準ファイバー空間のなす圏上の連続関手について

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九州大学学術情報リポジトリ

Kyushu University Institutional Repository

準ファイバー空間のなす圏上の連続関手について

酒井, 道宏

九州大学数理学研究科数理学専攻

https://doi.org/10.11501/3150921

出版情報：Kyushu University, 1998, 博士（数理学）, 課程博士バージョン：

権利関係：

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-�

. ""'· -

THE FUNCTOR ON THE CATEGORY OF QUASI-FIBRATIONS

MICHIHIRO SAKAI

ABSTRACT. For any continuous functor on the category of pointed topological spaces we give a continuous functor on the category of quasi-fibrations. This yields the existence of continuous functors , e.g. the fibrewise mapping space the fibrewise mapping cone, the fibrewise reduced product space and the localization functor.

1. lNTRODUCTIO

We work over the base space B. By a

fibrewise pointed space

_,we mean a space X together with maps

p

:X --+Bands : B--+X such that

pos

₌^1. We refer to

s

as the

section

and

p

the

projection.

We regard B itself as a fibrewise pointed space, with the identity as section and projection. We regard Ax B

,

for any pointed space A, as a fibrewise pointed space with section given by the basepoint of A. If Xi

(

ⁱ₌^1,

2)

is a fibrewise pointed space with section Si and projection

Pi,

a fibrewise pointed map

cP

: xl--+ x2 is a Inap such that

c/Josl

_{= 52}and

P20cP

₌

Pl·

Fibrewise pointed homotopies

are defined similarly. A fibrewise pointed homotopy into the fibrewise constant map

s2op1

is called a

fibrewise pointed nulhomotopy.

The fibrewise product ·X1 XBX2 is defined as a fibrewise pointed space with section given by

(s1, s2).

The

fibrewise wedge

_{X1 V BX2} is defined as the subspace of X1 ^XBX2 consisting of pairs

(x1, x2)

such that

x1

₌

s1oP2(x2)

or

X2

₌

s2op1(x1).

Let

Top�

be the category of fibrewise pointed topological spaces over a connected CW-complex B and fibrewise pointed continuous maps over B. We write simply f :X --+ Y for n1orphis1ns in

Top�.

Through

out this paper a space X stands for a fibrewise pointed topological space over B which has a homotopy type of CW -co1nplex and a map f ^:_X" --+ Y means a fibrewise pointed continuous 1nap over B between fibrewise pointed spaces X and Y over B.

In this paper we say the notation used in the category

Top�.

Let X and Y be spaces in

Top�.

For a fibrewise pointed subspace A of X over B let X/ BA denote the fibrewise quotient space. The fibrewise smash product over B is defined by X 1\B Y =(X xB Y)/B(XVB Y). In par

ticular, ��X = (51 x B) 1\B X is the fi brewise reduced suspension space of)(. vVe denote by_)( *By the fibrewise join over B. CEX denotes a fibrewis cone space. The fibrewise mapping space over B is denoted by map�(X

Y)

_or_yx. In particular n�x = map�(.-,1 ^X^BX) = X51 xB

l

(4)

is a flbrewise loop space. 'vVe denote by �B :

)

( ^----+ .\ ^a .\ the fi

brewise diagonal n1ap over B and V 8 : .\ V 8 .\ ----+ .\ the fibrewise folding n1ap over B. vVe denote by c8 : .\'" ----+ Y the fibrewise constant tnap over B. Let f be a n1ap in

Top�.

Then we denote by

f�(f)

the fibrcwise rnapping cone. A fibrewise pointed hotnotopy relation over

B

is denoted by ^'::::aand the set of the fibrewise pointed homotopy classes over B is denoted by

[X Y]�.

This paper is organized as follows: In section 2 we give a contin

uous functor on q-Fib the category of quasi-fibrations so that it be

comes the extension of any continuous functor on

Top ..

the category of pointed topological spaces. In section

3

we define the genaralized Hopf hornomorphism and the Whitehead product in

Top�

and obtain an anologue of the classical Hopf construction associated with a van

ishing Whitehead product. In section 4 we introduce James' work on the LS-category over B

[13).

In section 5 we give a projective space for an A00-space in

Top�

and obtain the reration between the LS-category over

B

and it. In section 6 we define the higher Hopf-invariant in

Top�

and show that it gives the obstruction for LS-category over

B.

The author would like to express his gratitude to IVIasayoshi Ka

mata and N orio I wase for valuable conversations and encouragements at Kyushu University, without which this work could not be done.

2. THE FU CTOR ON THE CATEGORY OF QUASI-FIBRATIOt S In

[9),

James studied

Top(2)

the category of continuous maps be

tween spaces.

A rnorphism between

(p1 : X1-+B1)

and

(p2 : X2-+B2)

in

Top(2)

is given by a pair of maps

(

f :

X1---tX2,

g :

B1---tB2)

with the following commutative diagram:

We denote by q-Fib the full subcategory of

Top(2)

whose objects are quasi-fibrations over a simplicial complex and have the locally trivial property i.e. , the following holds:

Let p: E---tB is a quasi-fibration with fibre

F.

Then for any simplicial n1ap fer :

6cr----+ B

there exists a homeomorphism cPcr : 6cr ^XF----+

J;

E 'uch that

p' oc/Jcr

⁼

p11,

where

6cr

denotes a canonical simplex and p' :

.f;E---t6..cr p11

:

6.crxF-+6.cr

are first projections, respectively. Then we have the following result.

2

(5)

Theoren1 2.1. For any continuous

f

unctor (J): Top.-+ Top,.,, thrr r.r

ists n continuous functor 1?(2): q-Fib-+q-Fib which enjoy . .:; thP following properties.

(1) <P(

2

)(E)b

� (J)

(

Fb

) for

bEE

( 2) T h r e exists � (

2

) ( f, g) such that

(j)

( 2) ( f, g )

=

( � ( 2 ) ( f, g) , g) (3) <P(2)(B

X

F -+ B)�(B

X

<P(F) -+ B).

Proof.

Let

p:

E-+B be a quasi-fibration with fibre F. Niaking use of local triviality, we give E a topological structure. Then we rnay regard

E as an attaching space. We define <P(2)(E) by U <P(F b)· Similarly

bEE

we give <P(2)(E) a topological structure. So we may regard <P(2)(E) as an attaching space. Next we give a rnorphism. Consider the following cornmu tati ve diagram:

Then we define �(2)(f, g): <P(2)(E1)-+<P(2)(E2) by �(2)(f, g)I<P(2)(Er)b

=

<P(f)I(Er)b. We only show that �(2)(f,g) is continuous. Let /1 :

6q-+ B1 be a simplicial rnap. Then there exists a simplicial rnap f2 : 62-+B2 such that the following diagran1 commutes.

D1 ----+ D

2 Br ^� B2

By the local triviality we obtain that �(2)(f,g) is continuous. This completes the proof.

0

Corollary 2.1. map

�

)

CE) ��) n� )

*B;

r�(f))

JB and RB are con

tinuous functors in the sense of

<P (2).

Rernark 2.1. Let p ^:

E -+ B

be a fibre space with local triviality.

