Delooping controlled pseudo-isotopies of Hilbert cube manifolds

Topology and its Applications North-Holland

26 (1987)

175-191

DELOOPING CONTROLLED -_ --_ _ * . _T_r._l _c. CUISD

175

PSEUDO-ISOTOPIES

OF HILBERT

IVIAIYIPULU3

C. Bruce HUGHES* Department

qf Mathematic.s,

Received 2 August Revised 20 January

Vanderbilt

Unimmity,

controlled

Subj. Class.:

simple

TN 37235, USA

1985 1986

This paper is concerned with the controlled isotopy theory of a Hurewicz tihration p: E + compact polyhedron B. The main result is that equivalent to the loop space of the controlled

AMS (MOS)

Nashdle,

homotopy

Primary

57N20.

simple homotopy theory and controlled pseudoB from a compact Hilbert cube manifold E to a the controlled pseudo-isotopy space is homotopy Whitehead space.

57N37;

theory

Introduction Let p: E + B be a Hurewicz fibration from a compact Hilbert cube manifold E to a compact polyhedron B. In this paper we construct the controlled Whitehead space Wh( p: E + B) and the space of controlled pseudo-isotopies P( p: E + B) which turns out to be homotopy equivalent to the loop space of Wh( p : E + B). The homotopy groups of Wh( p : E + B) are the domain of controlled simple homotopy theory. In particular, rroWh( p : E + B) is the controlled Whitehead group. This is explained as follows. Recall that the problem of simple homotopy theory is to decide when a homotopy equivalencef: K + L between compact polyhedra is a simple homotopy equivalence. The Whitehead torsion 7(f) off lies in the Whitehead group Wh(%r,L) and is equal to zero if and only if f is a simple homotopy equivalence (see [9]). Results of Chapman [l] and West [19] relate this to Hilbert cube manifolds as follows: r(f) = 0 if and only if fx id: K x Q --f Lx Q is homotopic to a homeomorphism (Q denotes the Hilbert cube). It follows that the problem of simple homotopy theory is to decide when a homotopy equivalence between compact Hilbert cube manifolds is homotopic to a homeomorphism. * Supported

0166.8641/87/$3,50

in part by NSF Grant

0

1987, Elsevier

DMS-8401570

Science

Publishers

B.V. (North-Holland)

176

C. B. Hughes / Delooping

controlled pseudo-isotopies

If we are given a fibration p: E + B as above, then the problem of controlled simple homotopy theory is to decide when a controlled homotopy equivalence f: M + E (where

M is a compact

to a homeomorphism. is explained Theorem homotopy

further

1.

IfM

‘Hilbert cube manifold)

By ‘control’

we mean

arbitrarily

is homotopic small

with control

e control

in B. This

in the body of the paper. We can now state our first main result. is a compact

equivalence,

Hilbert

cube manifold

then there is a well-dejined

which vanishes if and only iff isp-‘(

F)-homotopic

and

element

f:M r(f)

+ E is a controlled in r,Wh(

to a homeomorphism

p : E + B)

for every

E >

0.

A pseudo-isotopy on E is a homeomorphism h : E x [0, l] + E x [0,1] such that h 1E x (0) is the identity. Pseudo-isotopies which move points an arbitrarily small amount when measured in B are formed into the controlled pseudo-isotopy space P(p: E + B) in Section 4. Here is our second main result. Theorem

2. 6( p : E + B) is homotopy

equivalent

to the loop space RWh( p : E + B).

For the special case that p is a bundle projection over euclidean space with compact Hilbert cube manifold fiber, Theorems 1 and 2 are essentially contained in [14]. Our approach to controlled simple homotopy theory differs from the theories of Chapman [5] and Quinn [17] in two aspects. First, we use Hilbert cube manifolds in our definition rather than polyhedra. Second, we require that our controlling map p be a Hurewicz fibration. The setting in [5] and [17] is much more general Chapman [6] and Quinn [17] have also studied controlled pseudo-isotopies, again in a more general setting. In the uncontrolled setting, Hatcher [12] defined a Whitehead space Wh(K) for a compact polyhedron K and showed (using a result of Chapman [3]) that RWh( K) is homotopy equivalent to the space of pseudo-isotopies on K x Q. In a future paper we will develop a method due to W.C. Hsiang in order to study the homotopy groups of Wh(p : E + B) and 9( p : E + B) where p is the projection map of a locally trivial fiber bundle and B is a closed manifold. It will be shown that Wh( p : E + B) is homotopy equivalent to the space of cross-sections of a certain bundle whose fiber was the object of study in [ 141.

