/•-
C1 ½(A)) •<n.
By(1) thereisa continuous
mapg:(X/•, ½(A))-+(In,•n)such
thatf = go½.
Let {•ala G,9,) be the setof all normalsequences
of opencovers
of X
satisfying
(1),(2) and(3).If each
cover
of•a isrefined
bysome
cover
of•13wewrite
•a<•13;
inthis
case
there
isacanonical
map
½•a:
then
Ca
•ø½•=Ca
7and
if Cadenotes
themap
from
X toX/•a defined
above
then
½a=½•o½13
when
•a<cb$3
(cf.[4]).Asisproved
in[5],{½•la•) defines
an
isomorphism
li_+m
{nn(x/•a,
Ca(A)),
(½a•)
*) -•Hn(X,A).
Letga:(X/•a, Ca(A))• (In' •n) be a continuous
mapdefined
above
such
that
f=gaøCa'
Then,
if•a< •13
wehave
g13
=gaøCa
•'
Therefore,
if f*: Hn(In,in)-• Hn(X,A)iszero,thenthereissome
a GCZsuchthat
HOPF EXTENSION
THEOREM
125
g•:Hn(I
n,in)-• Hn(x/(I)c•,
•c•(A))
iszero.
Since
gc•(Cl•o•(A))C
in, gc•is viewed
asa continuous
mapha:(X/(I)o•,
C1•bo•(A)
) -• (In,in) andwehave
gcr
=her
ø½o•
where
•bo•:
(X/(I)o•,
•bo•(A))
-• (X/(I)o•
,
C1•o•(A))istheinclusion
map.Since
•: Hn(X/(I)o•,
C1•o•(A))
-• Hn(X/(I)o•,•o•(A))
is
anisomorphism,
h•:Hn(I
n,in)• Hn(X/(I)o•,
C1•o•(A))
isalso
zero.
Moreover,
by(10)
dim(X/(I)o•)/C1
•o•(A)•<n.
Therefore, if the "if" part of Theorem 1 is proved for the case where X is
metrizable
andA is closedin X, thenthe mapha, andhencegc•,is inessential
and
consequently f is inessential.
Thus, as for the "if" part of Theorem 1 we haveonly to proveit for the case
where X is metrizable
and A is closed in X.
3. Proof of Theorem 1 for the caseof metric spaces.Let X be a metric space
andA a closed
subset
of X suchthatdimX/A •<n. Let G = { Gx I XC A) beany
locally finite normalopencoverof X. The followingargumentis givenin [2, the proof
of Theorem 2.2].
Thereexisttwocollections
{ PX[XCAo } and{ Hx I XGAo} of opensubsets
of X suchthatAo C A and
(11)C1PxCHxCGx,
AcnPx:/=½forXGAo;ACUP
x.
(12) {AChPxlXGA
o} issimilar
to {HxlxGA o}.
Since
dimX/A •<n, wehavedim(X-UPx) •<n andhence
by [3,Theorem
1.2]there
is a locally finite collectionV of open subsetsof X suchthat
(13) V is a refinement
of {Hx, X-C1Px } for eachXCAo and alsoa
refinement of G,
(14) X-UPxCU {VIVGV},
(15) order V •< n+l.
Let uswell-order
Ao suchthatAo = {X I X < % } for someordinal
o•o andletQXbe
theunion
ofallVGV such
thatV c3PX:/=½andV C/Po•
=• foreach
o•witho•< X.
The setsV GV for whichV FIPx=½ for all X<o•o shallbe denoted
by
{ V•I • < t3
o}. Letusput
WX= PXU QXforX < o•o.
Thenby (13) wehaveWx C Hx forX< o•o.
IfX 1 <'"<Xr<O• o, v1 <'"<Vs<t3o and
126
KIITI
(i=ln wxi)
n
MORITA
vvi
) . 9,
and if this intersection does not meet A, then r+s • n+l. Indeed, we have then s • 1
by (12) and hence
(i•lWXi)
• (iOlVvi)=(i=QXi)
• (i=Vvi)
and hencer+s • n+ 1 by (1 b). Let us put
U= (Wx,V/•I X(• o,/•(go•.
Then by (14) U is a cover of X and is a refinement of G.
