approximating codimension 3 embeddings

Annals of Mathematics

Approximating Codimension 3 EmbeddingsAuthor(s): Richard T. MillerSource: Annals of Mathematics, Second Series, Vol. 95, No. 3 (May, 1972), pp. 406-416Published by: Annals of MathematicsStable URL: http://www.jstor.org/stable/1970867 .

Accessed: 21/11/2014 23:52

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

Annals of Mathematics is collaborating with JSTOR to digitize, preserve and extend access to Annals ofMathematics.

http://www.jstor.org

This content downloaded from 165.190.89.176 on Fri, 21 Nov 2014 23:52:54 PMAll use subject to JSTOR Terms and Conditions

http://www.jstor.org/action/showPublisher?publisherCode=annals

http://www.jstor.org/stable/1970867?origin=JSTOR-pdf

http://www.jstor.org/page/info/about/policies/terms.jsp


Approximating codimension 3 embeddings By RICHARD T. MILLER

The purpose of this paper is to prove

THEOREM 1.* Let k < n- 3 and s > 0. Then if H: Dk R"R is a topological embedding, there is a piecewise linear embedding G: Dk RI that s-approximates H.

In 1965 Homma [7] gave a proof of Theorem 1 for k < (2/3)n - 1. It orginally appeared that his proof worked for codimension 3, but in 1968 Berkowitz found a counterexample to the technique in the range (2/3)n < k < n - 3. Berkowitz subsequently [1] extended Homma's arguments to prove the theorem for k < (3/4)n - 5/4.

We quote

THEOREM 2. Let k < n - 3 and s > 0. Let Mk and Qn be piecewise

linear manifolds and let H:Mk Qn be a topological embedding. Then there is a a > 0 such that if G1, G2: Mk Qn are piecewise linear embeddings and each 3-approximates H, then there is a piecewise linear s-ambient isotopy of Qn that takes G1 to G2.**

This theorem was proved by Bryant and Seebeck [5]. The author has an independent proof.

The preceding theorems together imply

THEOREM 3. Let k < n - 3 and s > 0. Let Mk and Qn be piecewise linear manifolds and let H: Mk ) Qn be a topological embedding. Then there is a piecewise linear embedding G: Mk , Qn that s-approximates H.

THEOREM 4. Let k < n - 3 and s > 0. Let Mk and Qn be topological manifolds and let H: Mk , Q be a topological embedding. Then there is a a > 0 such that if H1 and H2: Mke Qn are locally flat embeddings and each (-approximates H, then there is an s-ambient isotopy of Qn that takes H, to H2.

Theorem 4 is a consequence of Theorem 2 and Theorem 14 of [10]. Theo- rem 1 and Theorem 4 together imply

THEOREM 5. Let k < n - 3 and s > 0. Let Mk and Qn be topological

* Theorem 1 has also been announced by A.V. Cernavskii [6]. ** In the theorems, if M is not compact, then a and s are functions from M to (0, 0o),

the given embeddings are proper, and the embeddings in the conclusion are also proper.



CODIMENSION 3 EMBEDDINGS 407

manifolds and let H: Mk ) Qn be a topological embedding. Then there is a locally-fiat embedding G: MkI Qn that -approximates H.

In Theorems 2 through 5, the manifolds may have boundaries if we require that the interior of M maps into the interior of Q.

Berkowitz and Dancis [2] showed Theorem 5 for Mk a simplicial complex and k <(2/3)n - 1. Bryant [4] uses Theorem 1 to get Theorem 5 for Mk a simplicial complex and k ? n - 3.

Let RI c R2 c - * - c Rn be the standard inclusions, and let [0, 1] be the unit interval in R1. Define Dk = ([0, 1])k. (This includes Dk in Dk+l as a face.) If X is a set in R", let N,(X) be the open a-neighborhood of X in Rn.