2. Approximate

fibrations

and other preliminaries

The Hilbert cube will be denoted by Q. Hilbert cube manifolds, or Q-manifolds, are locally compact, separable metric spaces which are locally homeomorphic to Q. For Q-manifold basics, including the notion of Z-sets, see [2]. In this section and throughout the rest of the paper we fix a Hurewicz fibration p: E + B (i.e., a map with the homotopy lifting property for all spaces) where E

C.B. Hughes

is a compact topological

Q-manifold manifold

Let X and

and

/ Delooping

B is either

with a handle

Y be compact

177

controlled pseudo-isotopies

a compact

polyhedron

or a compact

decomposition.

metric ANRs,

let f: X + Y be a map, and let F > 0. We

say that f is an e-fibration provided that given any Z and maps G : Z x [0, l] + Y and g : Z + X for which G(z, 0) =fg(z), then there exists a map G : Z x [0, l] + X such that G(z, 0) = g(z) and f6 is E-close to G. Iff is an &-fibration for every e > 0, then f is an approximate jibration. Approximate fibrations were introduced in [lo]. We will use A to denote the standard n-simplex for a given n. A jiber preserving (f.p.) map is a map which preserves the obvious fiber over A (or over some other n-cell). Specifically, if p : X + A, CT:Y + A, and f: X + Y are maps, then f is f.p. if af= p. Usually the maps p and CTwill be understood to be some natural projections and will not be explicitly mentioned. For fiber preserving Q-manifold results, including sliced Z-set unknotting, see [7], [8] or [ll]. We will need to use results about approximate fibrations from [ 151 and [ 161. Here is the main result from [ 161 (see [4] for the n = 0 case). Theorem

2.1. Let A and F > 0 be given. There exists a 6 > 0 such that if M is a compact

Q-manifold and f: M x A + B x A is an jIp. map so that f, : M + B is a G-jibration for each t in A and an approximatejbration for each t in aA, then there exists an$p. map j::MxA-,BxAsuchthatj:ise-closetof;fIMxaA=fIMxaA,and~:M~Bis an approximate jbration for each t in A. We will also need a relative Addendum

version

of this result which we now state.

2.2. If E is a Z-set in M and f 1E x A = p

x

id, then we can ,further require

thatfIExA=pxid. Proof.

There is a relative version of Theorem 2.1 in [16]. As stated, it would imply our Addendum if E = B x F where F is a compact Q-manifold and p = projection. However, the proof in [ 161 works equally well if p : E + B is the projection map of a locally trivial fiber bundle with compact Q-manifold fiber. To pass to the more general case where p : E + B is only assumed to be fibration, use Q-manifold theory tofindanf.p.maph:MxQxA + M x A close to projection such that h 1M x Q x aA is projection and h 1: M x Q x int A + M x int A is a homeomorphism. By sliced Z-set unknotting we may further assume that h 1: E x (0) x A + M x A is the inclusion. It follows from [7] that p(proj) : E x Q+ B is a locally trivial fiber bundle projection with Q-manifold fiber. Let 1: M x Q x A + B x A be an f.p. map such that p is close to f(proj), x is an approximate f(proj)l, andjIExQxA=(pxid)(proj).

fibration for each Then setf=fi-‘.

t in A, f 1M x Q x aA = 0

A special case of the following theorem is the main result of [13]. That result, together with a significant improvement, appears in [15] for finite dimensional manifolds. The proof given in [15] also works for Q-manifolds and we now state the result without further proof.