If B

is a

CvV

-complex) then Theorem 2.1 holds.

3. \NHITEHEAD PRODUCT AND HOPF CO STRUCTIO OVER

B

Let (p: E-+B) and (q: F-+B) be objects of q-Fib ov r B. In [9], [ 11]

^.]

ames proved that if a map ^¢

:

E-+

F

in

Top

� is a homotopy equivalence then ^¢ is a fibrewise pointed homotopy equivalence. Let

(.�-+B) be an object of

q-Fib

over B. By Theorem 2.1 and Corollary

3

(6)

2.1 we obLa.in Lhc following cotntnuLati\e diagra.n1:

J( )(b) � ^n<-Jxb

1 1

JB_x

_---+^JB

n,B�BX B B

1 1

B ---+ B

By Corollary 2.1 and the above argument, we obtain that jB is a fi

brewise pointed homotopy equivalence. Let

h2

:

JBX ----+lB(X 1\sX)

be fibrewise pointed combinatorial extension of a fibrewise pointed shrink

ing map

h2: (JsX,X)----+(X/\BX,s(B)).

We then have the generalized Hopf homomorphism over B

JIB : [ LJ�J{, ��X]�----+[��1{, ��(X 1\sX)]�

as the following composite,

[��!{,��X]� �[I<, D��X]�

(jB�

�

^.

^[J{, ^JBX]�

(h2)) [!{, ls(X /\EX)]�

(jB)•

[J{, D��(X /\EX)]�

� [��!{,��(X 1\sX)]�,

where

ad

denotes the adjoint map.

Let � =

(

^EB

^c2

^----+ ^{B and}^{r; =}

^r;'

^EB

^c2 ^----+

B be real vector bundles over B , where

c2 ----+

B is a 2-dimensional real trivial bundle over B.

Let

S(�)----+

_{B and}

S(r;) ----+

B be the sphere bundles associated with�

and r; respectively. 0 bviously

S ( �)

_and

S ( r;)

are spaces in

Top�.

Let

a= [f]

E

[S(�),X]�

and

b = [g]

E

[S(r;),X)�.

There is a natural homeomorphism

D(( EB c1 )/ sS((

EB

c1)

�

S(�).

Denote by

qB

_the

cornposite of the fibrewise quotient n1ap and the homeon1orphism

qB

:

D ( (

EB c

1 ) ----+ D ( (

ffi

c 1 ) I B s ( ( ^{EB c}

¹

⁾

^�

^s ⁽ ^�) ^.

The Whitehead product

[a b]B

is defined by

[a b]s [\7Bo(f xB g)o(qs xs qs) ls((ffiTJ'ffie:2)]

E

[S((

EB

r;'

EB

c2), X]�

4

(7)

where \7

a

is the fibrewise folding map. By n1aking use of the canonical bijection

() : [5(.) 1\a vV, �\]� ---+ [. '(.), .x·w]�,

the \Nhiteh ad product over B gives rise then to an operation

[S(�) 1\a vV,�Y]�x[(.�(TJ) 1\B vV,)C]�---+ [5((

w 77' ^<:2

) 1\B vV,)(]�.

For

a

E

[(5�) 1\B vV,)(]�

and (3 E

[(577) 1\B vV,X]�,

we define

[a,,G]�

⁼

e-1[B(a),()((J)]B·

Assun1e that

[a, (3] �

= 0. Then there exists a rnap F :

5 ( �) x a S('77) ---+ _xw

of type

(B(a), B((J))

in

Top�

by th sa1ne argurn nt with the ordinary case

[4].

Let f ^:

5(�) ---+ xw

and g

: 8(77)

^---+ x·w be representatives of

O(a)

and ()({3), respectively. We have a cornposite

(JB_X)W XB (JB_X)w (JBX)w,

where

iw

:

xw ---+ (JaX)w

is a natural injection and

pW

is induced by the fibrewise multiplication P:

laX Xa JBX---+ JBX·

Then we have

(3.1)

Let E

: (L:�X)E�W ---+ (D�L:�X)

be a natural homeomorphism. We consider an isomorphism

e-t ---+

[5(.), (D�L:�X)w]�

(5(.), (l:�X)E�W]�

[5(.) 1\B l:�W, l:�X]�.

This isomorphism combines the Hopf James map with the map

h:;'

as follows.

(3.2)

Following the method due to James

[ ]

let

d(

_'VI,_F)E

[S((

EB

77'

@

c3

)

^,

(

J

aX) w]�

be the difference elernent of vi and F. We define the

.5

(8)

Hopf con�truction ·ub�et overB associated with the vani�hing of

[n. J] �

as follows:

c(a,f3)

⁼

{<P(d( 'v!,F))/F

is of type

(

O

(

a

)

,O

(p)) } ( � [

� ^--.

( 1 �

EB

TJ 1

EB t: ³

) A a '\'a

^L.J

a

vv,

na )]a

LJ

a x a

The suspension n1ap

S: �)(w---+ (��)()::�w, S(f)

⁼

��f: L:�vV---+

._J��)(,

and the evaluation map

e

^: �)(

---+ n�x, e( x )(

t

)

⁼

[x, t],

satisfy the following relations

(3

^.

3)

·W ·W

ew

Ja

o t

E o

S.

Let x : W

---+ vV /\a vV

be a composition of the fibrewise diagonal map

6B : VV ---+ VV

X

B vV

and the fibrewise quotient map

qB

: W X

B

W ---+

vV 1\B

W. Then we have the following result.

Theorem 3.1. Suppose that

[a, ,G]if

= 01 a E

[S( �)/\a vV, _X]�

and

,G

E

[ S'( TJ) 1\B vV, X]�.

Then

HB(c(a,,G)) {��(a 1\B ,G)

o

L:�(1 1\B

T

1\B 1)

o

1 /s((EB171EBc:3) 1\BX}

B

^I ^I ₃

B B

(� _1rB(S(�

_EB

_TJ

_EB_t:

) AB z=Bw, z=B(x AB X))).

whe;e X : W

---+ vV 1\B

W is the fib;ewise diagonal class map and T is the switching map.

Proof. Let

f : S(�) ---+ xw

and g :

S(TJ) ---+ xw

be maps which represent

0(

a

)

and

O(,G)

respectively. The composite

(JBX)w

(JB(X 1\B X))w

maps

S( �) VB S( TJ)

into s

( B),

and the co1nposite induces a map r:

S(�) Va S(TJ)---+ (Ja(X /\a X))w.

Moreover the composite

S(�) xa S(TJ) !'-r Xw

� ^(JBX)w

� hw

(JB(X 1\B X))w

th fibrewise constant mapping

*B.

6

(9)

Let

q�

^:

5'(�) x

^a

8(11)

^--+

S(�) 1\B 5('7)

⁼

.-'((

^/

11'

--8 ^c-3

)

^{be the}

rrbrewise quotient tnap. Then

hiv o d( i'vf F) o q� = h.iv o

Lvf

= h2 o pW o ('iW XB -iW) o (.f XB g)

= iw oh:;! opw o (J xBg)

= -iw

0 r 0

q�.

