Annals of Mathematics

The Method of Infinite Repetition in Pure Topology: IAuthor(s): Barry MazurReviewed work(s):Source: The Annals of Mathematics, Second Series, Vol. 80, No. 2 (Sep., 1964), pp. 201-226Published by: Annals of MathematicsStable URL: http://www.jstor.org/stable/1970391 .Accessed: 28/05/2012 04:42

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The Method of Infinite Repetition in Pure Topology: I

By BARRY MAZUR

1. Introduction

In my thesis [4], [5] I introduced a method of constructing homeomorph- isms between spaces by a process of infinite modifications. With this method a Schoenflies theorem for all dimensions and the open star theorem [5] were proved.

OPEN STAR THEOREM. Let K be a finite complex, and let CK be the open cone construction on K. If CK is a manifold (a topological manifold), then it is homeomorphic with euclidean space.

Since that time, more general results have been obtained by methods some of which are analogous to, or extensions of, that original approach. (See the work of Brown [1], Morse [8], Stallings [10], and Kwun-Raymond [13], [14].)

Brown's was a more direct approach, and succeeded in proving the gener- alized Schoenflies theorem unencumbered by an assumption which my method imposed. As an easy consequence of his technique, he obtained the following generalization of the open star theorem.

THEOREM OF THE CONE. Let X be a compact metric space and CX the open cone construction on X. If CX is a manifold, then it is homeomorphic with euclidean space.

Morse employed the results of [4], but went on to remove the unnecessary technical assumption, obtaining also the generalized Schoenflies theorem. In later papers he proved Schoenflies-type extension results for parametered families of imbeddings [10] (as opposed to single ones), again using the same technique.

Stallings imbedded an improved version of the infinite-modification tech- nique in the apparatus which he used to prove the Poincare conjecture and obtained, as a by-product, the following sweeping generalization of the Schoenflies theorem.

UNKNOTTEDNESS THEOREM (Stallings). Let Sk be a locally flat topological sphere, imbedded in S". If n # k + 2, then Sk is unknotted.

REMARK. Stallings himself proved this theorem except in a few low dimensional cases which were cleared up afterwards [15]. He also obtained

202 BARRY MAZUR

results in the case of codimension 2. A few years ago, again using a technique of infinite modification, the

following theorem was proved. (See [3], [6].)

STABLE HOMEOMORPHISM THEOREM. Let f: M1 M2 be a tangential homo- topy equivalence between the two differentiable n-manifolds (compact, with- out boundary). Then there is a differentiable isomorphism

F: M1 x Rq - M2 x Rq

IP IP M1 M2

such that the above diagram is homotopy commutative, if q > n.

REMARK. A continuous map f is called tangential if

f*(t2) = tl

where tk is the stable tangent bundle class of Mk [3]. One is tempted to try to state similar theorems for combinatorial and

topological manifolds. However, the notion of tangentiality gives trouble, since a topological manifold has no tangent bundle. Milnor's recent theory of microbundles [7] allows one to state the combinatorial and topological analogues of the above theorem.

One therefore seeks a unified approach which will yield the stable homeo- morphism theorem in all three genres. It was with this object in mind that I decided to restate the method of infinite modifications in perfect generality.

The method, which should really be called the method of infinite repeti- tion hypostatized into a completely formal technique. In its essentials, it is a theory of canonical neighborhoods for arbitrary subspaces of arbitrary topo- logical spaces. (See ? 3.)

Standard applications of the technique fall out, quite formally, from the uniqueness theorem for canonical neighborhoods of X in Y. For example, the theorem of the cone for arbitrary paracompact spaces is a corollary of the uniqueness theorem applied to a point. (See Corollary B in ? 3.)

The object of this paper is to expose the formal aspect of the method. I have restricted myself to announcing only those applications that follow tauto- logically from the abstract theory. In a sequel to this paper, the results will be applied, using the theory of microbundles, to prove the stable homeo- morphism theorem stated above. As an added bonus for having worked functorially, I shall also obtain representability theorems for microbundles, and other applications to problems in the theory of microbundles. (In this

INFINITE REPETITION: I 203

connection, see Corollary 2 of ? 9.) In ? 3, five criteria are set up as prerequisites for well-working theory of

canonical neighborhoods. As formal consequences of the existence of such a theory, one has:

A. The open regular neighborhood of a triangulable pair X ' Y is (as a topological space) an invariant of the homeomorphy type of (Y, X).

B. Theorem of the cone, for X paracompact. C. The local fundamental groups of the complement of a locally tame

imbedding are topologically invariant. Theorems A, B had previously been obtained by Kwun [13]. See also Kwun

and Raymond [14] where a uniqueness theorem for mapping cylinder neighbor- hoods is proved. One obtains such a uniqueness theorem (Proposition 6, ? 9) from the theory of this paper as well.

In ? 4, the formal machinery of dilation spaces is introduced. A dilation space is a bit too general to play the role of a canonical neighborhood. How- ever, there is an operation of completion (replacing it by its infinite repetition space) which makes it more nicely behaved. This is developed in ?? 5 and 6.

By means of the notion of complete dilation space, one may define very regular-looking neighborhoods of arbitrary X in Y. Such neighborhoods are called dilation neighborhoods.

In ? 7, the strong uniqueness theorem is proven for dilation neighborhoods. In ? 8, it is shown that the theory of dilation neighborhoods satisfies criterion (3), stipulated in ? 3. Thus the theory is shown to satisfy all but the last criterion. This suffices for the applications we have in mind.