178

C. B. Hughes / Delooping controlled

Theorem

[0, l] + B

pseudo-isotopies

2.3. Let A and e > 0 be given. If M is a compact Q-manifold

A

x

x

[0, l] is anjIp. (over A

x

and f: M x A x [0, 11) map so thatf, : M + B is an approximate

jibration for each t in A x [0, 11, then there exists an jIp. homeomorphism H :M x A x [0, l] + M x A x [0, l] such that H ) M x A x (0) = id andfH is e-close tofo x id where fo=f~MxAx{O}. There are also two addenda which we will need. The proof of the first follows from the proof of Theorem 2.3. The second follows from Theorem 2.3 and sliced Z-set

unknotting.

Addendum

2.4. There exists a 6 > 0, 6 = 6( F, n), such that if we are additionally given an Jp. homeomorphism G : M x ?)A x [0, l] + M x aA x [0, l] such that G 1M x rlA x (0) = id andfG is S-close toJ;, 1x id, then we can further require H to satisfj H 1M x aA x [0, l] = G. Addendum

2.5. lf E is a Z-set in M and f ( E x A

require that H 1E

3. Controlled

x

A

x

x

[0, l] = p x id, then we can further

[0, l] = id.

simple homotopy

theory

We continue to let p: E + B denote a fixed Hurewicz fibration where E is a compact Q-manifold and B is either a compact polyhedron on a compact topological manifold with a handle decomposition. In this section we define the controlled Whitehead space Wh( p : E + B) as a semi-simplicial complex, study its homotopy relation from a geometrical point of view (Theorem 3.2), and define the torsions of certain controlled homotopy equivalences to E to be in the homotopy groups of Wh( p: E + B). As a result (Corollary 3.4) we obtain the proof of Theorem 1. In addition, we define two more ‘spaces’ homotopy equivalent to Wh( p : E + B) which will be useful in proving Theorem 2. We first need some definitions. Let p:X+ A be a map (where X is compact ANR), let f: X + E x A be an f.p. map and let E > 0. We say f is an f.p. (p x id)P’(e)-equivalence provided there exist an f.p. map g: E x A + X and f.p. homotopies F: X x [0, l] + X and G : E x A x [0, l] + E x A such that F, = id, F, = gf; G”=id, G, =fg, and the diameters of (p xid)f{F({x}x[O, I])} and (pxid) {G({y} x [0, 11)) are less than E for each x in X and y in E x A. If in addition X contains E x A, g is the inclusion, f 1E x A = id, and F is rel E x A, then we say f is an f.p. (p x id)-l(e)-sdr. In the case that n = 0 (so that we can drop the f.p. requirement) and f is a p-‘(&)-equivalence for every E > 0, then we say that f is a controlled homotopy equivalence. Approximate fibrations enter into this work because of the following fact: a homotopy equivalence f: X + E is a controlled homotopy equivalence if and only

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180

C.B. Hughes / Deloopingconrrolledpseudo-isoropies

Since j” is an n-simplex OS TV 1, such that (1) KO=id (2) K,IE

of Wh( p : E + B), there exists an f.p. homotopy

K, : M + M,

and K,=f; x A =id

for each t,

(3) fK,, 0~ t s 1, is a (p x id)-l(6)-homotopy. Finally, define F, : M + M, 0 d t d 1, by

F, =

G

if 0s tG$,

K3r-, 0 G,

if $< ts$,

We are now ready interpretation

q

if$atCl.

i C-9

for the first main

of the homotopy

relation

result

of this section.

in Wh(p:

It gives a geometric

E + B).

3.2. Let f:M + E x A and g : N + E x A represent the classes [,f] and [g] in r,,Wh( p : E + B), respectively. There exists an eO> 0, e0 = E~(B, n), so that the following are eguivaienf : Theorem

(i) Lfl = [gl; (ii) for every E > 0 there exists an jIp. homeomorphism h : M + N such that h = id on (ExA)u(Ex[O, l]xdA) and gh is Jp. (pxid)-l(e)-homotopic to frel(ExA)u(Ex[O, l]xaA); (iii) there exists anjp. homeomorphism h : M +Nsuchthath=idonExA,gh=f on E x [O, l] x iid, and (p x id)gh is e,-close go (p x id)f: Proof.