Since

(q�)*: 7r�(5(()/\B5(71), Z)

^{-t 1r}

�(S(�) xB5(71), Z)

is a rnonornor

phism, we obtain

( h r( )

*

(

d

(

LVI,

F))

= f.

It follows from

(3.2)

that

The composite

HB(c(a,{J)) = _¢ (

f

)

^.

�

,w

(5(�)_ 1\B VV 1\B 5(71) 1\B vV)w,

induces a map x'

: 5( �) 1\B 5( 71)

^-t

( 5( �) 1\B

w

1\B 5( 71) 1\B vV) w'

where

q� w (

^u

x B

^v

) (

^w

) =

^u

(

^w

) 1\B

^v

(

^w

)

^. We see easily that

B(a)

⁼

aw

⁰

B(ls(OABW): 5(�)

^-t

(S(�) 1\B vv)w

^-t

xw

and

B({J)

⁼

{Jw o B(ls(T7)!\Bw): 5(71)--+ (5(71) 1\B W)w--+ xw.

vVe then have

f

= (a 1\B {J)W ox'.

ote that

x'

is the following composition.

7

(10)

where T is the sw i t ching n1ap and \ _: vV

--+

^vV

/\a

I,V t he fibrcwisc d i agonal class tnap. vVe t hen calculate:

{fla(c(a,{3))}

=

{ fl a

0

¢ (

^d

(

^\If'

F)) }

=

{ ¢ o

^h

�v o (

^d

(

^vi,

F)) }

=

{¢or}

Thus we complete the proof. 0

Let

1 : I:�W --+ I:�W

be the identity map. If 6w : H*(W

/\a vV s(B))--+

_H*(

vV ,

s(B)) is not the zero homomorphism, then

Ha(c(1 , 1)) f. 0.

_{And so}

c(1 1)

�

[(I:�)3vV, (I:�)2vV]�

is non-trivial. Let

vV--+

_B

be a Hurewicz fibration over a simply connected space Band H*(W, s(B)) be a projective H*(B)-module with the finite type.

Then L. Smith

[16]

proved that the external tensor product induces an isomorphism

H"'

( vv,

^s_(B))⁰^H.

(a)

_H,.

( vv,

^s_(B))

--+

_H*

( -vv

^A

a vv

s

(

^B))^.

Therefore we have the following theorem:

(11)

Theoren1 3.2.

fl [{"'(vV, s(B))

^is^aprojective ff'"(B)-module with the

fin

itc type and the prod ucf

H*(vV .s(B))

^cH•(B)

H"'(vV, s(B))-+ H*(vV, .s(B))

i� non-trivial then

c(l,l)

is non-trivial in

[(��)3vV, (��r�vV]�.

4. FIBREWISE CATEGORY A D FIBREvVISE POINTED CATEGORY Firstly we introduce .Jarnes' work on the LS-category over

B [13].

By a fibrewise space, we n1ean a space X together with a rnap p : X

-+B,

called the projection. We do not require p to be a fibration, although this often be the case in applications. We regard

B

itself as a fibrewise space, with the identity as projection. vVe regard Ax

B

for any space A, as a fibrewise space under the second projection. If )(_i

(

ⁱ⁼

l, 2)

is a fibrewise space with projection Pi a fibrewise map cj;: X1-+X2 is a map such that p2ocj; ⁼ p1. Fibrewise homotopy are defined similarly. A fibrewise map cj; is said to be fibrewise constant if there exists a section

s2 ^: B--tX2 such that

cj;

₌ _s2op1. _Afibrewise homotopy into a fibrewise constant is called a fibrewise nullhomotopy.

Definition 4.1. A subset U of a fibrewise space X is fibrewise categoT

ical if the inclus,ion U-+ X is jibTewise nullhomotopy. H eTe U is regaTd as a fibTewise space with

P!u

as pTojection.

Example 4.1. Every subset of a fibrewise contTactible space is jibTe

wise categoTical. In paTticular1 if X is a euclidean vector bundle oveT B the associated spheTe bundle is fibTewise categoTical.

When

B

is a point -space, the fibrewise categorical subsets of X are precisely the categorical subsets of X in the ordinary sense. When

B

is contractible, fibrewise categorical subsets are categorical; if, moreover, the projection is a fibration the converse holds. In general, however, fibrewise categorical subsets are not categorical, nor vice versa.

Example 4.2. If

B'

is a subspace of

B

and X' = p-1 B' the restTiction U' = UnX' of a fibrewise categoTical subset of X'. In particulaT the Testriction of a jibTewise categoTical subset to a fibTe is categorical1 in that fibre.

Definition 4.2. The fibrewise category catBX of a jibTewise space X is the least numbeT m .�uch that theTe is a coveTing X by m +

l

fibTewise categorical open subsets.

Example 4.3.

catBX

= 0 if and only if X is fibTewise contractible.

cat

aX:::; 1 if X is the unreduced fibrewise suspension of a fibrewise space.

Example 4.4.

vV

e may Tegard 51 X S1 as a fibrewise space oveT ...,1 with

f

i

_T_tprojection p and section by inclusion i. Then

cat

51

(

^{S1 x}5'1) ⁼ l.

9

(12)

Example 4.5. LPt

I\.

drnolr n klein

bottle. 1\·

l-"

r

^e

gar

ded as ,1..,'1-bnnrl/(

ov r

^,-q.

T

h

e

n

cat5t (/\.)

⁼ l.

Exa1nple 4.6.

Let::;

xS

be a jibrcwise pointed space overS with fir. t projection and diagonal section1 where S

is

a sphere over of even di

mension. Then catsSxS

⁼ 1.

It follows that if B' is a subspace of Band

X'= p-1

B', then

catB,)C'�catBX.

In particular

cataX

is bounded below by the category of each of the fibres of

X.

Fibrewise category is an invariant of fibrewise homotopy type. J\!Iore generally, the following holds:

Fact

(

James

[

lO

])

.

If X fibrew.Zse dominates Y then catB)(?_cataY.

Example 4. 7.

Consider the product

Ax B 1

for any space

A.

If

V

is a catego1ical subset of

A1

then

V x B

is a fibrewise categorical subset of

AxB1

moreover

VxB

is open if

V

is open. Thus catB( A

xB

)�ca t

A1

in fact1 equality holds in v·iew of our previous remarks.

Definition 4.3. A

1ing

A

is said to be nilpotent if

An

=

0

for some positive integer n. The least such integer n ·is called the index of

A

and w1itten nil

A.

(If no s·uch integer exists1 the in dex is said to be infinite).

Consider an ordinary multiplicative cohomology theory H*. James gives lower bounds for fibrewise category.

Proposition 4.1

(

James

[

lO

])

.

Let H'B(X) denote the quotient of H(X) by the ideal generated by the sub1in g p H* (

B

)

.

then

Next we work over a fibrewise pointed space.

Definition 4.4.

We desc1ibe a subset

U

of a pointed space X (necessa1ily containing the section) as fib1ewise pointed catego1ical if the inclusion

U

-+X is fibrewise pointed nullhomotopy.

Every such subset of a fibrewise pointed contractible space is fibrewise pointed categorical. When B is a point-space the fibrewise pointed categorical subsets of

X

are precisely the pointed categorical subsets of

X.