Let H: Dk - RI be a topological embedding. Then there is a neighborhood Nof H(Dk) in RI and a uniformly continuous retraction r: N- H(Dk). We suppose lIx - r(x)ll ? 1/2, all xeN. Let r,: NORM be the homotopy obtained by moving each point x e N linearly along the line between it and its image under r in such a way that r,(x) = r(x) for t e x - r(x) , 1].

Let aj: Dn-? Dn be the deformation retraction *c.., xc ) = {identity if Xj ? t

(X1, *** Xi-it t, x;+l ... * * if x; < t . Let Atc: N-+ Rn be defined by

At(X) =rt(x) if t e [O. x - r(x)|]

- HOO ijx r(x)II oH-' if t e [|x - r(x) , 1] 1-i(x-r (x) hi

for each x e N. Suppose W is a set and ft: W N is a homotopy. We say fJ is a (j, a)-homotopy if ft is a a-approximation of *k of,. If W c N we assume that go is the inclusion.

All the theorems in this paper will be (&, s)-statements of the form "For s > 0, there is a a, > 0 such that if a, > a > 0 and if some hypothesis depending on a is. true, then some conclusion depending on s is true". It is then easy to construct a positive, monotone function s(a) for all sufficiently small a such that s(a) - 0 as a . 0, and such that the theorem is true when s(a) is substituted for m in the statement. The notation is advantageous in that it allows us to keep track of the property we want (namely, the existence of such a function) while forcing us to ignore the actual relation between a and a, which is irrelevant anyway.

It is clear that (j, a)-homotopies have the following properties.

LEMMA 6. 1. At I H(Dk) = Ho i oH`. There is a function s(a) such that

2. If x,yeN and IIx-y <a, then *jcf(x)- *(y) <e(a), all te



408 RICHARD T. MILLER

[0, 1]. The function s(a) may be modified so that if ft is a (j, a)-homotopy, then

3. Any a-approximation to ft is a (j, s(a))-homotopy. 4. ft(x) c Ne(a1(Ic(r(fo(x)))), all t e [O 1], all f0(x) e N(H(Dk)).

Suppose a simplex a collapses through its face z. If a is starred at Z (the barycenter of r) to obtain J, there is a simplicial retraction of a to bdy a - that takes ̀ to the vertex of a not in z. The map of a x [0, 1] to itself obtained from the identity on a x {0} U bdy a - z x [0, 1] and the simplicial retraction on a x {1} by starring at Z x {1/2} is a p.l. deformation retraction of a onto bdy a - z. Thus we may think of simplicial collapses as piecewise linear deformation retractions. If a complex has mesh < a, collapses to a subcomplex, and this collapse (thought of as a deformation retraction) is a (j, 5)-homotopy, we say that the complex is a (j, ()-collapse.

If S is a piecewise-linear manifold, and X is a subcomplex of S, then we denote an rth-derived subdivision of S by S followed by r primes. (E.g., the second derived of S is written S".) We let N(X, S") denote a second derived neighborhood of X in S, etc. Suppose a complex Y collapses to a subcomplex X. Then if Z is a subset of Y, we denote by image,,, (Z) the image of Z under the collapse considered as a retraction. Similarly, we denote by tracksx (Z) the track of Z under the deformation retraction.

The following theorem is proved in [9].

THEOREM 7. Let Q be a p.l. q-manifold. Let Y and X be subcomplexes of Q. If Y collapses simplicially to X, then there exists an isotopy Ot of N( Y, Q") into itself such that

(D1(N(Y, Q")) = N(X, Q"), such that

Ot is fixed outside N(N(vertici in (Y - X)', Q"), Q"') and such that if Z is a subcomplex of Y, then

$D1(N(Z, Q")) c N(imagey\x (Z), Q")

If, in addition, (Y - X) is contained in the interior of Q. then $t extends to an ambient isotopy of Q (also called qt) for which

~t (N(N(Zg Q") , Q"')) (- N(N(track (Z) y Q") , Q",) and Ot is fixed on the same set as above.