In ? 9, the most general category-theoretic interpretation is given to the preceding very formal arguments.

2. Terminology

Non-standard usage will be the following: Let f, g: B C be maps and AcB. Then

f = g(A)

means f, restricted to A, is equal to g, restricted to A. I use this as replace- ment for f/A = g/A.

The image of B under f will be denoted fB, no parenthesis. Idenotes the identity map of an understood domain. S is the closure of S.

All solid-arrow diagrams carry the assumption of commutativity. Non- commutative parts of a diagram will be indicated by broken-line arrows.

204 BARRY MAZUR

3. General criteria for a theory of canonical neighborhoods in pure topology

Theories of neighborhoods are extremely useful. In differential topology, one has the theory of tubular neighborhoods; in combinatorial topology, one has a theory of regular neighborhoods of subcomplexes. The content of these theories is given by two results:

( 1 ) The existence theorem. (Given any subspace, in the sense of differ- ential or combinatorial topology, it possesses a neighborhood, of the appropriate sort, in its ambient space.)

(2) The uniqueness theorem. (Any two such neighborhoods of a given pair are isomorphic.)

How generally can such a theory be constructed for arbitrary pairs of topological spaces Xc Y?

Given an arbitrary topological subspace X of a space Y, one would like to define the notion of a canonical neighborhood of X in Y. A canonical neighborhood U should be a topological neighborhood of X in Y satisfying certain properties (it might also possess some additional structure), such that these criteria hold:

(0) (Local nature). If Xc Yc Z where Y is open in Z, and if U is a canonical neighborhood of X in Y, U is also a canonical neighborhood of X in Z.

( 1 ) The notion is natural not only for topological spaces, but also for quite general topological categories. Hence, a formal proof in the theory should also be a proof in the categories of differentiable and combinatorial manifolds, and be valid for n-tuples of objects satisfying specific commutative diagrams, etc. See ? 9.

( 2 ) There is a uniqueness theorem. That is, any two canonical neigh- borhoods of X in Y are isomorphic over X. (In fact, one should be able to choose the isomorphism such that its germ, about X, is the identity iso- morphism.)

(3 diff) If Xc Y is a pair of differentiable manifolds, then the open tubular neighborhood of X in Y is a differentiable canonical neighborhood of Xin Y.

(3 comb) If X c Y is a pair of simplicial complexes, then the open regular neighborhood of X in Y is a combinatorial canonical neighborhood of X in Y.

More generally: (3 top) Let f: W - X be a continuous map, where W is paracompact, let

Y be the open mapping cylinder construction over X. Then Y is itself a

INFINITE REPETITION: I 205

canonical neighborhood of X in Y. (4 ) If Xc Y, let II*(X) be given by

flY (X) = proj limx wr*( V), -where V ranges over all topological neighborhoods of X in Y.

If the canonical neighborhood of X in Y exists, one would like the natural map

HIy (X) 7 r*(U)

to be an isomorphism. In good cases, then, the inclusion of X in U would be a weak homotopy equivalence; in better cases, a homotopy equivalence; and in even better cases, a strong deformation retract.

( 5 ) (Existence). There should be a theorem giving purely local criteria for the existence of canonical neighborhoods. For example, if X is locally tame in Y, then a canonical neighborhood of X in Y exists.

The theory of dilation neighborhoods to be given in ?? 4-8 satisfies all of the above criteria for a well-working theory of canonical neighborhoods, with the possible exception of criterion (5).

One may formally deduce many strong applications to pure topology from the first three criteria alone. (That is, under the assumption that a theory of canonical neighborhoods which obeys (0), (1), (2), (3), can be constructed.)

A. Topological invariance of open regular neighborhoods. Theorem. Let (X, Y) be a pair of topological spaces which is triangulable. (X is a sub- space of Y.) Then the open regular neighborhood of X in Y, considered as a topological space, is independent of the triangulation of the pair.

B. Theorem of the cone. Theorem. Let X be a paracompact space such that CX, the open topological cone over X, is a topological manifold. Then CX is homeomorphic with euclidean space.

C. Definition of local knot groups. Let f be a locally tame topological imbedding of X in Y, and x a point of X. Then one can define a group r(f, x) which plays the role of fundamental group of the complement of X in Y, in the vicinity of x.

There is difficulty in doing this directly. One is tempted to take a limit of knot groups of successively smaller neighborhoods about x; however, there is the problem of choosing base points in a coherent way.

Application (A) follows from (3 comb) and (2). Application (B) follows by applying the uniqueness theorem to the vertex

x of CX. By (3 top), CX is itself a canonical neighborhood of x in CX. By assumption, CX is a topological manifold, and therefore x has a neighborhood

206 BARRY MAZUR

in CX which is homeomorphic with RI, and which is therefore another ca- nonical neighborhood of x in CX (again by applying (3 top)).

To give a formal proof of application (C), I would have needed to be more explicit in (1). The idea is that, since f is locally tame in the vicinity of x, one may cut the situation down and assume the pair (X, Y) triangulable. Thus the couple (x, x) considered as a subspace of the couple (X, Y) has a canonical neighborhood (E, F) c (X, Y), the homeomorphy type of (E, F) being dependent only upon (X, Y). Thus one may regard F - E as a canoni-- cal local knot space of X in Y, about the point x. We would then take wr(f, x) to be the fundamental group of F - E (regarded as a bundle of groups over F-E).

