C OMPOSITIO M ATHEMATICA
I. M. J AMES
On the maps of one fibre space into another
Compositio Mathematica, tome 23, n
o3 (1971), p. 317-328
<http://www.numdam.org/item?id=CM_1971__23_3_317_0>
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317
ON THE MAPS OF ONE FIBRE SPACE INTO ANOTHER by
I. M. James Wolters-Noordhoff Publishing
Printed in the Netherlands
1. Introduction
The purpose of this note is to
study,
inspecial
cases, thePuppe
exactsequence of
ex-homotopy theory (for details 1,
see[4]).
Webegin by recalling
the basic notions of the category of ex-spaces and ex-maps, with respect to a fixed base space B.By
an ex-space we mean a space Xtogether
with apair
of mapssuch that pu = 1. We refer to p as the
projection,
to (f as the section, and to(p, 6)
as the ex-structure. LetXi (i
=0, 1)
be an ex-space with ex-structure(03C1i, 03C3i).
We describe amap f : X0 ~ X1
as an ex-map ifas shown in the
following diagram.
In
particular,
we refer to 03C31 03C10 as the trivial ex-map. We describe ahomotopy f : Xo - Xl
as anex-homotopy if f
is an ex-map at every stage. The setof ex-homotopy
classesof ex-maps
is denotedby x(Xo, X1).
Further
notions,
such asex-homeomorphism
andex-homotopy equivalence,
are defined in the obvious way.
Let B be a
pointed
space, withbasepoint e
E B. A functor 0 can bedefined,
asfollows,
from the category of ex-spaces to the category ofpointed
spaces. If X is an ex-space with ex-structure(p, u),
then 03A6X isthe space
p - le
with 6e asbasepoint. If f : X0 ~ Xl
is an ex-map, whereXo, Xl
are ex-spaces, then4lf : WXO - 03A6X1
is the map determinedby
1 Theories of this type have been developed independently by J. C. Becker and J. F. McClendon, amongst others.
restriction
of f.
We refer to 03A6 asthe fzbre functor.
Note that 0 determines a functionwhere the codomain means the set
of pointed homotopy
classesof pointed
maps.
In some cases this function ~ is both
surjective
andinjective.
Forexample,
let A be apointed
space.Regard
thewedge-sum
A v B as anex-space with section the inclusion and
projection
constant on A. Then03A6(A
vB)
= A and we have at oncePROPOSITION
(1.2).
Let X be an ex-space withfibre
03A6X = Y. Then thefunction
is
bijective.
By a fibre
ex-space we mean an ex-space with a fibration asprojection.
When
Xo
andXl
are fibre ex-spaces there is a useful necessary condition for an element ofN(OXO, 03A6X1)
tobelong
to theimage
of 9. This condi- tion involves the braceproduct,
apairing
ofhomotopy
groups derived from the Whiteheadproduct
as follows. Let X be a fibre ex-space with ex-structure(p, s)
and fibre Y. Consider the short exact sequencewhere i : Y c X. Given elements
03B2 ~ 03C0*(B), ~ ~ 03C0*(Y)
we form theWhitehead
product [s* fi,
i *il ].
This element of03C0*(X)
lies in the kernelof p*, since p* i*
=0,
and soby
exactness there exists a(unique)
element{03B2, ill,
say,of x* (Y)
such thatThis
operation { , 1,
which we refer to as the braceproduct,
is studied in[5]
and[9],
where variousexamples
aregiven.
From(1.1), (1.3)
and thenaturality
of the Whiteheadproduct
we obtainPROPOSITION
(1.4).
LetXi (i
=0, 1)
be afibre
ex-spacewith fzbre Yi.
If
a E03C0(Y0, Y1) belongs
to theimage of
qJ thenfor
ailf3
E03C0*(B),
11 E03C0*(Y0).
Here the brace
product
on the left refers toXo
while that on theright
refers to
Xl.
In certain cases, as we shall see, the condition is sufficientas well as necessary.