THEOREM 8. Suppose S is a p.1. submanifold of R". Then there is a function s(a) such that if C is a (j, ()-collapse that is a subcomplex of a trian- gulation of S n N(H(Dk)) with mesh < a, and if Gi-1: Di`-+ bdy N(C, S")




is a p.1. embedding that 3-approximates HI Di-', then there is a p.1. embedding G5: Di - N(C, S") that s(Q)-approximates H I Di.

Proof. By the compactness of Dk there is a function s(y) such that the diameter of H(x, [t, t + AY]) is less than s(y)/2 for all x e Di-', t e [0, 1 - ] We suppose w'& = 1 for some positive integer w'. Let C(w a) be the image of the collapse C at time w -r. By slightly changing the time parametrization of the collapse (think of the time to collapse each simplex C as much less than y) we can further assume that C(w y) is actually a subcomplex of C for all integral w, 0 < w < w'.

We construct by induction on w a function y((3) and a sequence of p.l. embeddings Gi: Di` x [-1, wy] US such that

1. Gi (Di-' x {w-e}) - Gi (Di-' x -1, w-e]) n bdy N(CQw-), S") 2. GJ [(Gi-1)-1(N(d, S")) x {wy}] c N(imagec\c(wr) q, S") for each simplex

ee C. 3. G5 IDi-' x [0, wY] is an e(3)-approximation of HI Di` x [0, wyl.

We can find N(C, S"') - N(C, S") ' [-1, 0] x bdy N(C, S") p.1. homeo

onto such that p I {?} x bdy N(C, S") bdy N(C, S") is the identity. Let Ga = pc(id x Gi-1).

Suppose that G2A1 is defined. By property 1 we can choose a new third derived subdivision of S (leaving the first two alone) such that

Gi -,(Di-' x [-1, (w - 1)'r]) n N(N(C((w - 1)y), s"), S"') - N(C((w - 1)i), S") -Gi_-(Di-' x [(w - 1)y - Y', (w -1)])

for an arbitrarily small y'. Now we can apply Theorem 7 to the collapse C((w - 1)y) \ C(wy) to obtain a p.I. homeomorphism 5 of S. Define a p.I. homeo

identity x p.1l C: Di` x [-1, wY] -A Di-' x [1, (w - 1)-i] with '(Di-' x [(w - 1)Y, wy]) = Di- x [(w - 1)Y - Y', (w - 1)Y]. Then define G- Oo G' _o<* Properties 1 and 2 are immediately satisfied. We see Property 3 as follows. For sufficiently small 'Y', it is clear that G. I Di-' x [0, (w - 1)'Y] is so close to G&_1 that it too e(Q)-approximates HI Di` x [0, (w -l)]. The case GA IDi-' x [(w - 1)Y, wy] is just slightly harder. For each point x e Di-', Gj-1(x) e N(d, S") where e is some simplex in C. Notice that d c N3a(H(x)). By property 2 we know that

GA -1(x, [(w - 1)y - Y' , (w - 1)-i]) c N(N(imagec\c((w-l) ) $, S"), S"')

Consequently, for t e [(w - 1)Y, wy],




Gi (x, t) e '(Gi -,(x, [(w -l)-i--i' , (w l)-i]))

c N(N(trackc((W.l)T)\c(W.) imagec\0c((w-4)O) , S"), S"')*

Since C is a (j,a)-collapse, (track image d) in the previous line is contained in NI(*Wl),W](d)). This set is in turn contained in N,(H(x, [(w - 1)y, wy])) by Lemma 6, part 2, together with the fact d c N38(H(x)). Thus,

Gi (x, t) e N2aN,(H(x, [(w - 1), wy])) c Nl12(H(x, [(w - 1)7, w7])) c N6(H(x, t))

for all t e [(w - 1)y, wy] if s(y) and a(a) are chosen appropriately. This completes the induction step and also the proof of Theorem 8.

