On topological neighbourhoods

C ^OMPOSITIO M ^ATHEMATICA

C. P. R ^OURKE B. J. S ^ANDERSON

On topological neighbourhoods

Compositio Mathematica, tome 22, n

^o

4 (1970), p. 387-424

<http://www.numdam.org/item?id=CM_1970__22_4_387_0>

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387

ON TOPOLOGICAL NEIGHBOURHOODS by

C. P. Rourke and B. J. Sanderson Wolters-Noordhoff Publishing

Printed in the Netherlands

This paper is concerned with the ’normal bundle’

problem

^for

topolog-

ical manifolds:

Suppose

^{M" is a} proper,

locally

^flat submanifold of

Qn+r;

^{then what structure can be}

put

^{on the}

neighbourhood

^{of M in}

Q?

If r ~ 2 the

problem

^{has been solved}

by Kirby [22],

^who has shown that there is an

essentially unique

normal disc

bundle,

while if r ~ 3 then our

counterexample [38] showed

that the notion of fibre bundle is too

strong

a

concept.

The notion of

topological

^{block bundle}

[37; § 1 ]

^{seems in-}

applicable

^{since M}

might possibly

^be

untriangulable

^{and a}

triangulation

of M would be unnatural structure for the

problem.

^{The answer we}

propose here is the ’stable microbundle

pair’.

The idea of

using

^micro-

bundle

pairs

^to

classify neighbourhoods

^was introduced

by Haefliger [9, 10]

^{in the}

pl

^{case and we showed}

[35; § 5]

^{that his}

theory essentially

coincides with ^our

theory of pl

block bundles.

An r-microbundle

pair

^{is a}

pair ^eN

^ce

(1 Ir

^where

^EN

denotes the trivial microbundle of rank N. Two are

equivalent

^if

they

^are

isomorphic

^after

possibly adding

^further trivial bundles to both elements and the iso-

morphism

^{restricts to the}

identity

^{on the trivial} subbundles. The

equiva-

lence classes form a

good ’theory’

^with

classifying

space

BToPr

⁼

limn~~(BTopr+n,n).

^{To the manifold}

pair

^{M c}

Q

^{we associate the}

pair LM EB

vM c

03C4Q|M ~

^{vM , where vM} denotes any stable inverse to rm - Our main theorem

(in § 3)

^asserts that this association classifies the germ of

neighbourhood

^{of M in}

Q except possibly

^{in the case}

n = 1, q = 3 (and

n ⁼

2, q

^{= 4 if}

^{~M ~} 0);

these ommisions are due to the unsolved 4-dimensional annulus

problem.

The main work of the

proof

^{is a}

stability

theorem for 03C0i

(Topr+n,n)

which is contained

in §

^{2. This we reduce}

by

^{means of immersion}

theory

to a statement about

straightening

handles in the sense of

Kirby

^and

Siebenmann

[21, 23], keeping

^a

pl

^subhandle

fixed,

^{which is}

proved

ⁱⁿ

§

^{1. The}

proof

^{follows the}

Kirby-Siebenmann proof

for the absolute case

using

^the relative surgery

techniques

^of

[32].

^In

§§ 4,

^{5 and 6 we}

give

some

applications

of the main theorem.

In

§

^{4 are theorems} about existence and

uniqueness

^{of normal block}

bundles in the case that M has a

triangulation

^not

necessarily

^combina-

torial. The results hold _{for any}

type

^{of block bundle}

(open,

^{closed or}

micro)

^{and are the same as}

[33; § 4] (existence

^and

uniqueness

up ^to

isotopy)

^{in the}

following

^cases:

r ~ 5

^{or ~}

2,

^{r =} 4 and M is 1-connect-

ed, r

^{= 3 and M is} 2-connected

(the

^omitted cases are

again

^{due to}

4-dimensional

problems).

In

§

^{5 we} prove stable existence and

uniqueness

^of normal micro and disc bundles. The dimensions are the same as obtained in the

pl

^case

by Haefliger

^{and Wall}

[12] (improved slightly by

^Morlet

[31 ] and

^Scott

[40]).

However we need

codimension

^{5 for our results on}

microbundles,

^and

6 for disc bundles.

In

§

^{6 we} prove

analogues of smoothing theory

^for submanifolds. There

are two cases:

(a)

^{M and}

^Q

^{are both}

pl.

^{manifolds and we seek to}

isotope

^{M to a}

pl.

submanifold. This is

always possible

^{in an}

essentially ^unique

^{way if}

r ~ 3;

^{and if}

r ~ 2, n + r ~ 5

^{there is a} well-defined obstruction.

(The

codim

^{3 result was}

originally

^announced

by Bryant

^{and Seebeck}

[2] ] using

^a result of Homma

[15 ] unfortunately

^the

proof

of Homma’s result appears ^to contain some gaps. Several other alternative

proofs

^have

been

given.)

(b) ^Q

^{is a}

^pl.

^{manifold. Here we have the}

^analogue

of the Lashof-

Rothenberg

^result

[28].

^{M can be}

isotoped

^{to a}

pl.

submanifold if and

only

^{if the}

classifying

map M -

BTop,

^for the germ of

neighbourhood

lifts to

BPL,..

^If

r ~

^{3 the}

problem

is identical to the absolute

problem

of

finding

^a

pl.

^{structure and if}

r ~

^{2 the} map lifts in an

essentially unique

way

by

the result

of Kirby

mentioned above.

We are indebted to A.

Haefliger

^{for his}

unpublished preprint [9] ^and

for a

private

communication

containing

^his

arguments

^for

classifying

germs of

pl. neighbourhoods.

^{We are also indebted to R. C.}

Kirby

^for

a copy of his excellent and detailed notes

[21 ]

^on

triangulating

^manifolds.