Let N(U) (resp. N(U 5• A)) be the nerve of U (resp. U • A) with the weak
topology. Then by the above considerationeach simplex of N(U) which is not
containedin N(U (qA) hasdimension• n (that is, dim(N(U) - N(U • A)) • n).
Letf: (X,A)-• (In,in)bea continuous
mapsuch
thatf*: Hn(In,i n)-• Hn(X,A)
is zero. Then there are a locally finite normal open coverU of X and a continuousmap
g: (N(U),N(U(qA))-• (In,in)satisfying
thefollowing
conditions,
where
•b:(X,A)-•
(N(U), N(U • A)) is a canonicalmap (cf. [5, õ4]).
(15) f-• go•:(X,A)-•(I n,in),
(16) g*= 0:Hn(In,in)• Hn(N(U),
N(U5•A)),
(17) dim(N(U) - N(U 5• A)) • n.
Hence, by Theorem 2 for the caseof complexes,which is equivalentto Theorem
1 in this caseas was shown in the introduction and hasbeen proved already (cf. [9]),
g is inessential.Hence go•bis inessential.Therefore, by the homotopy extension
theorem of Borsukf is also inessential.Thus, the proof of the "if" part of Theorem 1
for the specialcaseof X beinga metric spacewith A closedis completed.Henceour
Theorem 1 is completely proved.
4. Some theoremson coveringdimension.A combinationof the argumentsin
õ2 with thosein õ3 yieldsthe "only if" part of the followingtheorem.
THEOREM 3. Let (X,A) be a pair of topological
spaces.Thendim X/A • n iff
for any finite normal open cover G of X there is a locallyfinite normal open coverU
of X suchthatU isa refinement
of G anddimN(U)/N(U hA) • n.
To provethe"if" part,let G TM
( Go, G1,"', Gs} beanyfinitecozero-set
cover
of X suchthat it is the inverseimageof a finite cozero-set
coverof X/A underthe
HOPF EXTENSION
THEOREM
127
quotient
mapof X ontoX/A.Without
lossofgenerality
wemayassume
thatA C Go
andA n Gi = • fori • 1. Thenbyassumption
thereisa locally
finitecozero-set
cover
U of X such that it refines G and dim N(U)/ N(U n A) • n. Then from U we can
construct
a finitecozerooset
cover
V = {Vo, V1,--., Vs} of X such
thatVi C Gi for
i = 0,1,-",s and that dim N(V) • n, by formingthe unionsof suitablemembersof U.
This provesthe "if" part of Theorem3.
COROLLARY 4. Let A be C*-embedded
in a topological
spaceX. Thendim
X -- Max(dim A, dim X/A).
PROOF. If dim X •< n, then dim A •< n and dim X/A •< n by [5, Lemmas5.8 and
5.14]. Conversely,
supposethat dimA•<n and dim X/A•<n. Let G be any finite
normal open coverof X. Then there is a finite normal open coverH of X which is a
refinement
of G with dimN(I-Ifl A) •<n. Let { Hi I i=l,-",p } be thetotalityof
H G I-I with H n A 4: 0- By Theorem3 we can find a finite cozero-setcoverV of X
suchthat V is a refinementof H anddim N(V)/N(V fl A) •< n. Let usput
UI=U{VcVIVr3A%o,
VCH 1 },
Ui=U{VCVI V flA%O,
V ½Hjforj<i, V CHi }
for 2 •< i •< p; let
V1,'", Vq
bethetotalityof V G V withV
Assume
that
r
L=n
X=luixn( tl=l
for 1•<i1<-.-<ir•< p and 1•<Jl<"'<js •<q' If LnA4:½ thens=0 and
r
RH.nA%O
and
hence
r+sr•n+l.IfLDA=0,
then
there
are•XofVfor
X=l •X
1 • X • r such that
•XCUix
foreach
Xand
M=X=l
nr V!n (t•
91
V!
svjg)•
Since
M c3A = • anddimN(V)/N(Vfl A) •<n, wehaver+s•<n+l. Thus,if weput
U= (U1,-",Up,V1,-'-,Vq
},
then U is a finite cozero-set cover of X and a refinement of H and hence of G.
Moreoverdim N(U) •< n. This completesthe proof of Corollary4.
The followingtheoremis obtainedasanotherapplicationof the arguments
in õ2.