LEMMA 9. Suppose that s3 is a p.l. manifold in RI for which there is a function 'Q3') such that for each complex Z of dimension < s - 4 in s n NN,(H(D')) there is a (j, '(e'))-homotopy of Z in S n N,(,, (H(Dk)). Then it follows that there is a function s(a) such that if C* is a (j, e)-collapse of dimension < s - 3, and if X is a complex of dimension < s - 4, both C* and X lying in S n N(H(Dk)), there is a (j, s(a))-collapse C in S n N)(,(H(Dk)) that contains both C* and X, and such that dimension (C - C*) < (dimension X) + 1.

Proof. The proof is by induction on the dimension of X and requires slightly stronger hypotheses and conclusions. Specifically, if Y is a complex in S of dimension < s - 3, then C - C* is in general position with respect to Y. For the inductive step we assume that the lemma is true for all complexes of dimension < r < s - 4, that, in particular, the (r - l)-skeleton X'r-1' of X is already in C*, and that C* either contains or is in general position with respect to each simplex of X.

Suppose C is a simplex in S for which bdy c C* C. Let Z = C Ubdyr track,* (bdy C). (Trackc* (bdy C) is the track of bdy C under the deformation retraction that corresponds to the (j, 3)-collapse C*.) Suppose f: Z x [0, 1] > S is a (j, 3)-homotopy of Z. Choose a number a so small that Au x [t, t + 8]) has diameter < a, all simplexes a in Z, all t e [0, 1 - a]. Define a collapse of Z x [0, 1] to Z x {1} U image,* (bdy C) x [0, 1] as follows. Collapse C x [0, 8] through C x {0}. Then collapse trackc* (bdy C) x [0, '8] to track,* (bdy C) x f0, 6} U image,* (bdy C) x [0, 8] by mimicking the collapse of track,* (bdy C) given by the collapse of C*. Collapse trackt (bdy C) x {0} to image,. (bdy C) x {0}. Now collapse Z x [,, 1] along the product struc- ture to Z x {1} U image,* (bdy C) x [,,1]. This completes the collapse of Z x [0, 1] promised above. We define another collapse (called d) of Z x [0, 1] by rearranging the order of the above collapse (but keeping the tracks the same) so that each simplex a e Z x [0, 1] is collapsed as soon




as possible after the condition N8(f(u)) n f(Z x [t, 1]) = 0 is satisfied. It follows that there is a function s(a) such that f(t) is a (i, s(a))-homotopy of C.

Now for the induction. For each r-simplex C in X we construct a set Z as in the preceding paragraphs. The hypothesis of the lemma gives a (i, '(a))- homotopy f of Z, and as before we obtain a collapse d of Z x [0, 1] such that f(d) is a (j, s"(a))-homotopy of C. We take the union of the Z x [0, 1] along track,. (bdy C) x {O}; and call this union W. Then e gives a collapse of W. The homotopy f defines a map of W into S (call this map f, too). Note that f(l) is a (j, s"(a))-homotopy of Xlrl. The singular set of the map f, denoted by f *, is the closure of the set of points in W not equal to their image under f '(f). If f is p.l. and in general position is in S, that f * is a complex of dimension < 2(r + 1) - s ? r - 2. We suppose that f is in general position with respect to itself, and relative to Z, with respect to C*, to Y, and to each simplex of X. Let

fC* = {w e W - Us trackcs (bdy C) I f(w) e C*} .

Then dimension f C* < (s-3) + (r + 1) -s= r-2. Let F = f(track (f * U f C*)). Then dimension F < r - 1.

We now inductively construct a sequence of complexes Cr. ***, C1 in S such that

(1) Cq is a (i, eq(a))-collapse (2) Cq D f(tracke (f'-(Cq+i n f(W)))) U Cq+1 (3) Cq - (Cq+l U f(tracke (f-1(Cq+i n f(W))))) is in general position with

respect to Y, to each simplex of X, and to f(W) (4) dimension (Cq - Cq+,) < q.