It appears that
sphere-bundles play
aspecial
role inex-homotopy
theory just
asspheres
do inordinary homotopy theory.
LetOq (q
=1,2, ... )
denote the group oforthogonal
transformations of euclidean q-space.For m ~ q
weregard
the(m -1 )-sphere
S’ - 1 as anOq-space,
in the usual way. Given a
principal Oq-bundle
over B letEm
denote the associated(m-1)-sphere
bundle. When m > q weregard E.
as an ex- spaceby choosing
a cross-section of the bundle. When m >q + 1
wegive n(Em, X)
a natural group-structure, as describedin §
2below,
sothat
constitutes a
homomorphism.
We do notgive
a group-structure to03C0 (Eq+1, X).
Now consider the case when B is a
sphere,
say B = S’"(n
>1).
LetEm (m
=q+1, q+2, ···)
be associated with anOq-bundle
overSn,
asabove,
and let X be a fibre ex-space over S" with fibre Y.Let i,
E03C0r(Sr) (r
= 1,2, ... )
denote thehomotopy
class of theidentity
map and letdenote the operator
given by
Here the brace
products
are to beinterpreted
as in(1.4).
It follows from(1.7)
belowthat 03C8
is ahomomorphism
for r > q but this is not true, ingeneral,
for r = q. Our aim is to set up an exact sequencecontaining 03C8,
as in
(1.5),
the fibre function ç, and a third operatorwhich can be defined as follows. Recall
(see §
3 of[7])
thatas a
cell-complex,
where sm is the fibre and ,S" is embeddedby
thecross-section.
If f : Em+1 ~ X is
an ex-map such that4lf
is constantthen the
separation
elementd(f, c)
E1tm+n(X)
isdefined,
withrespect
to this
cell-structure,
where c denotes the trivial ex-map. Sincepf
= pcwe have
p*d(f,c) = 0
and sod(f,c) = i*03B2, by
exactness, where03B2 ~ 03C0m+n(Y). Conversely, given 03B2,
there exists anex-map f,
asabove,
such that
d(f, c)
=i*03B2.
We define0(p)
to be theex-homotopy
class offin 03C0(Em+1, X).
It is not difficult to check that 0 constitutes a homo-morphism
for m > q.Having
made the necessary definitions we are nowready
to state our main resultTHEOREM
(1.6).
The sequenceis exact.
It is
possible
to prove(1.6) by using
the methods of Barcus and Barratt[2].
However theproof we give shows,
in myopinion,
theadvantages
ofexploiting
theelementary properties
ofex-homotopy theory.
Before we embark on the
proof
it is convenient to make a few further observations.Suppose
that theoriginal q-sphere
bundle hasclassifying
element
03B2 ~ 03C0n-1(Oq). By (3.7)
of[7]
the braceproduct
in the case ofE., 1 is given by
where
S*
denotes thesuspension
functor andJ03B2
E1tn+q-l(Sq)
is definedby
theHopf
construction in the usual way.Suppose
that Y = S"(r ~ 1), regarded
as apointed 0,-space,
and that X is ther-sphere
bundle withcross-section associated with a
principal 0,-bundle. Then {ln, lr}
=Jy, similarly,
where03B3 ~ 03C0n-1(Or)
is theclassifying element,
and hence{ln, 03B1} = Jy o Sn-1*03B1,
where a E03C0m(Sr). Thus
Where the relevant information on the
homotopy
groups is available wecan calculate the kernel and cokernel of
03C8,
for a range of values of m, and hence calculate03C0(Em , X)
to within a group extension.For
example,
take n =2, q
= 2. TakeEm
to be the(m -1 )-sphere
bundle associated with the
Hopf
bundle over S2.Using
standard resultson the
homotopy
groups ofspheres
we find thatn(E,3, E6) ~ Z2, in
this case. If instead we take
Em
to be the trivial(m-1)-sphere
bund le wefind that
03C0(E8, E6) ~ Z2 ~ Z2 .