4. Dilation spaces

In this section we shall study pairs of topological spaces Xc E. Every- thing will have meaning if X, E are considered, more generally, to be n-tuples (or diagrams) of topological objects. (See ?,9.) The space X will be fixed in the discussion to follow. All spaces will contain X, and all maps f: E1 E, will be assumed to be the identity mapping on X.

__f El - E2 F\/7

-\ /- X

The word map itself will mean open topological imbedding. Consider triples of the form (E, X, i) where i is a map

E -E

-\ /- X

DEFINITION 1. With respect to the triple & = (E, X, i), a map f: E -Er

will be called extendable (it should really be &-extendable) if there is a map fP such that

E

E -*E' f

is commutative.

INFINITE REPETITION: I 207

DEFINITION 2. A subset Yc E will be called &-bounded if: (a) Y is closed, (b) i Y is closed.

LEMMA 1. The union of a finite number of bounded sets is bounded. PROOF. If Y1, ***, Yc E are bounded, clearly (a) Y = Y1U U Y is closed in E,

and (b) iY=iY1U * * i Y is closed in E. An invertible map f: E E is one possessing an inverse map f -1 (hence

surjective).

DEFINITION 3. An 6-automorphism is an invertible map

ai: E ) E

which is the identity map outside a bounded set K c E. Notice that

LEMMA 2. If y is an &-automorphism, 7-1 is again an &-automorphism.

We shall also show that the composite of two &-automorphisms is again an &-automorphism. (See Lemma 6.)

DEFINITION 4. A mapf: EVE' will be called bounded (or, (S, V)-bounded) if f carries bounded subsets of E to bounded subsets of E'.

LEMMA 3. The composition of any two bounded maps is bounded. PROOF. Immediate. Notice. Any compact set is bounded. If E is compact, then any bounded

set is compact, hence all maps are bounded.

PROPOSITION 1. Let h: E E be a bounded map, a: E E an s-auto- morphism. Then there is an &-automorphism a*: E E defined by the formula

y*h = ha .

PROOF. In fact, take y *(x) = ha(y) for x = h(y) e hE

and

7*(x)-x for x e E-hE. Let K c E be a bounded set outside which y is the identity.

We shall first show a* to be 1: 1. The map a* preserves the sets hE and its complement, E - hE. Since

y* = 1 (E-hE); * = hyh-1 (hE).

208 BARRY MAZUR

we have that a* is 1: 1 on both parts, hE and E - hE. Thus a* is 1: 1. We shall next see that Y* is an open imbedding. It suffices to show a* an

open immersion. This is a local matter and obviously true on the interiors of hE and E - hE. We may restrict our attention to the set hE E- hE

which is hE n E - hE since h is open. But

hEnE - hE c E - hK,

and *- 1 (E - hK). Since both K and h are bounded, hK = hK and there- fore E - hK = E - hK is an open set. Collecting our observations, we have that hE n E - hE is contained in the open set E - hK on which a* is the identity map. It follows that v* is an open immersion in the vicinity of hE n E - hE. Thus v* is an open imbedding.

Since hK is bounded and Y* = 1 (E - hK), we have shown that Y* is an &-automorphism.

DEFINITION 5. A dilation space E = (E, X, i) is a triple satisfying: (1) There is an &-automorphism h such that hi2 = i. (2) (Stretching maps). Let K c E be bounded. There is a bounded map

s (a stretching map for K) with these properties: (i) s=1(i2E), (ii) Kc siE.

(3) (Compression automorphisms). Let U be a neighborhood of X. Then there is an &-automorphism c such that

ciE c U.

(4 ) (Compression maps). Let U be a neighborhood of X in E. Then there is a map K: E > U which is the identity on an open set containing X.

REMARK. Property (4) will be used only once, to deduce the final part of the corollary of the uniqueness theorem (? 7). Most of the above properties may be weakened somewhat.

In what follows, assume & = (E, X, i) to be a fixed dilation space, and all notions are relative to &.

LEMMA 4. The map i is bounded. PROOF. Assume Y c E bounded. We must show that i Y is bounded.

Since i Y is closed (by boundedness of Y), it suffices to show i2Y closed. By property (1) of dilation spaces, there is an 6-automorphism h sending i2 Y to i Y, which is closed. Consequently i2 Y is closed.

LEMMA 5. &-automorphisms are bounded. PROOF. Let y be an &-automorphism. We must show that -i Y is bounded

INFINITE REPETITION: I 209

when Y is. Since Y is a closed set, Y a surjective homeomorphism, it follows that Y Y is closed. Since i is bounded (Lemma 4), Proposition 1 applies, giving an &-automorphism Y* such that Y*i = at.

Consequently, i(y Y) = y*(i Y) is closed, since i Y is closed. Thus y Y is bounded, concluding Lemma 5.

LEMMA 6. Let y1, y2 be &-automorphisms. Then the composition, Y2Y1 is again an &-automorphism.

PROOF. Let K1, K2 c E be the bounded sets outside which 71, 72 (respec- tively) are the identity. Clearly Y2Y1 is an invertible map which is the identity outside K2 U y1K1. Since K1 and Y1 are bounded (Lemma 5), y1K1 is a bounded set. Therefore K = K2 U Y1K1 is bounded (Lemma 1).

The type of maps which will be used in the sequel will be called extendably bounded maps.

DEFINITION 6. An extendably bounded mapf: E-> E' is a map possessing an extension f' which is bounded.

LEMMA 7. The map i is extendably bounded. PROOF. The identity map is an extension of i. Since it is invertible, it is

bounded.