We say a subset T of f(W) is saturated if T = f(tracke (f'-(T))). Condition (2) implies that the part of Cq n f(W) that is not saturated lies in Cq - Cq+1, and conditions (3) and (4) imply

(5) the part of Cq n f(W) that is not saturated has dimension < q - 2. An essential consequence of (5) is that C1 n f(W) is saturated.

By induction on the lemma, there exists a collapse Cr that satisfies properties (1)-(4) with Cr+1 = C*. In fact, we can assume Cr D F. This starts the induction. Suppose C,, , Cq+l are already constructed. Apply the lemma to the case where "C*" = Cq+1, "X" = f(track (f'-((Cq+l - Cq+2) nf f(W)))), and "Y"= YUf(W) U (the simplexes of Xnot in Cq+l). Then properties (1)-(4) are satisfied by Cq = "C". Define C** = C1. Define the set C to be C** U f(W). Note that C immediately satisfies all the required conclusions of the lemma except the one about collapsing. We now show there is a function




s(a) and a way to collapse C that is a (j, s(a))-collapse. Notice that since c* * n f(w) contains the singularities of f, the map f I ((W) - f (C* *)) is an embedding. Since C** n f(W) is saturated, the collapse d also gives a collapse ** of W to f1-(C**)(Zx {1})(C* x [0, 1]). Since fI(W-f-1(C**)) is an embedding, f(l**) is a collapse of C to C**. Let C** be the collapse of C* * with the time parameter delayed by the amount v. We define the collapse of C to be the collapses C>** and f(e**) run simultaneously. For sufficiently large a(a), this is actually a collapse, since the only possible trouble could come from a simplex of f(W) - C** blocking a collapse in C***. This does not happen because, with the delay a, by the time the collapse in C* * reaches a simplex, all the simplexes of f(W) -C** in a neighborhood of it have been removed.

Finally we check that C is a (i, s(a))-collapse. Tracks of points under C divide into precisely three classes: those that start in C* *, those that start and finish in C - C**, and those that start in C - C** but finish in C**. The first two types are the restrictions of the (j, Td*(a))- and (j, s"(a))-homotopies C** and f(**). Tracks of the third type remain in C** from the time they first cross into C**. Until the crossing time, they are carried by f(d**), and thereafter by C**. This tells us how to define s(a) so that the collapse of C is a (j, s(a))-homotopy.

COROLLARY 10. Suppose there is a function '(3') and a set {a}d > 0 of collections of p.l. s-manifolds without boundary in RI with each element of cS contained in N,(H(Di)). Suppose further that if 3 e Si, then for each complex Z of dimension < s - 4 in 3, there is a manifold Sg in , such that 9z n N, (H(Di)) = S and such that S, contains a (j, e'(e'))-homotopy of Z.

Then there is a function s(a) such that if S* e 35, if C* is a (j, e)-collapse of dimension < s - 3 in S*, and if X is a complex of dimension < s - 4 in S*, then there is a manifold S in ,(, such that s A N(H(Dj)) = S* and such that S contains a (i, s(a))-collapse C that in turn contains both C* and X, and dimension (C - C*) < (dimension X) + 1.

Furthermore, if Xc S* - N(,)(H(Dj-1)), then C - C* lies outside Na(H(Dj-1)).

Proof. The first two paragraphs of the statement of Corollary 10 arise from switching from the single manifold S of Lemma 9 to a sequence of collections of manifolds. It is easy to check that the proof given for Lemma 9 works here also.

The last paragraph of Corollary 10 is verified as follows. The construction of Lemma 9 requires a finite number , (that depends only on dimension




S and dimension X) of (j, e'(3'))-homotopies, the first with domain X and the subsequent ones with domains in the tracks of those preceding them. Thus, they are all contained in N(,)ie,(,)i[O, 11(X), where * This set misses N,(H(D5-1)) if X lies outside N,(,)(H(Di1-)) for sufficiently large s(a).