We

plan

^{a further} paper which will contain the technical details of

defining transversality

^for

topological

^manifolds

(Hudson

^showed

^{[17] ]}

that a local definition is

inadequate).

^{This is done}

by examining Whitney

sums

(defined

ⁱⁿ

^§

^{3 of this}

paper) along

^{the lines of}

[34; § 3]

^and

[39];

we then define M to be

’germ

transversal’ to W in

Q

^if

along

^{M n W the} three _germs of

neighbourhoods

^{form the}

Whitney

^sum

decomposition.

A relative

transversality

theorem in case dim M n

W ~

^{5 can then be}

proved using

^local

pl.

structures which exist

by Kirby

and Siebenmann’s results.

0. Preliminaries

We use the same basic scheme of notation as in

[35; § 0].

^{Rn denotes}

Euclidean _n-space and In the double unit cube

[-1, +1]n.

^{aln =}

[ - a, 03B1]n, ^{sn-i =}

^{DI". d n c Rn is} the standard

n-simplex

with vertices v0, v1 ··· v,,. There are natural inclusions Rn ^c

Rn+r,

^{In c}

In+r;

^and

identifications Rn Rr =

R n+r,

^{In Ir =}

^In+r.

Microbundles

We refer to Milnor

[30]

^{for basic results on} microbundles. We recall that if M is ^an unbounded manifold then _iM is the microbundle with total space

M x M,

zerosection

4M

^and

projection

03C01

(the projection

^{on the}

first

factor).

^{We often}

write i(M)

^{for iM. If Mis bounded then we define}

zrz ⁼

03C4(M+)|M

^where

M+ =

Mu open collar.

Suppose 03BEn, ~n+r

^are

microbundles. We

say 03BE

^{is a subbundle of} q and

write 03BE

^c 11 if

B(03BE) = B(~), E(03BE) ~ ^E(~),

^{at least in some}

neighbourhood

^of

B(03BE),

and for ea(,h

x E

B(03BE)

^there exists microbundle charts h : U x Rn -+

E( ç), 9 :

^{U x}

^Rn+r

^-

E(~)

^{with x E int}

^U,

^{so that}

The trival bundle an of rank n is defined

by

^the

diagram

An inverse

to 03BE

^{is a}

pair (~, t)

^{where il is a} bundle with the same base as

03BE

^{and t:}

E(03BE

⁰

q) - E(eN)

^{is a} trivialisation. Inverses are

unique

up ^to stable

isomorphism of il

and bundle

homotopy of t,

^see

[30].

A-sets and _groups

We refer to

[36, 37]

^{for the}

theory

^of

semisimplicial complexes

^and

groups without

degeneracies.

^The

A-group Topn (resp. PLn)

^{has as}

typical k-simplex

^a germ of

homeomorphisms (resp. pl. homeomorph- isms)

defined in a

neighbourhood

^of

^jk

^x

{0}

^and

^satisfying

(ii)

^{u commutes with}

projection

^on

^dk.

Applying

^the

classifying

functor of

[37; § 1]

^we

get classifying

spaces

BToPn, BPL.

^{which are} Kan 0394-sets.

BToPn

classifies n-microbundles with base a CW

complex

^{and there is a} universal microbundle

03B3n/BTopn. (We

recall from

[36]

^{that there is a natural}

bijection

^between

homotopy

classes of

d -maps [S(X), ^Y]

^and continuous maps

[X, |Y|],

^{where X}

has the

homotopy type

^{of a}

C W-complex,

^{Y is a Kan d-set and}

S(X)

^is

the

singular complex.

We will denote both these sets

by [X, Y]).

^All

topological

^{manifolds have the}

homotopy type

^{of CW}

complexes

^and

thus

BTop,,

classifies n-microbundles over manifolds. Similar remarks

apply to BPLn .

We define

0394-subgroups Topnr+n, Topr+n,n

^of

Topr+n by

^{the conditions}

(iii)1

^and

(iii)2 respectively

where here Rn is identified with

{0} Rn

^{C Rr Rn =}

^{Rr+n, PLnr+n,} PLnr+n,n

^{are defined}

similarly.

^{There are natural} inclusions of all the

pl.

groups in the

corresponding topological

groups and of

Topr+n,n

ⁱⁿ

Topnr+n

^etc.

We now define two

suspension

maps s and s’

(both injective)

^{in the}

diagram

where

s( (J) : ^dk

^x

^Rr+n

^x

^Ri

is defined to be a x id and

is obtained from

s( (J) by reordering

the last coordinate into the

(r + 1 )-st

place

^{and the}

j-th

coordinate to the

( j + 1 )-st place for j

⁼

r + 1, ...

^n.

f is

the natural inclusion. The outside square commutes while the tri-

angles

^do not;

however,

it is easy ^{to see} that

they

commute up ^to homo-

topy

which is all we

require.

We obtain a

large diagram

^of inclusions:

where the vertical inclusions are

s’,

the horizontal _{ones s,} and we have defined

Top

^{= u}

Topr , Topr

^{= u}

Topr+n,n

^and

^Top

^{= u}

Topr . By homotopy commutativity

^{we have}

Top

^c

Top

^a

homotopy equivalence.

Again

^{there are} similar definitions for the

pl.

groups.

If X is ^a A-set then we denote

by ^X(k)

^the k-skeleton of X.

Main tools

Apart

^from microbundles the

principal

^{tools will be}

isotopy

^extension

and immersion

theory,

^{in both}

pl.

^and

topological categories.

^An

isotopy

of M and

Q

^is

locally

^{trivial if it is}

locally

the restriction of an

isotopy

^of

an open subset of

Q

ⁱⁿ

Q.

^All

embeddings

^{of manifolds will be}

locally

flat and all

isotopies locally

^{trivial. The}

isotopy

extension theorem

(for locally

^trivial

isotopies)

^is

proved by

Hudson-Zeeman

[20]

^{in the}

pl.