Example 4.8.

If

B'

is a subspace of

B

an d X' = p-1

B'

the 1estriction U' = UnX' of a fibrewise po-inted categoTical subset of X'. In particu

laT the re �tTiction of a fibrewise pointed categorical subset to a fibre is pointed catego1ical in that fibre.

Definition 4.5.

The fibrewise pointed category cat�X of a jib1ewise pointed space X i the least numbe1

^m

such that the1e is a coveTin g X by

^m+ 1

fibrewise pointed categorical open sets.

10

(13)

Exa1nple 4. 9. cat

�

-'( ⁼0 if nnd only tf _\'" is .fib rc wise pointed con

trnctibl . cat

�

_,:'( ⁼ 1 if.\ is th.e unreduced .fz'brewis pointed suspen..:;ion.

If B' is a subspace of B and _)(' ⁼ p-1 B' then

cat�:.\"' :Scat�)(.

In particular cat

�

.\' is bounded below by the pointed category of each of fibres of .X.

Fibrewise pointed category is an invariant of fibrewise pointed homo

topy type. Tviore generally, the following holds:

F

a

c

t

(

.

J

ames

[ l

O

]) If ^X

dominates Y, in the sense of fibrewise pointed hmnotopy, then cat

�

X'2::cat

�

Y .

.] arnes gives lower bounds for fibrewise pointed category.

Proposition 4.2

(

James

[

10

])

. cat

�

X2::nilH"'(X, B)+

1.

Example 4.10. LetS

x

S be a fibrewise pointed space overS with first

projection and diagonal section1 where S is a sphere over of even di

mension. Let

X

be the fundamental class in cohomology with integral coefficients. Then H5(SxS)�H(S)1 with index 21 wh·ile H(SxS,S)

is isomorph·ic to the kernel of the cup pTod·uct H"'(S)®H*(S)--+H"'(S)1

which contains

x

®

1

- 1®X1 of which the square is non-zero. This im

plies that S

x

S has fibrewise category at least two. By example 4. 61 we already know that S

x

S has fibrewise category one. Thus fibrewise cat

egory can be less than fibrewise pointed category1 when both are defined.

Definition 4.6. The fibrewise space

X

over B is vertically connected if

X

admits a section and all such sections are vertically homotopic.

James gives the comparison of fibrewise category and fibrewise pointed category.

Proposition 4.3 ( J ames

[

10

])

. Let

X

be a fibrewise pointed space over B with closed section. Suppose that

X

is vertically connected. Then

catBX

= ca

t � ^X

.

Proposition 4.4

(

James

[

10

])

. Let

Xi (i

⁼ 1,

2)

be a vertically con

nectedfibrewise space over B . Then

catB(X1xBX2)<catBX1 +catBX2·

Proposition 4.5 ( J ames

[

lO

])

. Let

X

be a vertically connected fibre

wise space over B. Then

catBX :SdimX + 1.

ext we say that fibrewise pointed category can be characterised in terms of the compressibility of the diagonal. Consider the fibrewise topological products xn

(

n =

1'

2 ...

)

of the fibrewise pointed space

X

with itself. Let

X[nJ cXn

denote the subspace such that the fibre over each point bE B is the fat wedge. Then xn contains the diagonal

.6.X

of

X,

while

X[nJ

contains the diagonal 6B of B.

Proposition 4.6

(

.James

[

10

])

. Suppa e that

X

admits afibrewise pointed categorical neighborhood of B. Then the diagonal 6 ^:

X--+

xn+r can be

11

(14)

compressed into

.\[n+l]

by a fibrcwi.se pointed homotopy if and only zf ca t�_'( :s;n.

Ren1ark 4.1.

If_';( admits a fibre wise pointed categorical neighborhood, then cat�.'\ :s;1 if and only U _';( is a fibre wise co-Hopf space.

Nioreover James gives the following property about sphere-bundle with section:

Theoren1 4.1

(James[l2]). Let X be a fibrewise pointed sphere-bundle over B. Then cat� X :s;1

+

catB.

5. FIBREvVISE PROJECTIVE SPACES FOR AN A00-SPACE OVER

B In this section we only consider the case that p : X --t B is a fibre bundle.and analyze the relation between the fibrewise pointed cate

gory and the fibrewise projective space.

Definition 5.1.

Let Px : X --tB1 py : Y --tB and pz : Z--tB be fi

brewise pointed spaces over B and f : X --t Y and g : Z --t Y fibrewise pointed maps over B. Then the fibrewise homotopy pull-back vV of j and g is defined by

W

=

{(z, x, l)EZxX xY1Il(O)

=

g(z), l(1)

=

f(x)L

projection pw : W --tB is given by pw(z, x, l)

⁼

pz(z) and section sw : B--tW is given by

^s

w(b)

⁼

(sz(b), Csy(b), sx(b)).

Then we have the following homotopy commutative diagram:

vV�X

Z�Y,

where pr1 and pr2 are first and second projections, respectively.

Definition 5.2.

Let Let Px : X --tB1 py : Y --tB and pz : Z--tB be fibrewise pointed spaces over B and f : X --t Y and gX --t Z be fibrewise pointed maps over B. Then the fibrewise homotopy push-out vV' zs defined by

vV

^{I =} ^X^X

[- 1 ' 1]

u y u

z I

^r-v)

(

^X

-1 )

r-v

f (

^X

) and (

^{X '}

1 )

r-v

g ( z) ) projection Pwl : vV'

^--7

B is given by

Pw

¹

( x, t)

⁼

p x (

^x

)

^,

for (

^x,

t) EX x [

^-1

, 1]

pw

¹

( y)

⁼

py ( y), for y E Y Pw�(.::)

=

pz(z) for zEZ

and section sw�: B--tvV' is given by s(b)

⁼

(sx(b) 0).

12

(15)

Then we have the following honwtopy COlDinutative dia.gra.1n:

z

�

_vV',

where iy : Y--+ vV' and iz : Z--+ vV' are inclusions.

Definition 5.3.

The fibrewise pointed map u : A--+X1 where A and

X

are fibrewise spaces o uer

B)

is a fibre wise co .fibration if ·u has the foLlowing fibrewise pointed homotopy extension property. Let f : )(--+ E be a fibrewise pointed map1 where E is a fibrewise pointed ) an d let g: A--+PB(E) be a fibrewise pointed homotopy such that p1og = _f _ou.

Then there exists a fibrewise po·inted ho·motopy h : X--+ PB (E) such that p1oh

⁼

f and hou =

_g.

An important special case is when

A

is a. subspace of X and

u

is the inclusion. In that case we describe

(

_X,

_A)

as a fibrewise pointed cofibred pair when the above condition is satisfied. Let

A

be a subspace of a fibrewise pointed X over B . Then there are two useful criteria for the inclusion i :

A--+

X to be a fibrewise pointed cofibration, the formulation of which involves a pair

(

_1/J,ht

)

,·where 1/J: X--+

[

_0,1

]

_{is a}

continuous function which is zero throughout

A

and

ht

:X -+X,

o::;t::;1,

is a fibrewise pointed homotopy such that

h0(x) = x

for all

xEX,

ht(a) =a

for all

aEA, tE[O, 1]. (

*

)

Lemma 5.1

(

_Crabb

[

₁₃

])

_.