LEMMA 11. For each 0 < j < k, there is a function s(a) such that if Ck, * * *, Cj and Bk, * , Bj are sequences of complexes in N,(H(Dk)) that for j < r < k satisfy the hypotheses

1. Cr is an (r, a)-collapse in bdy Br+, NN(H(Dr)), dimension Cr = r. 2. Bk = N(Ck, R ") and Br = N(Cr, (bdy Br+1)'), (Let bdy Bk+l = RA) then

for each h < j and for each complex X in bdy Bj+, n N(H(D5)), dimension X <h, there are new sequences Ck, **, C5, and B1, ***, Bf that satisfy conditions 1 and 2 where a is replaced by s(a) and satisfy the additional conditions

3. CicCx and XcCx. 4. If h < j, BT contains an (h, s(a))-homotopy of X. 5. If X lies outside N(,(H(Di-1)) then

CX r NN(H(D>-1)) - Cr NN(H(Di1-)) and

Bxr n N,(H(Dj-1)) - Br Na(H(Di1-)).

Proof. For a fixed k, the proof is by downward induction on j. We start with j = k. Recall that h I NJH(Dk) is an (h, a)-homotopy. Consequently, by Lemma 6, part 4, there is a function 'Q3') that has the property that for each complex Z c N (H(Dk)), the map Prk Z Z is a (k, e'(Q'))-homotopy lying in N,,(a,)(H(Dk)). Also, if Ck and X are the complexes referred to in the hypothesis of this lemma, we have that V I X is an (h, a)-homotopy. Let ,ct be a p. 1. approximation to PT I X that agrees with it at t = 0 and that is also an (h, a)-homotopy. Apply Corollary 10 with Si containing the single n-manifold Na(H(D k)), with C* Ck and with

X if h= k X= . X=

,180.l(X) if h < k .

Then set Ck = C and set Bx = N(Ck, Rib"). Properties 1, 2, and 3 are immediately satisfied. ,t is the homotopy required to satisfy condition 4. If X lies outside N2,(),(H(Dk-l)) and a is sufficiently small, then &[0,1](X) lies outside N2, (, (H(Dk-l)) and property 5 follows from the "Furthermore. . ." part of the conclusion of Corollary 10.

Now for the case j < k. We assume Lemma 11 for j + 1 and prove it




as stated. Once again we will show how to satisfy the hypotheses of Corollary 10, and then apply it to finish the proof. In particular, we will find a function '63') such that if Ck, ***, C' and Bk, ***, Bj are complexes that satisfy the hypotheses of Lemma 11 with a replaced by a', if h is an integer 0 < h < j, and if Zcz bdyBj+1 n N8 (H(Di)) is a complex of dimension < h - 1, then there are new sequences Z, C, C and Bk, ., BZ+1 Bj with Crz D C7 j + 1 < r < k, that satisfy the hypotheses of Lemma 11 with a' replaced by '(3') and such that there is an (h, e'Q'))-homotopy of Z in bdy B>j+1.

By induction on Lemma 11, there is a function s"(a") that satisfies the lemma in the case j + 1 where j + 1 < k. Recall that Cj+1 is a (j + 1, (')- collapse. It is easy to find functions y"x(a") and a"(s") such that the collapse of Cj+1 to Cj+'(a") ( = the image of Cj+1 at time a") moves points < -"(a") and such that Cj+3(a"l) n NN (8,)(H(Dj)) = 0. If the mesh of the triangula- tion of Bj+2 is small, then the collapse of BPj = N(Cj+1, bdy 3-j+2") \ Cj+1 moves points only a short distance. Thus, by choosing a fine enough mesh, we can suppose the composition of these collapses moves points < y"(('). Call this composition C.