^case

and

Edwards-Kirby [4]

^{in the}

topological

^{case. We also need} the theorem for cubes of

isotopies.

^{This is Hudson}

[16]

^{in the}

pl.

^{case, while the}

topological

^{case follows}

by combining

^{his methods with those of Ed-}

wards and

Kirby,

^{see also}

Kirby [21].

^From this last theorem we have

a Kan fibration

where p

^{restricts to the last n}

coordinates,

^{and the fibre is}

Top, ln, n -

We will need a

doubly

relative version of immersion

theory,

^{this is}

stated in

Corollary

^{2 of the}

appendix

^to this paper. The

pl.

^{version is}

stated but not

proved

ⁱⁿ

Haefliger-Poenaru [11 ],

^see the last three lines

of §

2. However it follows

easily

^{from what}

they

do prove

by analogous (rather simpler) ^arguments

^to those used in our

appendix. Incomplete

versions of

topological

^immersion

theory

^{have been}

given by

^Lees

[29],

Lashof

[27]

^{and Gauld}

[5].

1. Relative handle

straightening Definition of

^{the set}

Hk(n, i) for

^{k ~}

0,

^{n ~ i ~ 0.}

A

representative

^{is a}

pair (h, V)

where V is a

pl.

manifold and h :

dk

Rn ~ V a

homeomorphism

^{such that}

hl DAk X Rn u Ak x Ri is pl.

and

h|0394k ^{x Ri i s} pl. locally

^{flat. Two such}

(h1, V1)

^and

(h2, V2)

^are

equiv-

alent if there is a

pl. homeomorphism

q :

V1 -+ ^V2

^{defined in a}

neighbour-

hood of

hl (jk

^x

{0})

^{such that}

commutes up to ^a

topological isotopy

^{which is fixed on}

^{ojk X}

^{Rn ~}

^{jk x Ri}

and defined in a

neighbourhood

^of

^dk

^x

{0}

^I.

Now

identify ^dk

^with

^Ik by

^an orientation

preserving pl. homeomorph-

ism then an addition in

Hk(n, i )

is defined for k &#x3E; 0

by identifying ^Ik

with each of

Ik-1

^x

[ -1, 0 ]

^and

^Ik-1

^x

[0, 1 ]

^and

gluing

^{the two re-}

presentatives along ^Ik-1

^x

{0}.

^{This addition makes}

Hk(n, i )

^{into a}

group with ^zero

represented by (id., 0394k Rn).

^{This follows from}

Proposition

1.1 below and the definition of addition in

7rk(TOPm, q PLm, q).

By ignoring

conditions

on jk

^x R. we have a set

Hk(n);

however in this

case the

’equivalencc’

^{relation is not} transitive since the

composition

^of

q1 and _q2

might

^not be defined and we take the transitive closure of this relation. This ’absolute’ set is

essentially

^{the set of handle}

problems

considered

by Kirby

^and Siebenmann

[23]

^{and the first halves of 1.1 and}

1.2 are theirs.

There is a

forgetful

^function

f: Hk(n, i )

^~

Hk(n)

^{and a}

^suspension

s :

Hk(n, i)

^~

Hk(n

⁺

1,

^{i +}

1)

^defined

^by s(h, V) = (h

^x

^id,

^{V x}

R1 ).

PROPOSITION 1.1. There are

bijections

and

which commute with the

suspension

^and

forgetful functions,

^where

m = k+n and q = k+i.

The

proof

^{of 1.1 is}

postponed

^to

§

^2.

and

i, f ’further

^{n - i ~}

3,

^then

Hk(n, i) ~ Hk(n)

^{and we have a} commutative square

of isomorphisms

The cases

n - i ~

2 of the theorem follow

easily

^from

Kirby’s

^results

[22]

^on codimension 2

embeddings.

^The

proof

^for

n - i ~ 3 is in

^two

parts;

^{first we show}

that Hk(n, i)

⁼

0 or Z2 when k

⁼

3 and Hk(n, i)

⁼

^0,

if

k ~

3. This is done

by relativising

^the

Kirby-Siebenmann

^{’main dia-}

gram’ [21; 5.1].

^Then

secondly

^{we show}

by ’unwrapping’ [21; 5.2]

^that

Hk(n, i)

^is

actually Z2

^{in the case k = 3.}

Relativisation

of

^{the main}

diagram

Let h :

0394k

x Rn ~ Vm be a

particular

^{relative handle}

straightening problem.

It will be convenient to denote the

pl.

submanifold

h(dk x Ri)

by vq

and write h :

Ak

X R n, in vm,q a map of

pairs.

^Consider

diagram

^1:

Diagram 1

All maps ^are

pl.

^on boundaries and indicated

submanifolds,

^and

by

relative

collaring [6]

^we may also ^assume that all _maps are

pl.

^{in a}

neigh-

bourhood of the

boundary. Tn,i0

⁼

^{Tn, minus}

^{a disc}

pair;

^{a is an immer-}

sion of

dk

x

Tô

ⁱⁿ

^jk

^{x Rn which}

respects boundary

and immerses

A k

^x

Toi

in

d k

^x

R. ; Wm,q0

^is

^{d k}

^x

Tn,i0 ^with

^{PL structure induced from ha. All the}

maps ^on the left commute with a standard inclusion of

0394k In,i

in

dk Tn,i0.

^The construction of the

diagram

^is

exactly

^{as in}

[21;

pages

71-74] except

^{for two}

points

a)

^The construction

of

g. Let C ^c

¿jk be

^an _open

pl.

^{collar on}

^BAk

^defined

so that

hl C x

^{Rn is}

pl.

^{Now define}

Let i : Um,q ~

Wm,qc

^{be the} identification _map

(see figure 1).