128
KIITI MORITA
In caseA = •, by X/A we meanthe disjointunionof X andonepoint.Hence,if G is
open in X, then X/(X-G) = C1 G/Bd G.
THEOREM 5. Let G be a normal open cover of a topological spaceX. If dim
Ci G/Bd G • n for eachG G G, then dim X • n.
Roughly speaking, Theorem 5 asserts that if X is uniformly locally at most
n-dimensional
then X is at most n-dimensional.
Before proceeding to the proof of Theorem 5, we shall prove
LEMMA6. If { GXI X GA } isa discrete
collection
of opensubsets
of X andif
dim Ci Gx/BdGx•<n for each 3,GA, then dim Ci G/BdG •<n for
G=U {Gx I X GA}.
PROOF. The spaceC1 G/Bd G is homeomorphicto the quotient spaceobtained
from the disjointunionof C1Gx/BdGx by identifyingall the basepointsof
C1Gx/BdGX to a singlepoint,wherethe basepointof X/A is eitherthepointto
which all the points of A are identified or the point which is added by definition in
caseA = ½.HenceLemma6 isproved
easily.
PROOF OF THEOREM 5. If G and G' are open in X and G C G', then
dim X/(X-G)•< dim X/(X-G'). Since any normal open cover of X is refined by a
a-discrete cozero-set cover of X, by Lemma 6 we have only to prove the theorem for
thecasewhereG isa countable
normalopencoverof X; let G = { Gi I i • N} .
Let V be any countable normal open cover of X.
Thenthereis a normalsequence
qb={ Ui I i • N } of opencovers
of X satisfying
the following conditions.
(1)' U1 isa refinement
of G andof V,
(2)' order{ U,St(X-Gj,
Ui) I U GUi, U C Gj} •<n+l foreach
i >•2 andj •<i.
Here X/qb and ½: X -> X/qbhavethe samemeaningasin õ2.
Hence by the argumentsin õ 2 we have
(10)' dim(X/•- C1½(X-Gj))
•<n forj GN.
If weputI• = Int(Gj;
qb)(forthenotation,
cf.õ2),
thenH= { HjIj GN} isanopen
cover
ofXand
0-10(Hi)
=Hi.Since
0(X-Gj)
C0(X)-0(Hi),
wehave
C3(C1½(X-Gj)[j
•N } =•.
By the sumtheoremon dimension,we concludefrom (10)' that dim X/cb •<n. The
theorem follows readily from the last inequality.
HOPF EXTENSION THEOREM
129
Combining the argumentsin the proof of [5, Theorem 5.2] with Theorem 1, we
have the following theorem, which is an extension of [5, Theorem 5.2].
THEOREM 7. Let (X,A) be a pair of topologicalspacessuch that dim X/A = n,
andletf: (X,A)-->(In,i n) beacontinuous
map.
If f isessential,
soistheproduct
map
f x 1.'(X,A)X (I,[) •(I n*!,I
), wherefX 1 isdefined by(fX 1)(x,t)=(ffx),t)for
x GX, t GI andI n•-l --I n X I.
THEOREM 8. If dim X/A --n, then
dim(XX D/(A X I LJX X i) --n•-l.
PROOF. Since (X/A) X I • (X X I)/(A X I), by [5, Theorem 5.7 and Lemma
5.14] we have
dim(XX I)/(A X I U X X i) • dim(XX I)/(A X I)
=dimX/A+l
=n+l.
On the other hand, by [5, Theorem 5.1 ] and the remark at the beginningof the proof
of [5, Lemma 5.13] there are a closed subset B of X containing A and an essential
mapf: (X,B)• (In, in). Hence
by Theorem
7, f X 1:(X,B)X (I,•)-• (In+l,in+l) is
essential
andconsequently
Hn+I(xX I, (BX I) LJ(X X i)) 4:• byTheorem
1.Since
(B X I) U (X X i) D (A X I) CJ(X X i), this shows that
dim (X X I)/(A X I t_JX X I) • n + 1. Thus, Theorem8 is proved.
COROLLARY
9 ([5, Lemma5.13]). Let (X,xo) be a pointedspaceof finite
dimension. Then dim SX -- dim X-t-l, where SX is the reduced suspensionof X.
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Tokyo Universityof Education
Tokyo, Japan
Received June 2, 1975