Apply Lemma 11 to the sequences Ck, *., Cj+ and Bk, * , Bj+l with X replaced by C1(Z). (Here is where we use the lemma inductively.) We obtain new sequences C 1 (Z), C , 1 j>(Z and B13k(z, , ' where Ck(Z, is an (r, s"(('))-collapse. We define C; = cj1(z) and Brl'z= Br. Let f3t be the (h, s"(a'))-homotopy provided by property 4. By property 5, Cr nr N8(H(Di)) =

NN (H(D')) and B7r N~a (H(Di) = B3 n N,, (H(Dj)), j + 1 < r < k. Thus we can add to the new sequences the old complexes Ci and Bj, and thereby lengthen the new sequences. Let ,8~ be the homotopy of Z obtained by composing the homotopies C, and ,8 in such a way that C, is traversed in a very short time interval, and ,8 is traversed in the rest of the time. If the first time interval is sufficiently short, the homotopy 8,8 is a y"(a')-approximation of ,C~ and so Lemma 6 provides a function d" such that 8, is an

(h, 1"(al"(a')))-homotopy. Notice that ,3[0,1](Z) c N2 ( (H(Dj)) We proceed to push /,3 into bdy B1+1, fixing 80 Find functions y'(a') and

a'(2V") such that BJz+(a') n N2N,,(H(Di)) = 0, and such that the collapse of CjZ+1 to Cj+'(a') moves points less than y'(a'). Let y-1 be the inverse of the ambient isotopy of bdy BjZ+2 that Theoroem 7 assigns to this collapse. 0' moves points < 2y'(a'). Let p0 be a p.l. isomorphism that fixes Cj+1 and takes B4+1 onto N(B4+1, bdy Bz"'),. Let p1 be a p. L map of N(B1>+1, bdy BJz'2) onto B+?, that is the identity on B4+1 and takes N(B4+1, bdy BJ"2') = Bz onto bdy BJz1. We can suppose that pip, I bdy B4 1 = identity. We can choose




the third derived subdivision of B>z , such that both p0 and p1 move points less than a', and such that p,(,[0,1](Z)) is still contained in N$ (,,(H(Dj)). Define ,t to be p1o0-1op,(S,). Thus 8,(Z) C bdy B>z. Also S, is a (2&' + 2y'(J'))- approximation to ht and so, by Lemma 6, is an (h, V"(23' + 27'(3')))-homotopy. Define s'(3') = d"(23' + 27'(Q')) to finish this part of the proof.

Now return to the statement of Lemma 11 as it is written. If h = j set Or = Cr and BR3 = Br. If h < j, use the previous paragraphs to construct new sequences Ck, * * j1, Ci and BR, ..., Bx >, Bj where x is an (r, e'())- collapse, and to construct an (h, e'(a))-homotopy Ad of X in bdy B> . Let 8, be a p.l. approximation to o that is also an (h, e'(J))-homotopy of Xin bdy3B>Y .

We now apply Corollary 10 (recall the first paragraph of the proof of Lemma 11, case j < k) with Si = {bdy Bj+1 n N(H(Dj)) I there are sequences Ck, **, Cj+1 and Bk, *.., Bj+j that satisfy the hypotheses of Lemma 11}, with 3(Cor. 10) = '(a), with C*(Cor. 10) = Ci, with

3*(Cor. 10) bdy BI+1 0 N(H(Dj)) and with

(X if h i X(Cor. 10) if h <

lI%,l](X) ~if h < j Corollary 10 then yields sequences C, *.., CO+1 and Bx, ..., Bf+1. Ci = C (Corollary 10) is contained in bdyB>Y+1 n N,)(H(Dj)). LetBfT =N(C , bdy(B +l)"). The sequences Ck, **, Ct+', Ci and Bx, *** B>,1 BRI satisfy Lemma 11.