2. The
Puppe
sequenceLet
(K, B )
be aCW-pair,
such that B is a retractof K,
and let p : K - B be a cellular retraction. Weregard
K as an ex-space with the retractionas
projection
and the inclusion as section. Let EK denote thecomplex
obtained from the union of K I and B
by identifying (x, t)
E K I withpx E B if either
x ~ B or t = 0, 1. A
retraction of IK on B isgiven by
(x, t)
H px. Wegive
IKcell-structure,
in the obvious way, so that B isa
subcomplex
and the retraction is cellular. We refer to IK as thesuspension
ofK,
in the ex-category. The r-foldsuspension (r
=1, 2, ···)
is denoted
by 03A3rK. Suppose,
forsimplicity,
that K islocally
finite.Let
(pxK,
for any ex-space X, denote thefunction-space
of ex-mapsK - X, with the trivial ex-map as
basepoint. By taking adjoints,
in theusual way, we
identify
thehomotopy
groupnr(CPxK) with 03C0(03A3rK, X).
Thus
03C0(03A3rK, X) (r
=1, 2, ···)
receives a naturalgroup-structure.
This group is abelianfor r ~
2 but not, ingeneral,
for r = 1.Now suppose that we have a
locally
finitecomplex
K’containing
K asa
subcomplex
and supposethat p
can be extended to a cellular retractionp’ :
K’ ~> B. Let K" denote thecomplex
obtained from K’by identifying points
of K with theirimages
under p. Letp" :
K" ~ B denote the retraction inducedby p’.
Weregard
K’ and K" as ex-spaces, with the retractions asprojections
and the inclusions as sections. Thenare ex-maps, where i is the inclusion
and j
is the identification map.Consider the maps
given by
functionalcomposition. By
astraightforward application
ofthe
covering homotopy
property we obtainTHEOREM
(2.1 ). If
X is afibre
ex-space then i * :~X K’ ~ qJxK
is afibration.
Notice that the
fibre,
over the trivial ex-map, can be identified withqJxK" by
meansof j *.
Hence thehomotopy
exact sequence of the fibra- tion can be written in the formThis is an
example
of thegeneralization
toex-homotopy theory
of thenotion of
Puppe
sequence. An alternativeapproach (see §
7 of[4])
is toconstruct a sequence of ex-maps
and
apply
thefunctor 03C0 (, X).
The
Puppe
sequencegives
useful information if one of the three domain ex-spaces is ex-contractible. Forexample
we haveCOROLLARY
(2.2). Suppose
that K" is ex-contractible.If
X is afibre
ex-space then
is
bijective, for
r >1,
andis monic.
Here the term monic is used to mean that the kernel of i * is
trivial;
it is not true that i * is
injective,
ingeneral.
In §
1 we aregiven
aprincipal 0 -bundle
overB,
and consider the associated(m-1)-sphere
bundleE. (m
=q, q+1, ···).
Weregard Em+1
as the fibre
suspension of Em
in the usual way(see §
7 of[7]).
Let m > q.Then
Em
admits a cross-section and so can beregarded
as an ex-space.The
suspension EEm
isdefined,
asabove,
and the natural ex-mapis an
ex-homotopy equivalence,
as shownin § 6
of[4].
Weidentify
03C0(Em+1, X)
with7r(ZE., X)
under thebijection
inducedby
r, where Xis any ex-space. Thus
03C0(Em+
1,X)
receives agroup-structure,
for m > q.3. The
operators
in the sequenceLet D"
(r
=1, 2, ···)
denote the r-ball boundedby Sr-1.
Choose amap br :
D" Sr which is constant on Sr-1 andnon-singular 2
on theinterior of D’. Given q, we
regard
Dm andSm-1
asOq-spaces for m ~ q,
in the usual way, and choose
bm
to be an0 -map.
We recall that an
0 -bundle
over S"(n ~ 2) corresponds
to a map T :Sn-1 ~ Oq,
in the standard classification.Let m ~
q. Writeg(x, y) = T(y) · x (x
Esm-l,
y ESn-1)
so that g :sm-l
X Sn-1 ~sm-l.