We first show that condition (i) impIies condition (ii). Since [f] = [g], there by f: t& is an (n+ 1)-simplex in Wh(p: E + B) which we can represent of II? E x A x [0, l] where we have a libering p : A? --f A x [0, l] and an embedding in Q x A x [0, l] with the following properties (among others): (1) f=f on p-‘(A x(0)) = M; (2) f==g on p-‘(Ax{l})= N; (3) T= n on p-‘(iid x [0, 11) = E x [0, l] x aA x [0, 11. Trivialize p by finding an f.p. homeomorphism k: E x [0, I] x A x [0, I] + fi such that k ( E x [0, 11 x aA x [0, l] = (k ( E x [0, l] x aA x {0}) x id and use sliced Z-set unknotting to get k = id on E x A x [0, 11. (Here and throughout the proof we are assuming n 3 1. Then n = 0 case is similar, but easier.) Now use Theorem 2.3 to find an f.p. homeomorphism H : E x [0, I] x A x [0, l] + Ex[O,l]xAx[O,l] such that H=id on (ExAx[O,I])u(Ex[O,I]xAx{O})u (E x 10, 11x aA x 10,11) and (p x id)TkH is E-close to (p x id)((fkj {0}) x id). Let h be the homeomorphism given by the composition (kl)-’ M-Ex[O,l]xAx{O}=Ex[O,l]xAx{l) ~Ex[O,I]xAx{l}

AN.

E x [0, l] x A x

C.B. Hughes

The asserted

homotopy

M -

(kl)-’

/ Delooping

controlled

181

pseudo-isotopies

from f to gh at time t, 0 d t s 1, is given by the composition

E x [0, l] x A x (0) = E x [0, l] x A x {t}

HI

-tEx[O,I]xAx{t}~p~‘(Ax{t})~ExAx{t}=Exh. Since condition (ii) obviously implies condition (iii), we are left with showing that condition (iii) implies condition (i). To this end use Lemma 3.1 to find an f.p. homotopy F, : M + M, 0 G t s 1, such that (1) F,=id and F, =f, (2) F,lExA=id, (3) fF,, 0~ t s 1, is a (p x id))l(S)-homotopy where S > 0 is small, (4) fF,IEx[O,l]xaA=~. Consider the homotopy ghF, : gh =A 0 s t c 1. It is rel E x A and it is constantly equal to n to E x[O, l]xdA. Moreover, it is a (p xid)-‘(2eo+6)-homotopy since (p x id)ghF, is e,-close to (p x id)fF,. Now let j: M x [0, I] + Q x A x [0, l] be a sliced Z-embedding such that j = id on (Mx{O})u(ExA~[O,l])u(Ex[O,1]~~A~[O,1])andj(Mx{l}=h. Define F: M x [0, l] + M x [0, l] by setting F(x, t) = (F,_,(x), t). Define a homotopy F, : id = F, 0~ s s 1, by setting 6$(x, t) = (F(,_,,,(x), t). Finally, define G: j( M x [0, 1)] + E x A x [0, I] by setting G = (gh x id)Fj-‘. Note that (1) G=fonj(Mx{O})= M, (2) G=g onj(Mx{l})=N, (3) GIExAx[O,l]=id, (4) G~Ex[O,l]x~Ax[O, l]=n. We now want to use Theorem 2.1 to deform G to an (n + 1)-simplex of Wh( p : E + B) showing [f] = [g]. For this, we need to show that G is an f.p. sdr with small control in B x A x [0, I]. To this end, let g, : id N = g, 0 s s s 1, be an f.p. homotopy with small control in B x A coming from the fact that g is an n-simplex in Wh( p : E + G,:j(Mx[O,l)]+j(M~[0,1]), O~ssl, by Then define B). Gs = j( h-‘g,h x id)F%jj’. Then G, : id = G, 0 s s s 1, and the homotopy GG, has small (depending

on 6 and eO) diameter

in B x A x [0, 11.

The following result gives a way to represent E x A by elements of rTT, Wh( p : E + B).