^Next

identify

^the

one-point compactification

of um,q with

0394k Tn,i by

^a

homeomorphism

^{which is}

pl.

^{on U and the}

identity

^on

^dk

^X

^In,i. Finally define g

^{to be the}

one-pair compactification

^{of i.} We claim that wm,q has a

pl.

^structure

extending

^{that of}

Wm,qc

^{and so that}

glAk x Ti

^is

pl.

Looking

^{at the end of}

Wm,qc

^{we see} that it is

enough

^to prove ^a relative form of the

hauptvermutung

^for

sm -1 X R,

stated in

proposition

^1.3

below,

^and

proved

^at the end of the section.

Figure ¹

b)

The construction

of g’.

We need

g’ =

g ^on

Ak

^x

Ti as

well as on

8(Ak X ^Tn)

^{as in}

^{[21 ].}

^When

it is

possible

^{to find}

g’

^{at all the extra condition can} also be satisfied.

This follows from ^a result of

[32],

^stated in 1.4 below:

PROPOSITION 1.3.

Suppose ( W m, Wq)

^{is a}

pl. manifold pair

^{and that} there is a

homeomorphism

^{h :}

( Wm, Wq) ~ (Sm-1 ^{x R, Sq-1} x R)

^such

that

hl

^Wq

is pl.

^{Then there exists}

a pl. homeomorphism

^{h’ : W’}

^{sm -1}

^x

R, extending h| W q, provided

^{m ~ 5 and}

m-q ~

^3.

PROPOSITION 1.4.

Suppose

^{h :}

Qm,q ~

^{Wm, q is a}

homeomorphism of pairs of compact pl. manifolds

^{such that}

hl8Qm u Qq

^is

pl. Suppose further

that h is

homotopic

^rel

8Qm

^{to a}

pl. homeomorphism,

^{then h is}

homotopic

rel

8Qm u Qq

^{to a}

pl. homeomorphism provided

^{m ~ 5 and}

m - q ~

^3.

PROOF OF THEOREM 1.2. Let

(h, h)

^{be the relative}

^problem

^considered

above and _suppose

n - i ~ 3

^and

n + k ~

5 and further that

(h, V)

^is

straightenable

^{as an absolute}

problem.

^Then

by

^{1.3 and 1.4 we can}

construct the

complete

^{relative main}

diagram

^{for h. H is the}

identity

^on

~(0394k

^x

2In)

^and

^{dk x 21’}

and hence is

isotopic

^{to the}

identity

^{rel these}

subsets

by

^{an Alexander}

isotopy. Restricting

^{to a}

neighbourhood

^of

A k

x

{0}

^{which is embedded in}

^Ak

^{x Rn}

throughout

^the

isotopy

^{shows that}

h is

straightenable

^{as a relative}

problem.

Now suppose

(h, Tl)

^{is un-}

straightenable

^{as an absolute}

problem,

^{it is therefore}

unstraighte nable

as a relative

problem;

^but

by adding

any

unstraightenablerelative problem

we

get

^a

straightenable

^absolute

(and

^hence

relative) problem.

^Thus

Hk(n, i) = 0

^if

^{k ~} 3, k + n ~ 5, n - i ~ 3

^{and 0 or}

Z2

^{if k = 3. It}

remains to show that

H3(n, i )

⁼

^{Z2 in}

this range.

Consider the commutative

diagram (diagram 2):

ex

Diagram 2

The indicated

isomorphisms

^come from 1.1 and

[21;

^theorem

12].

Horizontal maps ^are

suspension

^and

diagonal

maps

forgetful

^functions.

They

^are

homomorphisms by

^{1.1. We have} shown that all the groups ^are 0 ^or

Z2,

^{we will construct a function a which makes the}

diagram

^com-

mute, it then follows that all the _groups are

Z2

^{and the}

homomorphisms

are

isomorphisms.

DEFINITION OF a. We reverse the

unwrapping

construction

[21;

p.

79].

Let

h:03943 R2 ~ V represent

^an element of

H3(2). Identify 03943 x I2

^c

03943 R2

with

03943 I2 {1}

^c

~(03943 I3)

^{and let}

V0 = h(03943 I3).

^Glue

~03943 I3

to

V0

^{via h on}

~03943 I2 {1}

this forms

Wo,

^{which has}

pl.

interior since h is

pl.

^{on this}

subset,

^{and we have a}

homeomorphism

h’:03943 I2 {1}~~03943 I3 ~ W0

^{which is}

^pl.

^on

~03943 I3.

Now

int( Wo)

^{is a} contractible

pl.

manifold and hence

RS (Stallings [42]).

In

int( Wo)

^{choose a}

pl.

^ball

^BS sufficiently large

^{to contain}

h’(03943

^x

{0} x {1}

^u

^a.£13

^x

{0}

^x

[0, 1])

in its

interior,

^{and denote}

C5 = (h’)-1B5

^c

ô(d3

^x

I3).

^{Now extend}

h’IC5

^to

g:B6 = 03943 I3 ~ B5 I by

^two conical extensions. First extend over

aB6 using

^{the fact that}

ôB6 - int C’

is a ball

by

^the

Schoenflies thecrem and second extend over

B6 using

the standard cone

structure on

B6.

See

figure

^2. Observe that

g|03943 {0}

^is

^pl.

^since

h’|~03943 {0}

^was

pl. Finally identify ^R3

^with

int 0394I3

and

restrict g

^to

03943 03B5I3

to

complete

^the definition of

a(h),

^e

being

^{chosen so that}

g|~03943

^x

^8I3

^is

pl.

^It is easy ^to check

(cf ^[21;

^p.

83])

^that

03B1(h)

^is

equivalent

to the

suspension

^of

h,

^as

required.