LetEm
denote the space obtained from the union of
Sm-1
x Dn andSm-1 by identifying points
ofSm-1
xSn-1
with theirimages under g,
so that theidentification map
is a relative
homeomorphism.
Write03C0h(x, y)
=bny,
where x ~sm-l,
y E
D",
so that 03C0 :Em ~
,Sn. We recall(see §
3 of[7])
thatEm,
with thisprojection,
can be identified with the(m -1 )-sphere
bundlecorrespond- ing
to T, i.e. the(m- l)-sphere
bundle associated with thegiven Oq-bundle.
If we
replace sm-l by Dm,
in thisconstruction,
we obtain the associated m-ball bundleEm
withprojection
03C0’, say. Weregard Em
as asubspace
ofE’m,
in the obvious way, so that 03C0 =03C0’u,
where u :Em
cE’m.
Let
Fm , F’m
denote the spaces obtained fromEm , E’m by collapsing sm-l
to apoint.
Thenp’u
= vp, as shownbelow,
where v is the inclusionand p, p’
are thecollapsing
maps.Write sy
=h(e, y), where y
ED",
so that s : Dn ~Em .
Weregard Fm
asan ex-space with
projection
p =np -1
and section J =sb-1n. Similarly
2 Here, and elsewhere, it is unnecessary to specify orientation conventions since the validity of (1.6) is independent of the signs of the operators.
we
regard Fm
as an ex-space with ex-structure(p’, cr’)
so that v is anex-map.
Using
the method describedin § 3
of[7]
we can endow these spaces with cell-structure so as tosatisfy
thepreliminary
conditions of§ 2. Consider, therefore,
thePuppe
sequence associated with thepair
(F’m, Fm).
Recall thatF"m,
in the notationof § 2,
is obtained fromFm by identifying points of Fm
with theirimages
under theprojection
p.Now p
determines a
homeomorphism p" : E§l’ - F;’;,
whereEm
denotes the space obtained fromE. by identifying points
ofEm
with theirimages
under the
projection
03C0. We endowEm
with ex-structure so as tomake p"
an
ex-homeomorphism.
The0m-map bm :
Dm ~ sm is constant on sm-l and so determines anex-homeomorphism
betweenE§l’
andEm+1. By composing
this with the inverseof p"
we obtain anex-homeomorphism 03B2: F"m ~ Em+1.
Now03A6Fm
is apoint-space,
and4lF£
= Sm -03A6Em+1,
where 41 denotes the fibre functor. We use
03A603B2
toidentify 4lF£’
with Sm.Let co denote the
composition
where j
is the identification ex-map. Then 03A603C9 = 03A6 and henceas shown in the
following diagram
where X is a fibre ex-space with fibre Y.The fibre of
F.
has been identified with ,Sm. We embed the base space S" inFm by
means of the section. Let Sm v Sn denote the union of these twospheres,
with the standard ex-structure.If,
inF’m,
weidentify points
of Sm v Sn with their
images
under theprojection,
we obtain an ex-space which is ex-contractible. Thus(2.2) applies
toi*,
as shownbelow,
where03C4 : Sm Sn
~ F’m.
By using (1.2)
9 isbijective,
on theright
of the abovediagram,
and sowe obtain
LEMMA
(3.2). The fibre function
is
bijective for
m > q, monicfor
m = q.The next
step
is to set up abijection
betweenn(Fm, X)
and03C0(Sm+n-1, Y). Certainly Fm
andSm+n-1 Sn,
as spaces, have thesame
homotopy type;
however ingeneral they
do not, as ex-spaces, have the sameex-homotopy
type. Situations of this kind can be dealt with as follows. Consider the induced fibrationp*X
overF..