Let A be a closed subspace of the fibrewise pointed X over

B.

Then the following conditions are equivalent.

(1) The inclusion i

: A-+

X is a fibrewise pointed cofibration.

(...) There exists a pair (

1/J,

ht) as in (

^*

) such that (a)

1/J-1

(

₀

) _=A)

(b) ht(x)EA; whenever t>?/J(x).

(3) There exists a pair (

1/J,

ht) as in (

*

) such that

(a)

·1/J-1

(

₀

) _=A)

(b) h1(x)EA) for all xEX uch that 1>1/J(x).

13

(16)

Len1ma 5.2 (.Ja111e

[13]).

Let ^dJ

:

.\

---+Y be

a

fibre

wise pointed map.

Th n the

folio

wing seq'Uencc is e.ract:

--t

[��y vV]�

-�

[2:�X, vV]�

--t

[f�(J) vV]�

--t

[Y, vV]�

--t

[.X, vV]�.

Let

(X1, A1)

and

(X2, A2)

be pairs of space over B with

i1 : A1---+X1

and

i2 : A2---+X2

the inclusions. We denote by

ni1

and

ni2

the fibrewise rnapping fibre of

i1

and

-i2.

For given

j1

:

Z---+X1

and

!2 : Z---+_X"2,

we can define son1e fibrewise homotopy fibres and fibrewise pull-backs:

ni1,ft = {(z,lx1)EZxBL(XI)Ifl(z) = lx1(0), lx1(1)EA1}, n.i2,h = {(z,lx2)EZxBL(X2)if2(z) = lx2(0), lx2(1)EA2},

where

L(-)

denotes the space of fibrewise free paths on the space -.

Similarly, for maps

il XB-i2 : Ar XBA2---7Xl XBX2,

k :

xl XBA2UA1 XBX2 ---+X1xBX2

and

(J1,J2) = (frxBf2)o6.z: Z---+X1xBX2,

we can define

nilXBi2 = {(lxl, lx2)EL(Xr)XBL(X2)Ilxl(l)EAr, lx2(1)EA2} =nil xBni2'

1 ^{fr(z) =} ^lx1 ⁽⁰⁾ )

= (z, lx1 lx2)EZxBL(X1)xBL(X2) J2(z) = lx2(0) . Ux1, lx2)Enk

Then we have the natural projections Pl:

ni,xBi2, (h.h)---+ni,,jl

and

P2: nilXBi2, (h.h)---+ni2.h

given as follows:

p1(z, lx, LxJ = (z, lx, ), p2(:::, lx,. (y2) = (

z

lx2)

^·

vv show the following:

14

(17)

Len11na 5.3.

Let ( .

\

1 , At) and (.\2, A.2) be fibrewise connected fibre

wise pointed cofibrerl pairs and

Z

a fibre wi �e connected space o

u

r

^B

wdh mapsf1

: Z-+./\1

and f2 : Z-+.\2. Then the fibrewise hornotopy p

-u

l l - b a ^c

k f2k,(ft ,h)

of (/1, f2)

^:^Z-+^.�

1

^X

BX 2 ^and

^{k :}

�\'"1

^X

aA.2UA.1

^X

BX2

-+X1

xB .. "'\2 has naturally fibrewise pointed homotopy type of the fibrewise homotopy push-out ofpl :

f2i1x8i2,(ft,h)-+f2i1.ft

andp2:

f2i1x8i2,(ft,h) -+f2i2,h.

nil X ai2,(fl ,h) PI

nit,ft

---+

P2

1 1

ni2,h ^---+^! nk,(ft .h) ^---+

xl X aA2UA1 XBX2

.1 1

^k

z (h, h)

xl xBX2

Proof. Let vV denote the fibrewise pointed push-out of

p1

and p2.

Then we define a fibrewise pointed n1ap

:

^vV-+Dk,(ft ,h) by the fol

lowing:

<J?((z, (a1, a2),l), t)

₌

(z, ( a1 ,l " (

l +

t)), (l',l"1+t))

<J? ( (z, (a 1, a2), l), t)

₌

( z ( l' (

₁

- ^t), _a2), ^(l' _1-t, ^l"))

if

t<O

if

t�O,

where

l': l-+A1

^and

l": l-+A2

are paths such that

l'(O)

⁼

f1(z), l'(1)

⁼

a1, l"(O)

⁼

J2(z), l"(1)

⁼

a2

^and

l

⁼

(l', l"),

respectively.

Moreover maps

l\-t

^and

l'\+t

are paths such that

l�-t ( s)

⁼

l" ( ( 1 - t )

s

)

and

l"1+t(s)

⁼

/"((1

⁺

t)s),

respectively.

By making use of the property of the fibrewise pointed· cofibration

(

see Lemma

5.1),

we define a fibrewise pointed map

w _:

Dk,(ft,h)-+W as following:

Let

( 1/J, ht)

^and

( 1/J' h�)

be pairs with the above conditions. Let

U

⁼

1/J-1([0, 1))

and

U'

⁼

1/J'-1([0, 1)).

Then

h(x1, 1/J(x!))EU

^and

h(x2, 1/J'(x2)) EU'.

So we define a fibrewise pointed map W as follows:

w(- (x a ) (l' l"))

⁼

{ ⁽

^z,

^(l'(1), ^a2), ^(ft, l"), 1/J(x))

^if

1/J(x)

^<

1

�,

1 2 (z, a2, l")

^if

1/J(x)

⁼ ¹

w(.:, (al, a2) (l' l"))

⁼

((z, (a1, a1), (l', l")),

O)EDitXBi2,(ft,h)

x[-1 1]

(l', l"))

^_-

{ (z (a1,1[1'(1)), (l' , l'') -1/J'(x))

^if

1/J'(x)

^<¹

w(z _(al, x2)

( z a1 l )

^if

'!j/ _{( x)}

= 1,

15

(18)

where

�

is a path such that

l' ( 4t)

� ( t)

⁼

h (

:r

1 , ( 4 t - 1 ⁾ _·if ⁽

:z: I

) ⁾ h(h(x1, 1/J(xl)),

₂

t

_-

1)

O<t<-

1 - -4 1 1

-<t<-

4- -2

-<t<1, 1

2- - and also

l''

is a path such that

t"(4t)

l1'(t)

=

h'(x2, (4t- ^1)�'(x2)) h'(h'(x2, �'(x2)),

₂

t - ₁₎

O<t<-

1 - -4

1 1

-<t<-

4- -2

-<t<l. 1

2--

Remark 5 ^.1^.

If x1 E A1) then [i

=

l'. Similatly if x2 E A2) then _l''

=

l".