Figure 2

This

completes

^the

proof

of 1.2. We note for future reference

(§ 4)

that we could have

constructed g

^{to be}

pl.

^{on all of}

^{ôd 3 I3} by

^the

regular neighbourhood theorem,

^and also that the _range

of g is a pl.

^ball.

PROOF oF 1.3. The

case m ~

^6.

Let 03BE

^{be a} normal block bundle for Wq c Wm. We have e the trivial normal block bundle for

Sq -1

^c

Sm -1

and we can

homotope

^{h rel Wq to a} map h" : Wm ~

sm-l X R

so that

h’’|E(03BE)

^{is a block}

homotopy equivalence of 03BE

^{with e x R}

(see ^{[37; § 3]}

for

definition).

This follows

by

^{an easy induction}

^argument using

^homo-

topy

extension and the fact that h has local

degree

^{1. Then}

h’’|E(03BE)

determines a map g : Wq ~

Gr/PLr, r

^{= m - q, which we} claim is nul-

homotopic.

Consider s _{o 9 :} Wq ~

G/PL,

^the

suspension of g,

^this

factors via

Top/PL by

^{a standard}

argument

^{since we started with a}

homeomorphism. s

o g is therefore

nulhomotopic

since the natural map of

Top/PL

ⁱⁿ

G/PL

^{is zero on}

homotopy, by [23]

and the fact that

03C03(G/PL)

^{= 0}

(see

^{also Wall}

[43]).

^{Hence g is}

nulhomotopic by stability

of

r/PLr (see [35; 1.10]).

It follows that h" is

homotopic

^{rel Wq to} h"’ which restricts to a

block bundle

isomorphism of 03BE

^{with E x R. This}

provides

^a

pl. product

structure on

E(03BE)

which extends to all of Wm

by

Siebenmann’s relative

collaring

^theorem

[41 ].

^{We now have a}

pl. isomorphism

which extends

h|

^{Wq. But}

^Mm-1

^{is a}

pl. sphere by

the Poincaré theorem and the

pair (Mm -1, sq -1 ) is

^unknotted

by

^Zeeman

[45 ].

^{The construc-}

tion of the desired h’ is now easy.

The case m ⁼ 5. The case q ⁼ 1

presents

^little

difficulty

so we concen- trate on the _{case q} ⁼ 2. It is easy ^to

verify

that any

pl. self-homeomorph-

ism of

Si

x R extends to

S4

x R

if q ~

² and it suffices to find some

pl.

homeomorphism

^of

pairs ^{W5,2 ~ S4,1 x R.} By

^Wall

[43] ^W5

^is

pl.

homeomorphic

^with

^S4

^{x R and we assume that}

^{W5 = S4}

^x

R;

^{we have}

to unknot

M2 - h-1(S1 x R)

ⁱⁿ

^{S4 x R.}

^{We show how to}

^isotope

^M

to meet each

sphere ^S4

^x

{n}, n~Z,

^{in an essential} circle and the result follows from the 2-dimensional

pl.

^annulus theorem and

unknotting S1

^{x I} in

S4 I (Hudson

^and Lickorish’ concordence extension theorem

[19]).

^The method for each

S4

is the same.

By transversality ^S4

^{n M}

can be taken to be a finite number of circles. We show how to

pipe

^two

neighboring

^circles

together

and the result follows

by

induction. Choose

points p, q

^{on each circle and arcs}

03B1, 03B2

^{in M and}

^S4 joining

^{them and}

not

meeting

^other intersections. Then the circle oc

u 13

spans ^a 2-disc D which meets M and

S4 only

^{in a u}

fi.

^A

regular neighbourhood

^{of D is}

a 5-ball.

B5 meeting ^S4

^{and M} in unknotted subdiscs

B4, ^B2

^{which in}

turn meet in 2 arcs

ab,

cd say. It is ^a trivial ^{matter to}

isotope ^{B 2 in B5}

rel

boundry

^{so as to}

replace ab,

^cd

by

^{arcs ac, bd}

(or ^ad, bc)

and this has the effect of

piping

^{the two}

original

^circles

together.

2. The

stability

^theorems

Before

proving

1.1 it is convenient to define a new set

H’k(n, i).

^A

representative

^{is a}

homeomorphism

^{h :}

^{Rk x}

^{Rn ~ V where V is a}

pl.

manifold, h|cl(Rk-0394k) Rn

^is

^pl.

^and

h|0394k Ri

^is

pl. locally

^flat.

(h, V) ~ (g, W)

^{if there is a}

pl. homeomorphism q :

^{V ~ W defined in}

a

neighbourhood

^of

^dk

^x

{0}

^{such that}

commutes up ^{to a}

topological isotopy

^{which is}

pl.

^on

cl(Rk-0394k)

^{x Rn ~}

03B4k x Ri

and defined in a

neighbourhood

^of

^{d k}

^x

{0}

^{x I.}

(h, V) - (g, W)

if

(h, V) = (hl, Vi) z (h2, V2) ~

^{··· ~}

(h1, Vl) = (g, W).

^{The set of}

equivalence

classes forms

Hk(n, i); Hk(n)

is defined

similarly.

^{There are}

obvious

surjections 03C81 : Hk(n, i)

^-

H’k(n, i )

^and

03C82 : Hk(n) ~ H’k(n)

defined

by ’adding

^{a collar’.}

PROPOSITION

2.1. 03C81 and 1/1 2

^are

bijections.

PROOF. We have to show

injectivity. Suppose (h, V) z (g, W) ; ^{let q}

be as above

and st

^the

isotopy of qh

^{to g ; so we have so -}

qh

and s, = g.

Now

by

^two

applications

^{of the}

pl. covering isotopy

^{theorem we can}

find a

pl.