Thesection of X over Sn determines a cross-section of
p*X
over S’n. Theextensions over
Fm
of thispartial
cross-sectioncorrespond
to the ex-maps ofFm
into X.Similarly
the verticalhomotopies
ofcross-sections,
relsn, correspond
toex-homotopies. By
standardtheory (see [1 ])
such cross-sections are classified
by
elements of03C0m+n-1 (Y)·
Thecorresponding result,
in oursituation,
is thatis
bijective, where 03BE
isgiven
as follows.Let f : Fm ~
X be an ex-map of class y E03C0(Fm, X),
and let c :Fm ~
X denote the trivial ex-map. Let i : Y ~ X. Thenwhere the
separation
element is defined with respect to thepair (Fm, S").
Let 1 :
Sm+n-1 ~ Fm
be amap 3
ofdegree
1 such that03C1l : Sm+n-1 ~
Snis
nul-homotopic. By
consideration of the induced fibrationl*03C1*X
over Sm+n-1 we obtain the relationwhere E
1tm+n-l (Pm)
denotes thehomotopy
class of l.We
identify Sm+n-1
with theboundary
of Dm x D" in the usual way.Let
denote the identification map used in the construction of the m-ball bundle
E’m. By restricting p’h’
to theboundary
of Dm x Dn we obtain amap
Notice that H is constant on
Sm-1 Sn-1.
Also H maps Dm xSn-1
into S"’ andSm-1 Dn
intoFm.
Let03BA ~ 03C0m+n-1(Sm)
denote the classof the map
3 We use the phrase which follows to mean that 1 determines a homeomorphism when S" c Fm is collapsed to a point.
which agrees with H on Dm x
Sn-1
and is constant onSm-1
x D". Also let 03BB E 03C0m+n-1(Fm)
denote the class of the mapwhich agrees with H on sm-l x Dn and is constant on Dm x
sn-le By (3.9)
of[10]
the class of H in03C0m+n-1(F’m)
isequal (with
suitable conven-tions) to
where
r, j, v
are the inclusion maps and where im, i" E03C0*(Sm S")
arethe classes of the inclusion maps of
Sm,
Snrespectively.
But H is nul-homotopic,
since h’ extends over Dm xD",
and so we conclude thatIt is easy to check that
1,
asabove,
is a map ofdegree 1,
and thatis
nul-homotopic.
Moreover it follows from the basictheory
of theHopf
construction(see [10])
thatwhere (xe
03C0n-1(Oq)
denotes thehomotopy
class of T and the braceproduct
is defined withrespect
toEm .
We arrange our orientation conventions so that x =yn, lm}.
The next step in the
proof
of our main theorem is to establish thatas shown in the
following diagram, where
is definedby
means of thebrace
product,
as in(1.5).
Let
y’ E 03C0(F’m, X).
Write~03B3’ =
17 En.(Y), v*(y’)
= y E7r(F., X). By naturality 03B3’*03C4*[ln, lm] = [s*ln, i*~] = i*(ln, ~}, by (1.3),
where s :Sn ~ X denotes the section. Also
Y’j*(K)
=i*~*(03BA)
=i*~*{ln, i.1,
aswe have
just
seen. Now compose both sides of(3.5)
withy’
and we obtainthe relation
by (3.4). Since i*
isinjective
this proves(3.6).
It follows at once from
(3.3) by naturality
thatas shown in the
following diagram,
where 0 is definedby
means of theseparation
element asin §
1.Now we are
ready
tocomplete
theproof
of(1.6).
As we have seen,the
Puppe
sequence associated with thepair (F’q, Fq)
can be written inthe form
We use ~
and 03BE
toreplace
the terms in the middleby homotopy
groups of Y. We recallthat ç
is monic for m = q,bijective
for m > q ;also 03BE
isbijective
for m ~ q. Hence and from(3.1), (3.6)
and(3.7)
we obtainthe exact sequence
of §
1.4.
Proper
cross-sectionsThe above
theory
can be used togive
an alternativeproof
of the main result of[3],
which concerns thefollowing problem.
Let B be apointed
space. Let
Xi (i
=0,1)
be an ex-space of B with fibre03A6Xi
=Yi,
say.Suppose
we have an ex-map p :Xi - Xo which,
as anordinary
map, constitutes a fibration with fibre Z, say, over thebasepoint
ofYo
cXo.