Theorem 3.3. There exists an .eO> 0, E,, = E”( the following data: (i) a locally trivialJiber bundleprojection p (ii) anfp. (pXid)-‘(F,,)-equivalencef: M morphism k : E x [0, l] x aA + p-‘(dA)

q

certain

homotopy

equivalences

B, n), such that whenever

to

we are given

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‘L’E t?iiiiwia~

C. B. Hughes

4. Controlled

pseudo-isotopy

/ Delooping

controlled

pseudo-isotopies

185

theory

In this section we will define the space CP(p : E + B) of controlled

pseudo-isotopies

on p : E + B, the fibration of the previous sections. We also show how to represent certain pseudo-isotopies on E x A by elements of v,,P( p : E + B). This is analogous to the torsion

construction

in the previous

section.

Finally,

we will discuss the group

on rr,,P( p : E + B).

structure

Let h: E x [0, l] x A -+ E x [0, l] x A be an f.p. (over

A) homeomorphism. For each tin A, let h,=h~:E~[O,l]~{t}+E~[0,l]~{t} and continue to let T:EX [0, l] + E denote projection (we also use v to denote projection E x [0, l] x A + E x A). If E > 0, we say h is an n-parameter e-pseudo-isotopy on p: E + B provided hIEx{O}xA=idandd(pnh,,pm)<&foreachtinA. An n-parameter controlled pseudo-isotopy on p : E + B is a homeomorphism h :E x [0, l] x A x [0, ~0) + E x [0, l] x A x [O,OO) such that (1) h is f.p. over A x [0, ~0); (2) h/Ex{O}xAx[O,co)=id; (3) there is a decreasing sequence cO, F, , e2, . . . of positive real numbers converging to 0 (called a controlling sequence for h) such that for each integer i 2 0, h ( : E x [0, l] x A x {u} + E x [0, l] x A x {u} is an n-parameter E,-pseudoisotopy on p: E + B whenever u 3 i. Let CP(p: E + B) denote the semi-simplicial complex of controlled pseudoisotopies on p: E + B; that is, the n-simplices are n-parameter controlled pseudoisotopies on p: E + B. This complex satisfies the Kan condition and the homotopy groups will1 be based at the identity E x [0, l] + E x [0, 11. The next proposition shows how to turn an &-pseudo-isotopy into a controlled pseudo-isotopy if F is small enough. 4.1. There exists an F > 0, F = e(B, n), so that if h : E x [0, l] x A + E x [0, l] x A is an n-parameter e-pseudo-isotopy, then there is a homeomorphism H : E x [0, l] x A x [0, ~0) + E x [0, l] x A x [0,00) such that: (i) H is jp. ozler A x [0,00);

Proposition

(ii) (iii)

H = id on (E x (0) x A x [0, a)) u (E x [0, l] x A x (0)); (h x id) H is an n-parameter controlled pseudo-isotopy.

Given h : E x [0,11 x A + E x [0, l] x A as above, note that (p x id)nh is Eclose to (pxid)rr and (pxid)~hIEx{O}xA=pxid=(pxid)~(. It follows from [16] that there is an f.p. (over A x [0, 11) approximate fibration g: E x [O, 11 x A x [0, l] + B x A x [0, l] such that: (1) glEx[O,l]xAx{O}=(pxid)~h; (2) glEx[O,l]xAx{l}=(pxid)rr; (3) g(Ex{O}xAx[O,l]=pxid; x 1ZJ1‘1s &‘-close to (p x id)vh for each u in [0, ~0) where the (4) gtExCO,llxA size of F’> 0 depends on the size of E. Proof.