For an elen1ent

( z, a1, l') EDi1

.h, the composition of W and is the following:

Similarly for an element

(z,a2,l")EDi2,J2,

=

(

^z

(a1, l''1+t(l)) (t' l''1+t), -�(l"(1

+

t)))

if

�(t"(1

₊

t)) < 1 ^t:S;O

(

^z,

a1, l')

_if

�(l"(l

₊

t))

⁼ _1,

^t:S;O

(

z

, ( �

¹_

t ( 1 ₎ _{, a} ² ⁾ ( [i

^1-

_t _l") _·1/J' _{( l'} ⁽ ^{1 - t) )} ₎

if

if' ( L' ^{(1 - t )} ⁾ ^< 1 ^, ^t 2

₀

( z a2 l")

if

�' ( L' ⁽ 1

_-

^t))

⁼

1, t 2

_0.

16

(19)

So we rnay clell.ne a fibrewise pointed honlotopy ff ^:nk,(]l .]2) ^X[ -i-nf..·.(ft ,]2) by

the

following:

For

an

elen1ent (

z,

a1, l')EDi1,]1, H(z, a1, l',

s

)

=

{ (z, (a1,,CJ,(o)(1)), (I', ^CJ,(o)) ^, (,P'(h(z))- l)s- lj/(h(z))) if 1/;',(J2(z))

< 1

(

z,

a 1

,

l) 1f </J (f2(z))

= 1,

Similarly for an element (

z,

a1, l") E Di2

^,h,

H

(

z,

a2, l" , s)

=

{ ^{( z, ( C}

^!J

^�;) ⁽ ^{1), a} ²⁾ ^{, ( C}

^!J

⁽

^z⁾

^,l")

^,

^{( 1 - 1/;} ^(!1 ⁽

^z

^{))) s}

⁺

^1/; ⁽ ^/J ⁽

^z

⁾⁾⁾ ^if ^1/; ^(!1 ⁽

^z

⁾⁾

^<

¹

(

z,

a2, L ) 1f 'ljJ ( f 1 (

z

))

=

1,

=

(

_z,

(a1, [1'1+st(1)), (l', l''i+st), -</J(l " (1

₊

st))) if </J(l"(1

+

st))

<

1, t:SO

(

z,

a1, l') if </J(l " ( 1

+

st))

=

1, t:s;O

(z, (fl-st(1), a2), (f'l-st, l"), </J'(l'(1- st))) if </J'(l'(1- st))

<

1, t2::0

(

z,

a2, l") if </J'(l'(l- st))

= 1,

t2::0.

ext we require the composition of

<P

and

W.

w( (

.

) (L' l"))

=

{ ⁽

^z,

^(i'(1- <P(x1), a2), (l'1-w(x1), L")) if 'ljJ(xi)

^<

1

o

z, x1, a2 , ,

(

z,

(l'( ) 0

,

a2 , ) (

Ct'(o),

L " ))

^.

tf ·1/J ( x1 )

=

1,

(20)

So we 111ay define a fibrewise pointed homotopy G : vV

I---+

_vV_{by the}

following:

{ ⁽ ^:

^,

^(! ^: ^((¢'(

^xl

⁾ ^-, ^� ^�15

^{+ 1}

^- ^¢'(

^xt

⁾⁾

^,

^a2, ⁽ ^l'

^{( W(x,)-}

^;f)'+

^1_^W(q),

^l")

^tf

^1/J(

^�1

⁾ ^:

¹

(

^�,

( ^{l ( s)} ^, ^{a2), ( l}

^s,

^{l ) )}

^{1f 1/J}

(

^xI

)

^- ^{1 ,}

G(z, (a1, a2), (l', l") s) = (

^z,

(a1, a2), (l', l")), G(z,(a1,x2),(l',l"),s)

⁼

{ ⁽

^Z

^, ^{( (a 1} ^,

^{It

⁽ ^{( -} ^�

⁺

^{1j/ (}

^X

^{2 ) )}

⁵^{+ 1}

^-

^!/;I

⁽

^X

2 ) ) , (l'

, {It ( ^_t + W I ( x 2 ) ) s + 1 -1/J ^I( x 2 )

) if ·!/;I (

X

2 )

< 1

(-:, (a1, l"(s)), (l', l�')) if �'(x2)

⁼^1.

Thus we obtain that

Wo�Bidnk,(f1

,/2) and

o W �Bidw.

This completes the proof. D

Definition 5.4.

The fibTewise pointed map cjJ

:

E--+F) wheTe E and F aTe fibTewise pointed space oveT

_B)

is a fibTewise pointed fibTation if cjJ has the following pTopeTty joT all fibTewise pointed spaces X. Let f

:

X---+ E be a fi'bTewise pointed map and let g

^:

X xI---+ F be a fibTewise pointed homotopy such that goi

⁼

c/Jo f. Then theTe exists a fibTewise pointed homotopy h

:

X xI---+ E such that hoi

⁼

f ^and c/Jo ^h

⁼

g.

Xxi �F.

Let

p

:

E--+X

be a fibrewise pointed fibration. Then the following sequence is exact:

---+

[A, D�F]�

_---+

[A, D�E]�

_---+

[A, D�X]�

---+

[A F]�

_---+

[A E]�

_---+

[A,X]�.

Definition 5.5.

An fibTewise pointed An -stTuctuTe on a fi'bTewise pointed space X consists of an n-tuple of fibTewise pointed maps

X =E1

_--+

E2

_---+

B

=Po

_--+

P1

--+

18

--+

En

(21)

such that

Pi: Ei

_�P

i

^-_l is

fibre

wise pointed .fibration fo

r

i

=

^{1. 2,}^·^·^· ^{. 11.}

together with a fibre wise pointed contracting ho moLopy

h : cg

En-l �En such that

h(C�En_1)CEi

Definition 5.6.

{mi}J

l�i�n is called a.fibrewise pointed An-form, on J�

·if 71/,i

^:

_f{i

_xxi�)( satisfies the following propeTties:

(1) _m2(*, s(b),

^x

)

⁼

·m ₂

(

*

^{, x,}

s(b))

^{= x)} for any

.rE�Y, *Ef\2 (2) mi(ch(T, s)(p,

^0'

)

^,

X11

^• ^•^• ^,

:z:i)

=

mr(p, X1,

^·^·^·

_,

Xk-1 ms(O' Xk ^· ^·^·

,

Xk+s-d Xk+s ^·^·^·

,

_Xi)

for

pEf{r, O'E!C

^(r⁺

s _{= i}

+

1) (3) mi(T, X11

^•^• ^• ¹

Xj-11 s(b)

_Xj+l,

_·

^·^· ¹

Xi)

=

mi_1 ( sj ( T), _x1,

^·

_·

^·

, Xj-l, Xj+l,

^·^·^·

, xi)

for

T _Ef

{ⁱ^:i ^>2.

Then

(X, { mi })

is called an An -space over

B.

Remark 5.2. Since we only conside1 the case that p :X �B is a fibre bundle) the following statement holds as well as the o1dina1y case:

A fib1ewise pointed space X ove1

B

^{has an}An -st1uctu1e over

B

^{if and}

only if

X

^{has an}^{An -folm}

{ mi}

ove1

B.

Definition 5. 7. The

X

-p1ojective i -space ove1

B

^p

i

(X))

i

^�ⁿ⁾ ^asso

ciated with an An -space ove1 B is the base space Pi of the de1ived An -st1uctu1e over

B.