LEMMA 8. An extendably bounded map is bounded. PROOF. f = fi where f' is bounded. Since i is also bounded, so is f.

LEMMA 9. An extendably bounded map f possesses an extendably bounded extension.

PROOF. Let h: E E be the invertible map such that hi2 = i. Let f' be a bounded extension of f. Set f " = f 'hi. Then f " is an extendably bounded map, since f 'h is an extension of f " which is bounded (since h is invertible, hence bounded and f' is bounded, by assumption). The map f" is also an extension of f, since

fadi = fPh i2 =f fi = f-

LEMMA 10. If f is bounded and g extendably bounded, then fg is ex- tendably bounded.

PROOF. If g' is a bounded extension of g, then fg' is a bounded extension of fg.

LEMMA 11. If Y: E-> E is an &-automorphism, and f: E E' is ex- tendably bounded, then ft is extendably bounded.

PROOF. By Proposition 1, since f is bounded, frt = y*f. Since y * is bounded, Lemma 11 then follows from Lemma 9.

210 BARRY MAZUR

PROPOSITION 2. Let f and p be extendably bounded maps.

f, p: E - E.

Then there is an extendably bounded map q: E - E such that

4 E fE

PT qT E E

is commutative, and

qEDfE.

PROOF. Consider the commutative diagram

E

E D)E

PTC E )E

where f', p' are extendably bounded extensions on f, p, respectively. Since & = (E, X, i) is a dilation space, applying property (3), there is a.

compression automorphism c, such that

cliE c p'E,

and another compression automorphism c2 such that

c2iE c (p')-C'ci2E .

Thus we have:

(2.1) p'c2iE c c'i2E c cliE c p'E .

Since ip' is bounded, by Proposition 1, there is an automorphism c3 such that

(2.2) C3ip' ip=C2

Similarly, there is an automorphism c4 such that

(2.3) C4i = ic1 .

Let, again, h be the invertible map such that hi2 = i (property (1)). Let s' be a stretching map for hc4-'c3iE. That is, it is a bounded map

such that (a') S' 1(i2E),

INFINITE REPETITION: I 211

(b') s'iE D hc-'c3iE. Taking s= c4h-'s'hc-', we have a bounded map s satisfying:

(2.4) (a) s = l(ic1i2E), (b) sicjiE Z c3iE.

To see (a), notice that

Sf = 1 (hc4 1jCij2E)

which follows from (a') and the fact

(2.5) hc4 2

To see (b),

sicjiE= sc4i2E c4h=C siE.

By (b'),

c4h's'iE D c4h- (hc4'c3iE) = c3iE .

We may now define the map q by

q =ffc3-'Sip'C2

The map q is extendably bounded since f', c-', s, i are bounded, p' is extendably bounded and C2 is invertible (applications of Lemmas 10, 11). Proposition 2 will follow when it is shown that

(2.6) qi =fp, (2.7) qEDfE.

PROOF OF (2.6). By (2.1),

ip'c2iE c icJi2E .

Thus, by (2.4a)

s = 1 (ip'C2iE)

Consequently,

qi f 'C3-'Sip'C2i -ff'C3-'ipC2i

which is, by (2.2),

qi =ffipfi = fp . PROOF OF (2.7). By (2.1),

icjtE c ip'E .

Hence by (2.4b),

c3iE c sip'E .

212 BARRY MAZUR

Thus

qE f 'c3 sip'c2E -f 'c-1sip'E D f 'c-'c3iE or qED fE.

5. Infinite repetition spaces

If E is a topological space and f: E > E is a map, we may form the iterated injective limit

f f E

This makes sense, for f is an open imbedding. E(f) will be called the infinite repetition space formed by E and f.

In general, one would expect that E(f) is a much nastier looking object than E itself. In the case of dilation spaces, that is not true. In fact, E(f) is always as well, if not better, behaved than E. (See Proposition 5, ? 6.)

Consider E to be the solid torus S' x B2 c R3 positioned in euclidean space so that 0 E S1 x B2. Let i: E - E be an imbedding which is the restriction of some radial dilation of R3 by a small positive number. Then E(i) is homeo- morphic with R3.

Consider also, the example of an infinite repetition space given in ? 8, Example 1, E(i) is homeomorphic with E.

The following ? 6 will develop the idea that passing from a dilation space E to the infinite repetition space E(i) is a way of ironing out the kinks of the dilation space. It is a kind of completion, removing all unwanted holes. (In this manner, canonical neighborhoods will be constructed!)

Given any space E, and map f, there is a natural inclusion

Ec E(f)

obtained by considering E as the first factor in the injective limit E(f).

LEMMA 12. If f is a homeomorphism of E onto E, then the natural inclusion map, E c E(f ), is an identification between E and E(f ).

PROPOSITION 3. Let (E, X, i) be a dilation space (or an arbitrary triple satisfying property (1)). Then there is an imbedding g,

/

X

PROOF. Let h: E E be an isomorphism such that hi = i2 (property (1) of dilation spaces). Then one has the imbedding of E(i) in E(h) given by

INFINITE REPETITION: I 213

E )E~-+E > ...

h h h

After Lemma 12, E(h) is isomorphic with E, and Proposition 3 follows. Hence, after Proposition 3, we may regard the infinite repetition space

E(i) of a dilation space to be contained in E as an open neighborhood of X.