^ambient

isotopy st

^{of W so that}

s’0 = id

^and

s’tqh|L

⁼

sIL

where L ⁼

ad k

^x Rn ~

d k

^x

Ri. Define q = s’1 q and st = s’1

^o

(s’t)-1

o st

then q

^is

pl. and S-t

^{is an}

isotopy

^between

qh and g

which is fixed on L.

This shows that the restriction of

(h, V)

^and

(g, W )

^to

^{dk x Rn}

^are

equivalent

ⁱⁿ

Hk (n, i),

^as

^required.

Now define

Ik(n, i )

^{to be the set of}

regular homotopy

classes of orientation

preserving

immersions h :

Rk Rn ~ Rk+n

such that h is a

pl.

^{immersion of}

^{Rk x Ri}

^{and of a}

neighbourhood

^of

cl(Rk-0394k)

^{x Rn.}

The

regular homotopies

^{are via such} immersions but defined

only

^{in a}

neighbourhood

^of

(0394k {0}) I. ^Similarly Ik(n)

^{is defined}

by ignoring

the condition of

dk x Ri.

Now h induces a

pl.

^{manifold structure on}

Rk x Rn;

denote this

pl.

^manifold

by (Rk Rn)h.

^{We then}

^get

^{a relative}

handle

problem ht,

^with

ho =

^{h. It is} easy ^{to see} that

(’idB (Rk

^x

Rn)h) ,:

(’id’, (Rk

^x

Rn)hE)

for small e and

hence, by compactness

^of

I,

^the

(’id’, (Rk

^x

Rn)h) 1’01 (’id’, (Rk

^x

Rn)h1).

^We therefore have well-defined functions _{CP1 :}

Ik(n, i )

^~

Hk(n, i)

^{and CP2 :}

Ik(n)

^-

Hk(n).

PROPOSITION 2.2. _(fJ1 and _CP2 are

bijections.

PROOF.

Surjectivity.

^Let

(h, V) ^represent

^{an element of}

H’k(n, i)

^then

by

a

collaring argument

^we may suppose h is

pl.

^{in a}

neighbourhood

^of

ôdk

^x Rn. Now Vits a contractible

pl.

manifold and therefore

pl.

^immerses

in

Rk+n by

^{an immersion a} such that ah is orientation

preserving.

Injectivity. Suppose (’id’, (Rk

^x

Rn)h0) ~ (’id’, (Rk x Rn)h1)

^{we have to}

construct a

regular homotopy

^between

ho

^and

hl.

Let q, st be

given by

the definition of ~

(notation

^{as in}

2.1).

^{We construct the}

regular

^homo-

topy

^{in two}

stages.

Stage

^1.

By collaring st

may be taken to be

pl.

^{in a}

neighbourhood

^of

ôd x Rn then

h,

st defines ^an allowable

regular homotopy

^between

hl oqandhl.

Stage

^2.

h o

^and

hl

o q ^are both

orientation-preserving pl.

^immersions

of

(Rk x Rn)ho

ⁱⁿ

^Rk+n

^{and are therefore}

regularly homotopic

^{since both}

manifolds are contractible.

PROOF oF 1.1. We have functions

defined

by restricting

the differential of an immersion to

dk {0}

^{and it}

follows from the

pl.

^and

topological

immersion theorems that these are

bijections.

The result now follows

using

^{2.1 and} 2.2 and the commutativ-

ity of 03C8i,

CPi

and di

^with

suspension

^and

forgetful

^functions.

THEOREM 2.3.

Suppose

^{r ~ 2 or k + r ~ 5} then inclusion induces an

isomorphism.

PROOF. Consider the

diagram

i2 and i4

^are

isomorphisms

^{for i ~ k}

by [9; 8.5] ]

^{see also}

[35; 5.4].

i3

^{is an}

isomorphism by

1.1 and 1.2. To

apply

the 5-lemma we need

il epimorphic.

^{This is true for i ~ 2}

since 03C0i+1(Topr, PL,)

^{= 0}

by

^{1.1 and}

1.2. For i ⁼ 2 however we have

03C02(PLr+k,k) ~ 03C02(PLr) ~ 03C02(PLr) ~

03C02(0r) ~

⁰

(see [35; 5.5] ]

^and

[8; 6.6]) ^{so i*}

^{is an}

isomorphism by

^an

easier

argument.

THEOREM 2.4.

Suppose

^{r ~ 3. Then}

TOPr/PLr -+ Top/PL

^{is a}

homotopy equivalence.

PROOF. This is immediate from 1.1 and 1.2.

COROLLARY 2.5.

Suppose

^{r ~ 3. Then}

Gr/Topr ~ GITop

^{is a}

homotopy

equivalence.

^Here

Gr

^{is the}

homotopy analogue Topr (cf ^{[35 ; § 0]),}

^which

has the

homotopy type of

the monoid

Gr of sel f homotopy-equivalences of Sr-1

PROOF. Use 2.4 and

[35; 1.10].

3. Classification of

r-neighbourhoods

^{of manifolds}

For

simplicity

^we work first with an unbounded manifold Mn. The bounded case introduces technical difficulties which will be dealt with at the end of the section. Let i : Mn ~

Nn + r

be a

locally

^flat

embedding

in the unbounded manifold N. The

pair (i, N)

^{is called an}

r-neighbourhood

of M. Two such

(i, N), (i’, N’)

^are

equivalent

^{if there is an}

embedding

h : N ~ N’ defined in a

neighbourhood of i(M)

and such that h ^o i ⁼ i’ . The set of

equivalence

classes is denoted

Nr(M)

^and called the set of germs of

r-neighbourhoods

^{of M.}

To the

r-neighbourhood (i, N)

^we associate the microbundle

pair

(i*7:(N),7:(M)).