Theorem 5.1. Let p ^:X �B be a fib1e bundle and X jib1ewise con

nected. Then cat

�

�m if and only if the canonical inclusion pm(n�X) CP00(fl�X)r::::.BX has a 1ight fib1ewise pointed inve1sion.

Let £m+l be the fibrewise hornotopy fibre of the inclusion X(m+ll�xm+l and pm the fibrewise homotopy pull-back of

1

[ l

{ lxiEX&,i=o,···m,bEB}

where X m+l

= (xo

... xm)EXm+l

(b)

^f

Xt

⁼

s

or some t

and 6.m+l denotes the diagonal. Then we have the following diagram commutes up to homotopy:

£m+l _---+^id £m+l

1 1

pm ^---+ x(m+l]

1 1

X

^---+^Clm+L ^x·m+l

19

(22)

Let us recall that

cat�j\ �rn

if and only if the diagonal n1ap _im+t is cmnpressible into )(lm+l]o The latter condition is clearly with the existence of a fibrewise hornotopy section of the projection pm----+ _X 0

Now we take Z = _)(. Y = xm,

ft

= idx

f2

= 6.m, Ar ⁼ s(B), and A2

_

_v_(m] _tl ₁ _n _ pm n. ^rv n. _ pm-1 d

- _/\. , 1en we 1ave Hk,(ft ,h)- , Hq ,ft-BCB, Htt.ft- an

the following fibrewise pull-back diagram:

Di2 ^---+ Dit ^xsi2,(!t ,h) ^---+ Dit.ft

Ptd 1

1 1

Di2 ^---+ Di2,h ---+ z

Since

f

= idx, and A1 = s( B), Di1 ,ft is fi brewise pointed contractible, and hence Di1 ^xsi2 ,(ft ,h) is fibrewise pointed homotopy equivalent with Di2 the fibre of Di2,h ----+Z , in the this case. Here i2 is the inclusion map xlml----+Xm, and hence Di2 is Em by definition. Thus we have fibrewise pointed push-out and pull-back diagram:

Em ^---+

1

s(B) ---+

pm-1

1

pm ---+ X XBX[mlus(B) XBXm

1 1k

6m+l

X � XxBxm

Hence pm has the fibrewise pointed homotopy type of a

(

_unreduced

)

fibrewise mapping cone of the canonical inclusion Em----+ pm-1,

m�

1.

Similarly using Lemma503, we have the following push-out and pull

back diagram:

D�XxBEm pr2

---+ Em pr11

1

DBX B ^---+ Em+1 ---+ X xBX[mlus(B) xBXm

1

^1k

s(B)

�

_XxBxm

Hence Em+t has the fibrewise pointed hmnotopy type of the ( unre

duced) fibrewise join of D�X and Em ⁰ Since Em+l and pm are continu

ous functors in the sense of

(2)

this implies that {(Em+l, pm);

m�O}

gives the fibrewise A00-structure for D�X in the sense of Stasheff

[16]

0

1.eo We have the following commutative diagram:

20

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fl�)\

--t

n �

^.^\ ^--t ^--t

n�.\

--t

n �

^..^\

1 1 1 1

fl�)(

^--t

E2(D�X)

^--t ^--t

E ( n�.�\)

^--t^:=.a

s(B)

1

Pt B ^nsx

1 1

^Poo8^nsx

1 B

^--t

P1(D�X)

^--t ^--t

P00(D�X)

^:=a

X,

where

p��x : £k+1(D�_X)--+Pk(D�X)

is fibrewise fibration,

£k+1(D�X) -7Ek+2(D�X)

is fibrewise nullhomotopy,

Pk(D�X)--+Pk+l(D�X)

is in

clusion for k�l, and

P1�BI:�D�X.

Thus pm has the fibrewise hon1otopy type of pm

( D�X)

the fibrewise

D�X

-projective m-space. This implies Theorem. 0

Re1nark 5.3. By Theorem 5.1} we obtain the following:

Let

(p : X

^---7

B)

be an object of q-Fib and

X

fibre wise connected.

If

cat

�

^X

:s; 1

then

I:�D�X �X

has a right fibrewise inversion.

Remark 5.4. We show that cat

��

^f{

=

^2. Assume that cat

��

^f{:::;1.

Then by Remark 5.2} there exists a right fibre wise inversion of ^ev}

f: I

^{

--+I:�D�I{}

where

B =51.

Let

g: I:�D�I{ -7I:DS1 xB

is defined by

g(

t/\Blb)

= (

t/\Blb,

b).

Then we have the following commutative diagram between quasi-fibrations:

I:DS1

^--t

I:DS1x{b}

1 1

I;BDB B B ^f{

� I:DS1

^X

B

1 1

^pr2

B _--t B

So

I:�D�I{

�B

I:DS1 x B.

^euof �B idr,· makes the following composition

H"(I<) � H"'(I:�D�I<) � H*(I<)

an identity map. In particular} ^ev"'

: H( I<)-+ H (I:�D�I<)

is an injec

tion} while

H(I:�D�I<)�H"(I:DS1xB)�H(I:DS1)®H*(S1)

has no tor ion and

H"' (I{)

has torsion. This is contradiction. Thus we have cat

�

I\�2. Since ^f{ is a .�ection ed 51-bundle over

51}

by making use of Theorem 4-1} we obtain that cat

�

I{:s;2. This completes the proof ⁰

21

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Re1nark 5.5. Remark 4.5 and Re mark 5.2 gives an e.z;runple snch that fib rP wi�e category and fibre wise pointe d category do not co inc ide

6. FIBREWISE HIGHER HOPF It VARIA1 T ^{At 0} FIBREvVISE POI:\ITED CATEGORY

Firstly we begin with the definition of the fibrewise higher Hopf in

variant. For a fibre bundle p

: X ---7B

with cat

�X

:;.m and

u�V ---7B

we fix the fi brewise homotopy sections

CJ( X) : X ---7 pm ( D� (_X"))

and

CJ(V): V---7I:DV

⁼

P1(D�(V))cPm(D�(V)).

Definition 6.1. vVe define a fibrew'is h·igher f!opf -invar-iant as

H�

:

[I;�V, X]�---7[<.--J�V, £m+1(D�X)]�.

For a map f :

2:::� V ---7 X)

we have the following commutative diagrams:

I;BV B

I;�V � X � pm(D�X)

---+

pm(D�X)

cr(E�V)

cr(X)

1 1

^e��x

1 pm(D�X)

---+

P00(D�X) �8 ^X

I;BDBI;BV B B B

^---7

pm(D�I;�V) pm(n�f)

pm(D�X)

=1 1 1

2::BDBI;BV

_---7

P00(D�I;�V) poo(n�j) _P00(D�X)

B B B

ev

_"BV 1 l�B l�B

LJB

^---7

LJ�v � X

By the above diagrams) the difference between

CJ(X)of

and

pm(D�f) oCJ(I;�V)

in

pm(D�X)

is given by a map

dc:n(X) (f)

:

I;�V ---7Pm(D�X)

which will vanish in

P00(D�X)�8X)

i.e. For the following co·mmuta- . d.

n�x dcr(X) (J')

twe 'lagram)

Cm

0

m �BCB.