THEOREM 1. Let f be extendably bounded, and regard E as a subset of the infinite repetition spaces E(i), E(f) imbedded in the natural manner as the first factor. Then, there is an isomorphism i such that

-\/- EWi - E(f)

E is commutative.

PROOF. Since f is extendably bound, it possesses an extendably bounded extension f'. We must define a commutative ladder

Em E E Em

l11 'AlI 4211 E f E f E f E f

so that the induced map *: E(i) - E(f) is surjective. This will be the case if

*/rEDfE forn>1. We construct the AnJr inductively taking f ' and AnJr (n > 1) to be the ex- tendably bounded map q given by Proposition 2 for p =

6. The operation of completion

Let 6 (E, X, i) be a triple satisfying property (1) of dilation spaces (see- ?4). We will construct a new triple &*, a kind of completion of &. If & is a dilation space, &* possesses most of the properties of a dilation space again (probably all), but this is a complicated matter to prove, and it need not concern us.

Construction of &*. Take E* E(i). Take i*: E* E* to be the map induced by the ladder:

ihi{ i2h2iI inhni{

214 BARRY MAZUR

where h: E E is an &-automorphism such that hi2 = i, whose existence is ensured by property (1).

To show that &* is independent of h, we must compare i* and i* obtained from two choices, ho and h1. But consider the isomorphism k given by kn- hl'h-7 for each n. Clearly ki* = i*.

Note. P almost satisfies property (1), again. That is, one may construct an invertible map h*: E* ) E* such that h*(i*)2 -i*. This is sufficient to enable us to construct the double completion &** = (E*)*.

Construction of h*. By property (1) and Lemma 6, i is extendably bound- ed. Define then, a sequence of &-automorphisms he such that h1 = h and

hn+l- = ihn Y

using Proposition 2. Set h* to be the map determined by the sequence {hn}. Then

h*(i*)2 =-t* -

DEFINITION 7. An arbitrary triple (E, X, i) will be called complete if E

is isomorphic to

E* = E(i) .

PROPOSITION 4. If S is a triple satisfying (1), then P* is complete. PROOF. Consider the diagram

E -+E ,***,E

ihi i2h2i n

E ,-E -+**,

ihi j2h2i "inhni

and take the diagonal sequence as a representation of the infective limit E*

But the diagonal sequence is obtained from the following sequence, by re-grouping every three terms:

i h j2 j h2 i hn jin+1 E E E >E- E * - * E E*--.

But re-grouped in another way, that sequence may be seen to have the same limit as:

INFINITE REPETITION: I 215

h2X3 h3i4 h4i5in hnin

which is exactly E(i) = E, since hnin+1 = i. Most dilation spaces are complete already, as follows from

PROPOSITION 5. Let s = (E, X, i) be a dilation triple satisfying these extra properties:

(1) E is the union of a countable number of e-bounded sets. (2) The stretching maps (given by property (2) of Definition 5) may be

chosen to be surjective. Then s is complete.

PROOF. After (1) we may take a sequence of abounded sets K, c K2 c* ...

such that

U>=K3 = E.

Let h be an s-automorphism such that hi2 = i.

Let us define sequences of surjective maps ffn}, {sn} inductively as follows: (a) fo = 1, so = 1; now assume fJn-a, sn-l defined, and (b) let sn be a surjective stretching map for

Ln = (f.-lh)-1K. .

Let fn = fin1hsn. Now set gn = fni, and we may obtain a mapping

E E E**-

gol 91l g2l E yE yE >

As usual we must show: (i ) gni = gn-1; because

gni = fni2 = fn-1hsni2 - fnlhi2 = gn-

Hence the {gn} induce a map g: E(i) - E. (ii) gnE D Kn; because

gnE = fniE = fn-1hsniE D fn-,hLn

= fn-lh(fn-lh)- Kn = Kn .

Hence g: E(i) - E is an isomorphism, and s is complete.

7. Dilation neighborhoods

By an imbedding of one triple (E1, X, ij) in another (E2, X, i2), I mean a map w: E1 - E2 (a topological open imbedding) such that

216 BARRY MAZUR

il1 X i2

E1 ->E2 w

is commutative.

DEFINITION 8. Let Xc Y be a pair of topological spaces. A dilation neighborhood, U, of X in Y is an open set of Y containing X, such that U is isomorphic to the completion of a dilation space over X.

PROPOSITION 6. If there is some neighborhood U of X in Y, such that the pair (U, X) is imbeddable in a dilation space, then a dilation neighbor- hood of X in Y exists.

PROOF. By property (2), the dilation space (E, X, i) is then imbeddable in U, hence in Y. The completion E* is itself imbeddable in E. Consequently, E* is a dilation neighborhood of X in Y.

UNIQUENESS THEOREM. Any two dilation neighborhoods U1, U2 of X in, Y are isomorphic. The isomorphism C: U1 U2 may be taken to be the identity on some open set containing X.

PROOF. Consider U1, U2 c Y, two dilation neighborhoods of X. Thus there are dilation spaces Ek, k = 1, 2 for which there is the isomorphism,

Ek(ik) Uk .

By the imbedding constructed in Proposition 3, we regard Uk as a subspace of Ek. Choose maps

Jk = ckik: Ek - Ek

such that E1 E2

/ I / I , .u, nu2, - E . >u2nu, n U--E

31 32

Then the maps f- j2Jl: El - El f2- ilj2: E2 - E2

are clearly extendably bounded. Let E1, 2 denote the injective limit space

31 . 3> E 2 31. 2

By re-grouping,

INFINITE REPETITION: I 217

El fE, El f , El- E2

=1 ,___02 1 1 3 2 3 1 2

E2 E2 E2 f

f2 f2 f2

we have the very useful tautology El (f1) = El =E2(f2)

which is the main point of our method!! Thus we have an isomorphism

U, '1 El(il) El(fi)

4' 2 E1,2

U2 E2(i2) - E2(f2)

where *, is the isomorphism given by Theorem 1 (k = 1, 2) which applies, since fk is extendably bounded.