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‘8.3

981

C. B. Hughes / Delooping

(2) H=id

on

187

controlled pseudo-isotopies

(E~{O}XAX[O,~))~(E~[O,~]~~A~[O,~))~(E~[~,~)~

A x (01); (3) (h x id)H is an n-parameter sequence beginning with F.

controlled

pseudo-isotopy

with a controlling

Define the torsion 7(h) of h to be the class of (h x id)H in ~,9( p: E + B). Note that r(h) is well-defined (i.e., independent of H) by Proposition 4.4. The next result of this section shows that r(h) is invariant under a small (measured in B) isotopy of h. The proof uses the techniques of the propositions above and is left to the reader. Proposition

4.5. There exists a y > 0, y = y( B, n), such that if h, h’: E x [0, l] x A + E x [0, l] x A are two n-parameter S-pseudo-isotopies (where S comesfrom the dejnition

of torsion)

such

that h = id=

h’ on E x [0, l] x~A and h and h’ are y-isotopic l]xaA) when measured in B, then T(h)=T(h’).

rel(Ex{O}xA)u(Ex[O,

Composition of maps induces a group structure on rr,p( p: E + B). The final result of this section shows that this group is abelian. Proposition

4.6. ~,@(p:

induced by composition

E + B) is an abelian of pseudo-isotopies.

group where the group operation

is

Proof. It suffices to show that if g, h: E x [O, l]+ E x [0, l] are two s-pseudoisotopies, then T(gh) = 7( hg) if E > 0 is smal1 enough. To this end, Iet S: E x [0, I] + E be a homeomorphism quite close to projection and let E, = S( E x [0, i]) and E, =

S(E x [;, 11). Use Z-set unknotting to find a small (measured in E) isotopy of h rel E x{O} to h’ where h’l S( E x (0)) x [0, I] = id. Then slide along the interval direction in S( E x [0, 11) to find a small isotopy of h’ rel E x (0) to h” where h”l E, x [0, l] = id. If the isotopies are small enough, then I = 7(h). Likewise, find g” such that r(g) = r(g”) and g”I E, x [0, l] = id. Then g”h” = h”g” and T(gh) = r(hg).

5. The homotopy

q

equivalence

.P( p : E + B) = a Wh( p : E + B)

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sasde[Io~ qD!qM [I ‘o]+ti Ou!doolaa

/ say8n~

2:'~

Ial

uo!pe~la~ aql

‘8.3

881

C.B.

Find

a sliced

Hughes

Z-embedding

/ Delooping

j: E

x

controlled

[0, l]

x

189

pseudo-isofopies

[0, l] + Q x [0, l] with image

M such

that j=id on (Ex{O}x[O, l])u(Ex[O, l]x{O})u((E,n E2)x[0, l]x[O, 11) and j = gh on E x [0, l] x { 1). Then cr*T(gh) = 7(f) where f: M + E x [0, l] is defined by f = $ij-‘. Define

Define

k: E x [0, l] x [0, l] + E x [0, l] x [0, l] by “. if (proj h(x, Y, 2t), t) k(x, Y, t) = 1 (proj g(h(x, Y), 2t - I), t) if f: M + E x[O, l] by f=

a*T(h).

“*a).

rrkj-‘.

It is clear

0s tC$, ;
7(f) = T(J)

and

7(T)=

0

Our goal is to show that a : P(p: E + B) + fiWh(p: E + B) is a homotopy equivalence. For this we need to show that CK*: r,,P( p : E + B) + n,,flWh( p : E + B) is an isomorphism for each choice of basepoint. However, the proof of the proposition above shows that we can just concern ourselves with the usual basepoints: the identitiy for P( p : E + B) and the projection for 0 Wh(p : E + B). We will do this in the next two propositions to complete the proof of Theorem 2. Proposition

5.3. ay*: T,,??( p : E + B) + r,,RWh(

p : E -+ B) is a monomorphism.

Proof. Let h : E + I? be an n-parameter controlled pseudo-isotopy representing the j. But this is already allowed in the definition of 0Wh( p : E + B). Therefore, we are only left with showing that f is a loop in Wh( p : E + B). Since f 1i x (0, 1) = T, we only have to show that f has the correct controlled sdr property. Since [f] = [ ~1 there is an (n + I)-simplex in fiWh( p : E + B) given by G : fi + E x A x [0, ~0) x [0, 112 where (among other properties): (1) there is a bundle projection ;: fi + A x [0, ~0) x [0, 112; (2) ExAx[0,~)x[0,1]2~n;i~QxAx[0,~)x[0,1]’; (3) if A = (aA x [O, ~0) x [O, 11’) u (A x [O, ~0) x {O, l} x [0, 11) u (A x [0, co) x [0, l] x {0}), then p-‘(A) = E x [0, l] x A; (4) ;?(Ax[O,~)x[O,l]x{l}=M; (5) G=idon E~dx[O,~)x[O,l]~; (6) G = 7~ on p-‘(A); (7) G =f on M. Since b is trivial, we can use sliced Z-set unknotting to find an f.p. homeomorphism k: i x [0, 112+ &? such that: (1) k = id on (E x (0) x A x [0, ~0) x [0, 112) u (E x [0, l] x dA x [0, ~0) x [0,11’) u (2) Use that: (1) (2)