Em+1(D�X)

_---7

Em+1(D�X)

nax

1

Pm8

1 I;BV B

^d�X)(J)

pm(D�X)

^---7

xrm+l]

n8x

1

em8

1 X

---7

xm+l

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Thus d�l(X\f) ^has

^a

^lift ^H7�: ��1/-tEm+l(D�.\'") to the total space of Sta.'he.ff's fibrewise fibration:

Rernark 6.1.

If

^X

is a

fibrewise

pointed suspension space1 then Hf can be regarded as the collection of all James Hopf invariants1 hf� ^j?_2.

Proposition

^6.1.

For a given structure map a-(X) for cat�X

=

m1 the map f makes the following diagram without p��x commute up to

fibrewise

pointed homotopy.

l:BV B � ^X

^----+^t ^vv

^� ^(L:�)2V

Hf!t(f) 1 1 cr(X)

l1B

^X

Em+1(D�X)

^�^p

B pm(D�X)

Em+t(D�i) 1 1Pm(n�i)

nBw n�w

e�+l

Em+1(D�W)

^---7

PmB pm(D�W) � pm+l(D�W)

^----+

W, where

^vV

denotes the fibrewise pointed map·ping cone of f� and i X

_-+^W₁

t.,m n�w are znc uszons.

·

l ·

Proof.

Let us recall that

H! (f)

is given by the difference between

a-(X)of

and

L:�D�foo-(V).

Then we have the following commutative diagram:

X

cr(E�V) 1 1 cr(X)

pm(nBBJ)

pm(D�L:�V) pm(D�X)

pm(n�f) 1 1Pm(n�i)

Hence the composition

pm(D�i)op��xoH!(f)

gives the difference be

tw n

pm(D�i)oo-(X)of

and

pm(D�i)oPm(D�J)oo-(V)�BCB·

Thus we obtain the commutativity of the diagram. D

23

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Ren1ark 6.2. By the romrntLtatiuity of the diagram in Proposition 5.1.

there e:-cists a map 0'1:

vV-+Prn+1(D�vV)

^gi^v^en by the homotopy deform- ing p

m (

D

�i) o

O'

( .

\"

)

to

P�1�w o3

ⁱⁿ

pm(D��V)

and by

\'lc(Em+t(n�X))o C({J)

in

pm+1(D�vV)}

where

{3

⁼

Em+t(D�i)oH!(f)

and we denote by

C�

the functor taking cones and x':

(C�(Em+1(D�vV)), Em+1(D�vV)) -+(Pm+1(D�vV), pm(D�vV))

be the fibrewise characteristic map of the attached cone of the fibrewise mapping cone space

pm+l (D� vV)

of p

�

�w.

Re1nark 6.3. Since the two maps

e:+l

^00'1and the identity ·idw are fibrewise pointed homotopic on

X}

the difference between them 1 :

�� V---+ vV

is defined by

O<t<-1 - -2 -<t<l. 1 2- - Here we know the fibrewise fibration:

n8x

D�X

---+

E2(D�X) � P1(D�X)

induces the following long exact sequence:

... ---+

[E�2V, W]�

---+

[E�V,E2(D�W)]�

---+

[E�V,P1(D�W)]�

---+

[E�V, W]�

---+

[V, E2(D�W)]�

---+

(**)

JVext let us consider the following fibrewise fibration sequence:

... ---+

n�£2

---+

n�vv

---+

E2

---+

P1

^---+

^w.

Since the map

n�vV -+E2�Bn�w*Bn�vv

is a fibrewise nullhomotopic}

a map id

n

Bw has a lift

D�vV -+D�P1(D�W).

lvforeover we have the

B

following commutative diagram:

[��v P1(D�W)J�

^---+

[��v, vv]�

�B

l

^�B

l

[V, D�P1(D�W))�

^----t

[V, D�vV]�.

Thus we have a splitting

D�vV -+D�P1(D�vV).

The long exact sequence

( ^**)

becomes a spit short exact sequence:

0-+ [LJ�V E2(D�vV)]�-+[��V P1(D � vV) ]�

-+

[ � � V, vV ]� -+ 0.

(

^{* * *}

)

'o 1 can be pulled back to a map Ia

: �V---+ P1 (D�W)cPm+l (��D�W).

Since the composition

2:�vvBvv -yVB�tO<T' vvvBvv

vB)

w

24

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can be deformed into an identity

map

^idw, we obtain the following co·mm·utatiue diagra·m (up to f£bre w·ise pointed homotopy):

vv

_---+

idw _vv

1 1idw

I;�Vv B vv

^\leo(-yv e(em+tocr'))

vv

T'O V BCT'

1 I

^\lBo(em+L V Bem+t)

pm+lv Bpm+l

^---+

pm+lv Bpm+l,

where

vV --+��vv B vV

is a fibrewise pinching map. Thus we may define a compresszon

o-(W) : W --+Pm+1(fl�W)

of the identity

idw

^{by the}

composition:

which gives the structure map for

cat�vV

:::;m ^{+ 1.}

Theorem 6.1. The following statement holds for

W

^with

cat�vV �m

^{+ 11}

m ⁼

cat�X.

cat�W �cat� X if H:; (f) is trivial fa� some choice of cr(X).

Proof. If the map

Em+1(D�i)oH!(f) : V--+Em+1(D�vV)

is trivial, then by Proposition 5.1, the structure map

o-(X)

is extendible on

vV.

By Remark 5.3 and the extension & of

pm(D�i)oo-(X),

we have the following comrnutative diagram:

vv

^---+

vv

1 1

��VVBW

^---+

��VVBvV

\l B 0 ('Yo VB <7)

1 1

^\lBobo v Bcr')

pm(D�W) � n8w pm+l(fl�W)

em

1 1

^em+L

vv

^---+

vv

Thus we obtain a cornpression

o-(vV) vV--+ pm (D�vV).

This implies

that

cat� vV:::;

^m. ^D

25

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223-255 ( L963).

[2] _Dold._A. and Thorn. R, Quasifaserungen und unendliche symmetrische Pro

dukte, Ann. of Ylath.(2) 67 (1958), _pp.2 5-305.

[3) _{Hall. I.};vr The generalized Whitney sum, Quart . .J. Math. Oxford.16, pp. 360- 384 ( 1965).

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don Math. Soc. 30, pp. 623-634 ( 1998).

[7) James. I. M, Reduced pmduct spaces, Ann. Math.62, pp. 170-197 (1955).

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[9) James. I. M, General topology and Homotopy theory, Springer ( 1984).

[10) James. I. M and Morris. J. R, Fibrewise category, Proc. Roy. Soc. Edinburgh.

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[11) _{James. I.}M, Handbook of Algebraic Topology, orth-Holland (1995).

[12) _James._{I. M,} Numerical 'invariants of fibrewise homotopy type, Homotopy Theory and Its Applications, Adem et al., eds, Amer. fath. soc. (1995).

[13) Crabb. M. C. and James. I. M, Fibrewise Homotopy Theory, Springer (1998).

[14) _lay._{J. P,}Fibrew,ise localization and completion, Trans. Amer. Math. Soc.268, pp. 127-146. (1980).

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.

E-mail address: msakai<Omath. kyushu. u-ac, jp

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