Unfortunately the isomorphism ;' obtained, between U1 and U2, is not necessarily the identity map in an open neighborhood of X. We must modify it slightly to obtain such an isomorphism ;. The isomorphisms obtained satisfy the following commutativity:

y

U1 ' U2

U U2~~

//~~~~

Ji /

".E, nE..

218 BARRY MAZUR

In the above diagram, the Ek are considered as subsets of the infinite re- petition spaces Ek(ik), Ek(fk) the imbedding being given, as usual, by identifi- cation with the first factor of the infective limit.

The 'pediment' of the above diagram is not commutative. The two symmetrical 'column' squares are commutative by Theorem 1.

Now let g, = h1cT1 where h1 is an &-automorphism such that h1 i =. Construct a commutative ladder of &-automorphisms

Eg? E1 El - * -

gj g2 g3

El - El El >

by applying Proposition 2 inductively, taking go to be given by:

g9ii = n1gn 1 -

Thus, the above ladder is commutative. The go induce an isomorphism,

g: Ej(ij) >Ej(ij) .

Set u = -'u1pgul. This is the identity in a neighborhood of X since g1j1 is. The strengthened version of the uniqueness theorem is thus proved.

COROLLARY. Let U1, U2 be two dilation neighborhoods of X (arising from distinct imbeddings of X in the spaces Y1, Y2 respectively). Let

q':U1 - U2

C\ /'

X

be an open imbedding (a map). (It need only be defined as a germ about X.) Then there is an isomorphism

4:U1 > U2

-\/-

X

whose germ about X is representable by Ip. PROOF. If the open imbedding qi is defined on all of U1, then we may

regard j' as an identification of U1 with an open subset of U2, and the unique- ness theorem of dilation neighborhoods implies the corollary.

If the open imbedding q is defined on an open neighborhood U c U1 of X, then we are faced with the necessity of finding a compression map X: U1 - U which is the identity in a neighborhood of X. Then setting j' = 9qX, we would

INFINITE REPETITION: I 219

have a map q' defined on all of U1 whose germ was that of T. Identify U1 with E(i). Let E c E(i) be the first factor, and set U' =

u n E c E. Let k: E - U' be a compression map which is the identity in a neighborhood about X. Consider the sequence

E E ***E**-

khij kh2i} khni{

U' = U' = *- U

This is commutative, since (kh"+1i)i = kh4(hi2) kh1hi. Also, since hi is the identity map on iE, we have that khi is the identity

in a neighborhood of X c E c E(i). The proof is concluded by setting X to be the map X: E(i) - U' c U induced by the above sequence.

REMARK. It is this corollary which will prove especially useful in study- ing the problem of representability of microbundles by open (and closed) cell bundles.

8. Examples of dilation spaces

The theory of dilation spaces has been treated on a completely formal level. It satisfies the criteria (1), (2) of ?3, and quite trivially satisfies criterion (4). In this section, in the course of which some general classes of dilation spaces are constructed, it will be seen to satisfy criterion (3 top) as well (Proposition 7).

Criterion (3 diff) is easily seen to be satisfied, and (3 comb) follows from [12] and the, combinatorial version of Proposition 6.

To prove Proposition 6 in the category of combinatorial spaces, one would have some added work ensuring piece-wise linearity of the maps constructed. If one carries over the proof word for word to the combinatorial case, one would find that his constructions would not yield piece-wise linear maps (they would be piece-wise quadratic or worse). But there are standard technical methods for ironing out these complications.

Example 1. Let p: A - X be a continuous map of topological spaces, where A is paracompact. Let E be the open mapping cylinder construction for p. Thus

E = {A x (0, 1)} U X/ -

where a x {0} - p(a) for all a e A. Let i: E - E be the map given by the identity on X and i(a, t) = (a, t/2) for (a, t) E A x [0, 1).

PROPOSITION 7. The triple (E, X, i) described above (A paracompact; X

220 BARRY MAZUR

arbitrary) is a dilation space. PROOF. In preparation for the proof, consider a real number r, 1/2 <

r < 1. Decompose the unit interval into the four subintervals determined by the sequence 0, 1/4, 1/2, (r + 1)/2, 1. Let Zr: I > I be the (piece-wise linear) homeomorphism which leaves 0, 1/4, (r + 1)/2, 1 fixed, sends 1/2 to r, and is linear on the four subintervals.

Let jA: A [1/2, 1) be a continuous function on A. Set a: En E to be the map y =(q9) defined as follows:

( i ) 7 = 1(X),I (ii) y(a, t) (a, ZWa)(t)) for a e A, t e I.

LEMMA 13. y = y(q') is an &-automorphism. PROOF. One sees easily that y is continuous, 1 :1, and possesses a con-

tinuous inverse. It is thus an invertible map. Therefore, K = yiE is bounded and, since y = 1(E - K), y is an &-automorphism.

Notice also, that y =J(i2E). Let K c E be a subset. Then K gives rise to a set K' c A x I as follows:

K' = A x {0} U ,8-1(K). Clearly, K' c A x [0, 1).