(E x [O, 11 x A x [O, 00) x (0) x [O, 11) u (E x [O, 11 x A x [O, 00) x [O, l] x IO},; k=j on Ex[O,l]~Ax[O,~)x[0,l]x{l}. Theorem 2.3 to find an f.p. homeomorphism g : .!? x [0, I]‘+ _!?x [0, 112 such g = id on the sets listed above where k = id and k = j; the map (p x id)Gkg : E x [0, 112+BxAx[O,~)~[O,l]~iscloseto(pxid)~ with the ‘closeness’ becoming arbitrarily small near infinity.

190

C. B. Hughes / Delooping controlled pseudo-isoiopies

Finally, consider to be the restriction (1) H = id on

the homeomorphism H : ,?? x [0, l] + E X [O, 11 which is defined of kg to i x (1) x [0, 11. Note that: (E x (0) x A x [0,00) x [0,11) u (E x [0, 11 X aA X [O, ~0) x [O, 11) u

(E x [O, 11 x A x [O, ~0) x (01); (2) H = h on E x [0, l] x A x [0, ~0) x (1); (3) (pxid)~H=(pxid)rr(kgI)=(pxid)G(kg]) It follows

that H is an (n + I)-simplex

which is close to (pxid)~.

in Y(p : E + B) showing

[h] = [id].

0

Proposition 5.4. a.+: n,B( p : E + B) --f rr,,O Wh( p : E + B) is an epimorphism. Proof. Recall from Section

3 that there are isomorphisms ni,: r,,L! Wh( p : E + B) + Thus, let ~,fi%( p : E + B) and Uq.+: ~,RW^h(p:E-+B)+~,&Wh(p:E+B). f: M + E x A x [0, l] represent the class [f] in rr,,fiWh(p: E + B) where: (1) MC QxAx[O, l] as a sliced Z-set; (2) f= id on E x A x [0, l] c M; (3) f=rron Mn(Qxd(Ax[O,1]))=Ex[O,1]xd(Ax[O,1]); (4) f is an f.p. (p x id)-‘(a)-sdr for every .a > 0. As we have done before, we can find a trivializing homeomorphism k : E x [0, l] x A x [0, l] + M such that k = id on (E x (0) x A x [0,11) u (E x [0, l] x aA x [0, 11) u (E x [0, l] x A x (0)). Also, use Theorem 2.3 to find an f.p. homeomorphism g : E x [0, l] x A x [O, l]+ E x [0, l] x A x [0, l] such that g = id on the set indicated above where k = id and (p x id)fkg : E x [0, l] x A x [0, l] + B x A x [0, l] is E-close to (p x id)rr for a given small E > 0. Consider the homeomorphism h : E x [0, l] x A + E x [0, l] x A which is defined to be the restriction kg 1E x [0, l] x A x (1). Note that h = id on (E x (0) x A) u (E x [0, l] x dA) and h is an n-parameter E-pseudo-isotopy. If E is small enough, then r(h) is defined in r$P( p : E + B). We will complete the proof by showing that a*r(h) = (fiq,)(%)([f I). Recall that T(h) is defined to be [(h x id) H] where H is a certain homeomorphism on ,?? Next recall the notation used for defining (~*r( h). The canonical homotopy id=(hxid)H

is used to define

a map

~:,6~[0,1]-+~~[0,1].