^The

isomorphism

^class

rel7:(M)

^{of this}

pair depends only

^{on the germ of}

(i, N)

^{and this}

gives

^{us a} well-defined function

K :

Nr(M) ~ [M, BTopnn+r]03C4(M)

where

[M, BTopnn+r]03C4(M)

^denotes

^homotopy

classes of sections of the fibration over M induced from the

fibratjon

by

^the

classifying

map

03C4(M) :

^{M ~}

BTopn .

PROPOSITION 3.1. K is a

bijection.

PROOF. K is

injective: Suppose K(i, N)

⁼

K(i’, N’)

then there is an iso-

morphism

^{h :}

(i*03C4(N), 03C4(M)) ~ (i’*,03C4(N’), 03C4(M))

^of microbundle

pairs

such that

h|03C4(M)

^{= id.}

^By

the immersion theorem there is an immersion h’ :

(N, M) ~ (N’, M)

defined in a

neighbourhood

^{of M such that}

h’|M

⁼ i’. This last condition

implies

^{that h is an}

embedding

^{in some}

smaller

neighbourhood

^{and thus}

(i, N) - (i’, N’).

K is

surjective: Suppose given

^a microbundle

pair (03BEn+r, 03C4(M)).

^We

have to construct an

(n + r )-manifold

^M with M c N and an

isomorph-

ism

(T(N)IM, 03C4(M)) ~ (çn+r, 03C4(M))

^rel

03C4(M).

^{We will construct N}

inductively

^{over an} open ^cover of M

using

the immersion theorem ^to match

overlaps:

Let {Ui}, {U’i}, i

⁼

^{1, 2 ···,}

be countable

locally

^finite open ^covers of M such that

Vi

^c

U’i

^and

(03BE, 03C4)|U’i

is trivial for all i. Let

and suppose

inductively

that there is ^an

(n+r)-manifold V ~

^{U and an}

isomorphism

^{h :}

(03BE|U, 03C4(U)) ~ (03C4(V)|U, 03C4(U))

^{which restricts to id. on}

03C4(U).

^{Let t :}

U’p

^x

(Rn+r, Rn) ~ (03BE|U’p, 03C4(U’p))

^{be a}

trivialisation,

^{and let}

Z ⁼ U n

U’p.

^{We can define a}

representation

cp:

03C4(Z Rr)|Z ~ 03C4(V)|Z

by

^the

following diagram,

^{in which the maps are}

suitably

restricted:

and it is

trivially

^{checked that}

~|03C4(Z)

^{= id. Thus}

by

^the immersion theo-

rem 9 is

homotopic

^rel

i(Z)

^to the differential of an immersion

cp’ : N(Z) ~ V, cp’IZ

⁼

id,

^where

N(Z)

^{is a}

neighbourhood

^{of Z in}

Z x Rr. Now

ç’

^{is an}

embedding

ⁱⁿ

N(Z)

ⁿ

N( Up)

^{for some}

^neighbour-

hood

N( Up)

^of

Up

ⁱⁿ

U’p Rr.

^Now define V’ = V ~

N( Up)

^identified

by cp’i

^and

U’ = ~ {Ui;

^{i =}

1, ···, p}

then it is

easily

^{verified that}

V’ ~ U’ has the inductive

property.

Now let

(i, N)

^{be an}

r-neighbourhood

^{of M and choose an inverse}

v to

03C4(M) (that

^{is to}

say 03C4(M)

^{~ v has a}

^preferred trivialization).

^Then

we can

identify

^the

pair (i*03C4(N)

^{EB v,}

T(M)

^EB

v)

^with

(i*1:(N) EB

^v,

8N)

using

this trivialisation. The last

pair

determines a

classifying

^map

c(i, N) :

^{M ~}

BTopr

Now the

isomorphism

^class

^relaN

of the above

pair depends only

^{on the}

germ of

(i, N)

^{and thus we have a well} defined function

THEOREM 3.2.

Suppose

^{n + r ~ 5 or r ~}

2,

^{then c is a}

bi jection.

PROOF. Consider the

following diagram.

Here si

^are

suspensions.

^The vertical sequences ^are fibrations. We now use the main

stability

theorem 2.3 to deduce _{that s1} induces

isomorphisms

on nk

for k ~

^{n+1 and hence}

by

^{3.1 that}

Nr(M)

^{is in 1-1}

correspondence

with lifts

of S3

^o

03C4(M)

^{over p2.}

^{But p2}

^{is fibre}

^{homotopy trivial;}

^{this is}

seen as follows. There is a retraction t :

BTop’r ~ BTopr

constructed

inductively

^{over skeletons}

by

^{means of}

inverses 03B6(k) for p*2(03B3(k))|BTop’r(k)

where

y(k)

^{is the universal bundle over}

BTop(k)t(k)

is then the

classifying

map for the

pair (03BE(k) ~ 03B6(k), s)

^where

(03BE(k), p*y(k»)

^{is the universal bundle}

over BTop’r.

It is

easily

^checked that t o 1 ~ id. Thus _p2 is fibre

homotopy

^trivial

by [3]

and it follows that

Nr(M)

is in 1-1

correspondence

^with

[M, BTopr] by

^the

correspondence (i, N) - t

o s2 ^o

K(i, N).

It remains to observe that this is

homotopic

^to

c(i, N) by

^stable

uniqueness

^{of in-}

verses.

The bounded case

Let M be bounded and

M+ -

^{M ~}

{open collar}

and recall that

03C4(M) = 03C4(M+)|M.

We then have an

isomorphism i(ôM) ^{E9 el}

^~

-r(M)lèM given by choosing

^{an inward}

collar,

^{and a} commutative

diagram

An

r-neighbourhood

^{of M is a}

pair (i, N)

where N is a bounded

(n+r)-manifold,

^{1 : M - N an}

embedding

^and

i-1(~N)

^{= ôM. Two}

such are

equivalent

if there is an

embedding

^{h : N -} N’ defined in a

neighbourhood

^of

i(M)

^{such that}

h-1(~N’) =

^{~N and h i = i ’. The set}

of

equivalence

classes is

again

^denoted

Nr(M).