LEMMA 14. If Kc E is bounded, then K' c A x I is closed. PROOF. Consider the commutative diagram:

E : A x [0, 1) A x I

K1 fi 11/2 1/2{

E< A x [0,1) A x I

Now iK is closed in E, by boundedness of K. Consequently (1/2) - K' =f8-riK is closed in A x [0, 1). Thus (1/2)- K' c A x [0, 1/2] is a closed set. Consider the homeomorphism 2: A x [0, 1/2] A x [0, 1]. Clearly, (2) *(1/2) -K' = K' a A x [0, 1]; thus K' is closed in A x [0, 1].

LEMMA 15. Let K be a closed set in A x I such that wK = A. Let A, a be the functions on A which are given as follows: r(a), a(a) are the smallest and largest values of t E I (respectively) such that (a, t) E K. Then z is lower semi-continuous and a is upper semi-continuous.

PROOF. One must show the sets:

{a I a(a) > r}, {a I 7(a) ? r}

closed for all r E R. Let Ar= A x [0, r], Ar = A x [r, 1]. Then

INFINITE REPETITION: I 221

{a a(a) > r} _ (Ar nK) {a I r(a) < r} -(Ar n K)

which are clearly closed. Refer to v, a as "minimum" and "maximum" functions (respectively) for K.

LEMMA 16. Let K c E be bounded. Then there is a continuous function 9: A t [1/2, 1) such that the &-automorphism y =(9) has the following property:

7iE D K.

PROOF. By Lemma 14, the set K' c A x I is closed, where

K' A x {O} U f-1K.

Let a be the maximum function for K'. One has wK' = A. Since the closed set K' is contained in A x [0, 1), one has a(a) < 1 for all a E A.

Let p: A - [1/2, 1) be the upper semi-continuous function given by

p(a) = max (1/2, a(a)) .

Let j' be a continuous function A: A - [1/2, 1) such that 9(a) > p(a) for all a e A, [2, Exercise X, p. 172: betweenness theorem for semi-continuous functions]. Here we have used paracompactness of A.

Take v = Yq(9). PROOF OF PROPERTY (1). Let h: I-n I be the homeomorphism, linear on the

subintervals given by the decomposition 0, 1/4, 3/4, 1, determined by requiring that it leave 0, 3/4, 1, fixed, and send 1/4 to 1/2. Let h: E-m E be the invertible map which is the identity on X, and which is 1 x h on A x [0, 1).

Clearly hi2 i, and h is an &-automorphism. PROOF OF PROPERTY (2). It is immediate from Lemma 16, for the con-

structed y plays the role of a stretching map for K. Since it is an &-auto- morphism, it is bounded.

PROOF OF PROPERTIES (3) AND (4). Let U c E be the open set containing X. Take K - A x {1} U /,-'(E - U) to be the closed set in A x L Clearly 7K = A. Let r be the minimum function for K on A. Clearly r(a) > 0 for all a e A. Take X(a) =min {1/2, T(a)}; thus X: A - (0, 1/2]. Applying the betweenness theorem to the lower-semi-continuous function X[2, p. 172] there is a continuous function 9 on A such that 0 < 9(a) < X(a) for all a e A.

Now set ca: In) I to be the (piece-wise linear) homeomorphism linear on the subintervals determined by the sequence 0 < 9(a) < 3/4 < 1, which leaves 0, 3/4, 1, fixed, and sends 1/2 to 9(a). As before, an &-automorphism c: Em E is determined by the map (a, t) - (a, Ca(t)) on A x I.

222 BARRY MAZUR

Also ciE c U. Thus c so constructed is a compression automorphism for U. This proves property (3).

Set 'Ca: I I to be the (piece-wise linear) homeomorphism linear on the subintervals determined by the sequence 0, p(a)/2, 1, which leaves 0, q'(a)/2 fixed, and sends 1 to 9(a).

In the usual manner, the {la}aEA determine a map K: E E. Clearly X = 1( y(q')i2E), and rE is disjointed from a, hence KE c U. This proves property (4).

Example 1'. A specific case of Example 1 is when E is an open disc bundle over a paracompact base space X, and i: E E is scalar multiplication by 1/2.

9. Remarks on the formal nature of the preceding arguments

Let T denote the category whose objects are topological spaces, and morphisms are locally closed topological imbeddings.

Then T has the following property: If

A1 i 2 in ..- Al 2 >A2 * **An 2

is a sequence of open imbeddings, the injective limit, A = lim- . A", exists. A topological category (C, f) is a pair consisting of a category C and a

functor f: C e T satisfying these properties: (1 ) The induced map

Ear: Home (X, Y) - HomT (fX, f Y)

is injective for all X, Ye Ob(C). (2 ) Let A' -y- B' be an open inclusion map in the category T. Let

B' = fB for B C Ob(C). Then there is an inclusion map A B in the cate- gory C sent to A' - , B' by the functor f.

In this case, the subobject A c jB will be called an open subobject. and the inclusion map j will be called open (or an open imbedding).

(3 ) The category C obeys the following condition: If

A1 2 A2 2 . ** -> An*

is a sequence of open imbeddings, the injective limit

A limn .oo An

exists, and the functor f: C J- commutes with the injective limit of such sequences of open imbeddings.

Examples of topological categories are the categories of differentiable

INFINITE REPETITION: I 223

manifolds, or combinatorial spaces whose morphisms are locally closed imbed- dings of the appropriate genre.