We then

need

a

sliced Z-embedding j: ,6 x [0, l] + Q x A x [0, ~0) x [0,11. To this end let fi: E x [0, l] + .r? x [0, l] be the f.p. homeomorphism such that fi 1g x (0) = id and fi 1E x { 1) = H which arises by phasing H out to the identity. Then set j = (kg x id)6. (Note that [0, ~0) and [0, l] have been interchanged.) Let ~=TJ-h;.~‘:Mx[0,co)+ExAx[0,1]x[0,co). Then a*T(h)=[f]. Let f’= f 1M x (0). We will complete the proof by showing T( f ‘) = [f 1. This suffices because [T] = (LJq.+)(Lti,)7(f’) by Lemma 3.5 and therefore ct*T(h) = [f] = (nq,)(fli,)[f 1. to In order to show that T(f ‘) = [f 1, we will show that f’ is f.p. homotopic f rel(E x A x [0, 11) u (E x [0, l] x d(A x [O, 11)) via a homotopy which is small when measured in B x A x [0, 11. To this end, let F,: id=f; OG t s 1, be the homotopy given by Lemma 3.1. Let & = i( E x [0, l] x A x (0) x [0, 11.It is the map which arises

C. t?. Hughes / Delooping controlled pseudo-isotopies

191

from the canonical

homotopy from id to h. Note that f’= r&‘k~‘. Our desired is close homotopy is &OgP’kP’F,:f’-f, 0~ d s 1. Note that (p x id)&,gP1kP’F, to (p x id)rg-‘km’F, because of the nature of the canonical homotopy. And (p x id)rrg-‘k-IF, is close to (p x id)fF, because of the way g was chosen. Finally, the 0 homotopy (p x id)fF, is small because of the way F, was chosen.

References [I] T.A. Chapman, Topological invariance of Whitehead torsion, Amer. J. Math. 96 (1974) 448-497. [2] T.A. Chapman, Lectures on Hilbert cube manifolds, C.B.M.S. Regional Conf. Ser. in Math. 28 (Amer. Math. Sot., Providence, RI, 1976). [3] T.A. Chapman, Concordances of Hilbert cube manifolds, Trans. Amer. Math. Sot. 219 (1976) 253-268. [4] T.A. Chapman, Approximation results in Hilbert cube manifolds, Trans. Amer. Math. 262 (1980) 303-334. [S] T. A. Chapman, Controlled Simple Homotopy Theory and Applications, Lecture Notes in Mathematics 1009 (Springer, New York, 1983). [6] T.A. Chapman, Controlled concordances, preprint. [7] T.A. Chapman and S. Ferry, Hurewicz tiber maps with ANR fibers, Topology 16 (1977) 131-143. [8] T.A. Chapman and R.Y.T. Wong, On homeomorphisms of infinite dimensional bundles III, Trans. Amer. Math. Sot. 191 (1974) 269-276. [9] M. Cohen, A Course in Simple-Homotopy Theory (Springer, New York, 1970). [lo] D.S. Coram and P.F. Duvall, Jr., Approximate fibrations, Rocky Mountain J. Math. 7 (1977) 275-288. [ 1 l] S. Ferry, The homeomorphism group of a compact Hilbert cube manifold is an ANR, Ann. of Math. 106 (2) (1977) 101-138. [12] A.E. Hatcher, Higher simple homotopy theory, Ann. of Math. 102 (2) (1975) 101-138. [13] C.B. Hughes, Approximate fibrations and bundle maps on Hilbert cube manifolds, Topology Appl. 15 (1983) 159-172. [14] C.B. Hughes, Bounded homotopy equivalences of Hilbert cube manifolds, Trans. Amer. Math. Sot. 287 (1985) 621-643. [15] C.B. Hughes, Approximate tibrations on topological manifolds, Mich. Math. J. 32 (1985) 167-183. [16] C.B. Hughes, Spaces of approximate tibrations on Hilbert cube manifolds, Compositio Math. 56 (1985) 131-151. [17] F. Quinn, Ends of maps II, Invent. Math. 68 (1982) 352-424. [18] E. H. Spanier, Algebraic Topology (McGraw-Hill, New York, 1966). 1191 J. West, Infinite products which are Hilbert cubes, Trans. Amer. Math. Sot. 150 (1970) l-25.