Lemma 12. For each s > 0 and each j < k, there are sequences Ck,* * *, Ci and Bk, *--, Bj such that

1. Cr is an (r, s)-collapse in N6(H(Dr)) n bdy Br+l. 2. Bk = N(Ck, R") and Br = N(Cr, bdy Br+1"). 3. There is a p.1. map (not embedding) gr-1: Dr- 1 Cr that s-approxi-

mates HID7-'. Proof. By downward induction on j. If j = k, let gl-1: Dk-l N0(H(Dk))

be a p. 1. a-approximation to H IDk-l. Apply Lemma 11 with j = k, with Ck(Lem. 11) = 0, and with X(Lem. 11) = gkl-(Dk-l). Then setting Ck(Lem. 11) = Ck(Lem. 12), Bx(Lem. 11) = Bk(Lem. 12), and s(&)(Lem. 11) = S satisfies this case of Lemma 12.

If j < k, the induction yields sequence Ck, **, C'+1 and Bk, *.., Bj+ satisfying 1-3 above where s is replaced by a. A construction essentially identical to that which builds the map ploo-lop, of Lemma 11 produces a p.l. map p such that pog'(Dj) c bdy Bj+1 and such that po Gi is a 7(J)-approximation of H I Di. Let g-1 = pogi I Di`'. Now apply Lemma 11 to the sequences




Ck, .., Cj+1 0 and Bk, *.., Bj+,, 0, with X (Lem. 11) =g5-1(D5-). Set s(y(a)) (Lem 11) = s. The sequences Ck, ** , Ci and B1, ** , BT then satisfy 1-3 above.

LEMMA 13. There is a function s(a) such that if Ck, * , C1 and Bk, * , B1 are complexes satisfying the conditions 1-3 of Lemma 12 with s replaced by a, then for each j, 1 < j < k, there is a p.l. embedding G': Di-? Bj that s(3)- approximates H I Di.

Proof. By upward induction on j. For j = 1, we have the map g':D' C1 c B, = N(C', bdy B2"). Since D' is a point, we can find a point G"(D") c bdy B, that is within a of g'(D'), and is therefore within 2a of H(D'). We apply Theorem 8 with C(Thm. 8) = C' (which is a (1, 3)-, hence a (1, 28)-collapse). Let G1 be the embedding provided by the theorem. Then G' is an SThm.8(2a) = s(3) approximation to H I D'.

If j > 1, suppose the embedding Gj-': Di`- Bj-, is an &'(a)-approximation to H I Do-'. Since Bi-, c bdy Bj, we can apply Theorem 8 once more with CThm.8 = C' and obtain an embedding Gj: Di Bj that SThm.S(Q3)) approximates HI D.

Lemma 13 with j = k and a such that s(Lem. 13) (a) = a is precisely the same as Theorem 1. Thus the proof of Theorem 1 is complete.

NORTHERN ILLINOIS UNIVERSITY, DEKALB, ILL.

REFERENCES

[1] IH. W. BERKOWITZ, Piecewise Linear Approximation of a Homeomorphism from One Com- binatorial Manifold into Another, Ph. D. thesis, Rutgers, 1968.

[2] and J. DANCIS, Piecewise Linear Approximation of Embeddings and Isotopies of Polyhedra, (Preprint).

[3] R. H. BING, Radial Engulfing, Proceedings of the Conference on Topology of Manifolds, Michigan State University, 1967.

[4] J. L. BRYANT, Approximations of Embeddings of Polyhedra, (Preprint). [5] and C. L. SEEBECK, Locally Nice Embeddings in Codimension Three,

(Preprint). [6] A. V. CERNAVSKII, Topological Embeddings of Manifolds, Sov. Math. 10, No. 4, (1969). [7] T. HOMMA, Piecewise Linear Approximations of Embeddings of Manifolds, (Mimeo notes,

Math. Dept., Florida State University, 1965). [8] , A Theorem of Piecewise Linear Approximation, Yokohama Math. J. XIV (1966),

47-54. [9] R. T. MILLER, Close Isotopies on Piecewise-Linear Manifolds, Ph. D. Thesis, University

of Michigan (1968). [10] , Close Isotopies on Piecewise-linear Manifolds, Transactions A. M. S. (October

1970).

(Received August 18. 1970)



approximating codimension 3 embeddings

Documents