It is easily seen that the methods of this paper are valid in an arbitrary topological category.

We may generalize the range of applicability of our methods a bit more by introducing the notion of proto-diagram. The reader may consider what follows to be make-shift, but I do not see a more natural statement, and I do need various cases of it for specific later applications.

The letter s will denote a class of topological spaces closed under the operation of infective limit for sequences of open imbeddings (e.g., the class of topological manifolds, or all topological spaces).

The letter m will denote a class of continuous maps closed under the operation of infective limit for sequences of open imbeddings.

That is, for all ladders

X1 ~L> y1 ai{ {1i

f211 X2 > Y2

a21 :

xn- y

X. Yn

an{

where the aj, fj are open imbeddings and the fj are of class m, then:

f = limie. fj: lim Xj > lim Yi

is again of class m (e.g., the class of vector bundle projections, of closed imbeddings, of open imbeddings...).

A proto-diagram a is a directed graph, each of whose vertices pj is labeled by a class of spaces swl, and each of whose directed line segments 1j is labeled by a class of maps my.

Given any proto-diagram 3, one may form the category T(Q3). An object Xof T3(a) will be a diagram of spaces and continuous maps, plus an isomorphism between that diagram and the directed graph a such that each space in the diagram X is in that class of spaces s which is the label of the corresponding

224 BARRY MAZUR

vertex of a, and each map in the diagram X is in that class of maps m which is the label of the corresponding directed line segment a. Each closed loop of the diagram X is required to be commutative.

A morphism h: X - Y of the category T(a) is a collection of morphisms hl, *..., h, of the spaces occurring in X to the corresponding spaces occurring in Y (where the correspondence between the two diagrams X, Y is given by comparing their respective isomorphisms with a). The morphisms hl, ***, h, are morphisms of T and, together with X, Y, make up a ladder-type diagram, which is assumed commutative.

Given any topological category C and a proto-diagram a, one may form, in the same way, the category 6(a). The functor f induces naturally a function

f (a): eCa) Ad () W The pair (C(a), f(a)) will be called a topological diagram category (of type 3).

Our methods make sense for topological diagram categories.

Example 1. Consider the proto-diagram a:

m m m m

where m denotes the class of closed imbeddings. Then T(a) is the category of topological n-tuples X1 c X2 c ... c Xn.

Example 2. Let a be the proto-diagram:

ml 5.

m3\ //m2

where s is the class of all topological spaces, m1 is the class of open disc bundle projections, m2 is the class of identity maps, m3 is the class of closed imbed- dings. Then the following is an object of T(a):

V x By > V

(E) x\\ xIO0}\ /1 V

where w is natural projection, and B mintDn c Rn is the open unit disc in R .

Let Xc E be the subobject given by

INFINITE REPETITION: I 225

V x {O} > V

(X) x 101\ /I

V

Let i: E E be the map given by

V Bn 1 x 1/2 V Bn

V \1 \

V > V

where 1/2: Bn - Bn denotes scalar multiplication by 1/2. Then d = (E, X, i) is a complete dilation space in the category 2(6).

Consequently, the corollary to the uniqueness theorem (?7) applies, yielding:

COROLLARY 1. Let Cp: E E be a germ of a map about X c E. Then there is an isomorphism 4i: E m E whose germ is a9.

Interpreting this in words, one obtains:

COROLLARY 2. Let w be an automorphism germ of the trivial micro- -bundles of dimension n over V,

V x R n

x ol/ \7r

V SD V

X{03\, 1 /X

Vx Rn

then there is a global isomorphism

V x Rn

x 1T/ \7r

V V x{O1\ /

V x R n

whose germ is 9. (This result is related to some theorems of Morse [8], [9].) HARVARD UNIVERSITY

226 BARRY MAZUR

REFERENCES

1. M. BROWN, A proof of the generalized Schoenflies theorem, Bull. Amer. Math. Soc., 6& (1960), 74-76.

2. J. KELLEY, General Topology, Van Nostrand. 3. S. LANG, Tangential homotopy equivalence, Seminaire Bourbaki, 1961. 4. B. MAZUR, On embeddings of spheres (Princeton University Ph. D. Thesis, 1959), Acta Math.,

105 (1961), 1-17. 5. , On embeddings of spheres, Bull. Amer. Math. Soc., 65 (1959), 59-65. 6. , Stable equivalence of differentiable manifolds, Bull. Amer. Math. Soc., 67 (1961),

377-384. 7. J. MILNOR, Theory of microbundles, Mimeographed Notes, Princeton University, Princeton. 8. M. MORSE, A reduction of the Schoenflies extension problem, Bull. Amer. Math. Soc., 66

(1960), 113-115. 9. , The dependence of the Schoenflies extension on an accessory parameter, J.

Analyse Math., 8 (1961), 209-271. 10. J. STALLINGS, Polyhedral homotopy spheres, Bull. Amer. Math. Soc., 66 (1960), 485-488. 11. , On topologically unknotted spheres, Ann. of Math., 77 (1963), 490-503. 12. J. H. C. WHITEHEAD, Simplicial spaces, nuclei and m-groups, Proc. London Math. Soc.,,

45 (1939), 243-327. 13. K. W. KwUN, Uniqueness of the open cone neighborhood, to appear. 14. and F. RAYMOND, Mapping cylinder neighborhoods, to appear. 15. H. GLUCK, Unknotting S' in S4, Bull. Amer. Math. Soc., 69 (1963), 91-94.

(Received February 11, 1963)