^{To the}

^pair (i, N)

^we

associate the bundle

diagram

which

depends

up ^to

isomorphism rel i(M) only

^on the germ of

(i, N).

This

gives

^a function K from

Nr(M)

^{to the set of}

homotopy

^{classes of}

commuting diagrams:

PROPOSITION 3.1 b. K is a

bijection.

PROOF. Follow the

proof

of the unbounded case but

split

^{into two}

cases. Deal first with the

boundary

^then relativise the bounded

argument

to deal with the interior

keeping

boundaries fixed. Details are left to the reader.

Now _suppose

( j, L)

^{is a fixed}

r-neighbourhood

of ôM and consider

r-neighbourhoods

which extend

( j, L)

^{under the}

equivalence

^of germ of

homeomorphism

rel L. Call this set

Nr(M rel L)

^{then the}

^proof

^{of 3.1 b}

shows

PROPOSITION 3.1 c.

Nr(M rel L) is in

^1-1

correspondence

^with

lifts of

03C4(M) in ^{BTopn +r}

^{which extend}

^s’2

^o

K(j, L).

We now stabilise 3.1b and c

along

^{the lines}

of 3.2.

^As

before

^{choose a}

fixed

^{inverse v to}

03C4(M)

^{then we}

get functions

where the last set denotes

homotopy

^classes

of

maps which extend

c( j, L)

on ôM.

THEOREM 3.2b. _{cl is} ^a

bijection provided

n + r ~ 6 ^or r ~ 2.

THEOREM 3.2c. _{C2 is} a

bijection provided

^{n + r ~ 5 or r ~ 2.}

PROOF. Consider the

diagram

The results follows from 3.1 b ^or 3.1 c the

triviality

^{of the}

right-hand

fibration and the

stability properties of s’1

^and si

(2.3).

^{Note that we}

get

one better dimension for 3.2c than 3.2b since we are

considering

^{a fixed}

map

K( j, L)

and thus do not need

stability

^for

s’1.

Whitney

sums,

product

^theorem and induced

neighbourhoods

We deduce three

simple

consequences of the main theorem

(3.2).

Suppose (i1, N1)(i2, N2)

^are

neighbourhoods

^{of M of} codimension rl, r2

respectively.

^{We can} form their

Whitney

^sum

(i 3 , N3)

^{of codimen-}

sion

rl + r2 uniquely

up ^to

equivalence provided n+r1+r2 ~

⁵

(or

^{6 if}

~M ~ 0):

^form the bundle

pair (1*1(Ni )

^{E9 v E9}

i*(N2) ~ 03BD,

^{a ~}

8)

^and

then define

(i3, N3)

^{to be a} member of the class classified

by

^this

pair.

Secondly,

suppose M ~

Ml

^{x I and}

(i, N)

^an

r-neighbourhood

^{of M.}

Then

provided n+r ~

⁶

(or

^{7 if}

8M1 i= )

^{we can find a}

neighbourhood (il, Nl )

^of

Ml

^{such that}

(i, N)

^is

equivalent

^to

(il ^+id, N1 I):

^Define

(il , N1 ) ta

^{be a}

neighbourhood

classified

by c(i, N)/M1

and then the result follows since

c(i, N)

^and

c(il

^x

^{id; N1} I)

^{agree on}

^Ml.

Thirdly,

suppose

(i, N)

^{is an}

r-neighbourhood

^{of M and}

f :

^{W ~ M}

a map of manifolds let co ⁼ dim W then

provided only r + 03C9 ~

⁵

(~

^{6 if}

^{~W ~} 0)

^{we can} define the induced

neighbourhood f*(i, N)

^of

co

uniquely

up ^to

equivalence

^{as the}

neighbourhood

classified

by c(i, N) ^{o f.}

4. Normal block bundles

We deal first with micro block bundles.

Analogous

results for _open and closed block ubndles will be deduced afterwards.

Tõpr(03BC)

^{is the}

topological analogue

^of

PLr(03BC) ^[33],

^a

typical k-simplex

^{is a} germ of block and

zero-preserving homeomorphisms

^{03C3 :}

^0394k

^{x Rr W} defined in a

neighbourhood

^of

^dk

^x

{0}.

^An r-microblock bundle will mean a

Tõpr(03BC)-

block bundle in the sense of

[37; § 1 ].

^{If K is the base}

of 03BE

^{then we write}

ç/K

^and

identify |K|

^{with the zero} section in

E(03BE). Isomorphism

^classes

of block bundles are classified

by homotopy

classes of maps in

BTõpr(03BC) (for

^{more detail see}

[37; § 1 ]).

We will define a map

inductively

^{over skeleta. We need}

PROOF. There is a

surjection Hk(r, 0) ~ 03C0k(Tõpr(03BC), PLr(03BC)) (see

^4.5

below)

^{and the result} follows from

[32; 2.6] and

^1.2.

Assume r ~

⁵

or ~

^{2 and use 4.1 to}

replace BTõpr(03BC)(k) ^by

^a

^parallel-

ized _open manifold

Tk by imbedding

^a

locally

^finite

simplicial complex (cf [33; § 2])

^{in some}

large

dimensional Euclidiean _{space and}

consdering

the interior of ^a

regular neighbourhood.

^{Then the}

pair (id, E( yr¡ Tk)

^is

an

r-neighbourhood

^of

Tk where yr

^{is the universal}

bundle,

^{and thus} determines a map

~(k) : Tk -+ BToPr.

^{We observe that}

by choosing Tk

^to be a

parallelized

open manifold ^{we can} take

~(k)

^to

classify

^the

pair