Write
q,p
= q.Then q : Y1 ~ Yo
can beregarded
as a fibration with fibre Z so that we have the situation indicated in thefollowing diagram,
where u, v and ui are the inclusions.
Under what conditions does the
fibration p
admit a cross-section? As in[3 ] we
describe such a cross-sectionf : X0 ~ Xl
as proper ifNote that proper cross-sections are ex-maps,
since p
is an ex-map.If f
is proper
then g : Y0 ~ Y1
constitutes a cross-sectionof q, where g = 03A6f.
As in
[3]
wedescribe g,
or its verticalhomotopy class,
as the typeof f.
We
approach
ourproblem by asking,
for eachcross-section 9
of q, whether p admits a proper cross-section oftype
g.Now let B be a
(pointed) CW-complex. Suppose
that(Xi, Yi)
is aCW-pair,
and that the section 03C3i i embeds B as asubcomplex.
Then(Xi,
B vYi)
forms aCW-pair,
and the answer to ourquestion
is in-dependent
of the choiceof g in
itsclass, by
thehomotopy lifting property.
Consider the
homomorphism
given by
03C30* on the first summand and io* on the second.Certainly
0 isan
isomorphism
for r ~2,
since ao is aright
inverse of po.Suppose
that 0 is also an
isomorphism
for r = 1. This is the case, forexample, if 03C01 (X0)
isabelian,
or ifTC1 (B)
is trivial. Under thishypothesis
we proveLEMMA
(4.2). Let f : X0 ~ Xl
be an ex-map such that03A6f : Yo - Y1
isa cross-section
of
q. Then there exists a proper cross-sectionf’ of
p such that03A6f’ = Of.
Since
pf : X0 ~ Xo
maps B vYo identically
we have(pf)*03B8
=0,
where 0 is as above and
Hence
(pf)* = 1,
since 0 is anisomorphism,
andso pf is
ahomotopy equivalence, by
Theorem 1 of[11].
Hencepl
determines ahomotopy equivalence
of thepair (Xo,
BYo )
withitself, by (3.1)
of[6], since pf
maps B v
Yo identically.
Therefore there exists an inversehomotopy equivalence k : X0 ~ Xo,
say, which alsomaps B
vYo identically.
Writef " = fk : Xo - Xl. Then f"
agreeswith f on
B vYo. Let lt : X0 ~ Xo
be a
homotopy of pf"
into theidentity
such thatlt(B v Yo)
c B vYo.
By composing
withh = f|B Y0
we liftltBB v Yo
to ahomotopy 1,’ : B
vY0 ~ Xl
such thatl’0
= h =l’1.
Weextend 1;
overXo
so as tocover
lt, using
thehomotopy lifting
property, and thusdeform f" into f’,
say. Then
f’
is a proper cross-sectionof p
and03A6f’ = lPf" = Tf,
asrequired.
To
apply (4.2)
we retum to the situation consideredin §
1, whereEm+1
is anm-sphere
bundle over S". The fundamental group isabelian,
since n > 1. We
apply (4.2)
with(Xo, Yo) = (Em+l’ sm)
and write(Xl, Y1) = (X, Y),
sothat p :
X ~Em+1 and q :
Y - sm. Hence and from the exactness of our sequence we obtainCOROLLARY
(4.3). Let y
E1Cm(Y)
be the classof
a cross-sectionof
q.Then p admits a proper cross-section
of
type yif
andonly if
In
particular,
suppose that Y is apointed O.-space,
andthat q
is apointed Om-map. Suppose
that X andE., 1
share the sameprincipal Om-bundle
and that p is the map associated with q. To obtain the main result of[3] ] from (4.3)
it isonly
necessary to convert the braceproducts
into the kind
of product
used in[3],
as shown in[5].
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(Oblatum 17-XI-1969) Oxford University,
Mathematical Institute 24-29 St. Giles OXFORD, England