APPROXIMATING EMBEDDINGS OF POLYHEDRA IN CODIMENSION THREE(J)

Transcription

1 TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Vlume 170, August 1972 APPROXIMATING EMBEDDINGS OF POLYHEDRA IN CODIMENSION THREE(J) BY J. L. BRYANT ABSTRACT. Let F be a p-dimensinal plyhedrn and let Ç be a PL q- manifld withut bundary. (Neither is necessarily cmpact.) The purpse f this paper is t prve that, if q p > 3, then any tplgical embedding f P int Q can be pintwise apprximated by PL embeddin-gs. The prf f this therem uses the analgus result fr embeddings f ne PL manifld int anther btained by Cernavskii and Miller. Intrductin. Recently Cernavskii and Miller have shwn that any tplgical embedding f a PL zzz-manifld M int a PL -manifld Q can be apprximated by PL embeddings prvided q - m > 3- (See [5], [6], and [10].) In this paper we use this fact t prve that a tplgical embedding f an arbitrary p-dimensinal plyhedrn int Q (fl - p > 3) can be apprximated by PL embeddings. This generalizes the results f Berkwitz [2] and Weber [13]- Our therem can be stated as fllws. Therem 1. Suppse that P is a (nt necessarily cmpact) p-dimensinal plyhedrn, that 0 is a PL q-manifld withut bundary (q p > 3), and that f: P * Q is a tplgical embedding. Then fr each cntinuus functin c P» (0, 00) there exists a PL embedding g; P»0 such that d(f(x),g(x)) < e(x) fr each x P. chse Mrever, if R is a subplyhedrn f P n which f\r is PL, then we may g s that g\r=f\r. The "mrever" part f Therem 1 is simply an applicatin f an istpy therem prved by Cnnelly [8], and we shall make n further reference t it. Definitins and ntatins. By a plyhedrn we mean the underlying space f a lcally finite simplicial cmplex embedded as a clsed subset f sme euclidean space (r manifld). We use the standard definitin f piecewise linear (PL) map and PL manifld. When we say that R is a subplyhedrn f a plyhedrn P, we shall always assume that R is a clsed subset f P. If /: P» 0 is a PL map, Received by the editrs Nvember 18, AMS 1970 subject classificatins. Primary 57C05, 57C55, 57C35. Key wrds and phrases. PL embedding, PL mapping, PL apprximatin, plyhedrn, PL manifld. (1) Partially supprted by NSF Grants and Cpyright 1972, American Mathematical Sciety

2 86 J- L- BRYANT [August then S, = Cl \x\f~ l f(x) x\ shall dente the singular set f /. The unit ball in euclidean «-space En is dented by Bn; I = [0, 1]. Given metric spaces X and Y, a cntinuus functin e: X (0, <x>), and tw mappings f, g: X Y, we define d(f, g) < e t mean d(f(x), g (x)) < e(x) tt each x X. (We ambiguusly use d t dente the metric f any space under cnsideratin.) If K and L are lcally finite simplicial cmplexes, then K N L means that K cllapses t L in the sense f [12]. In particular, we require that the cllapse f K t L determine a prper defrmatin retractin f \K\ nt L ("prper" meaning that inverse images f cmpact sets are cmpact). If H is a subcmplex f K, then we use the ntin f the trail f H, tr H, under the cllapse K>» L as defined byzeemanin Chapter 7 f [16]. The imprtant prperties f tr H ate (i) dim tr H < 1 + dim H, (ii) dim L n tr H < dim H, and (iii) K\LutrHNL as prved in Lemmas 44 and 45 f [16J. If e: \K\ (0, <») is cntinuus, then the cllapse f K t L is called an e-cllapse, written K \ L, if it gives rise t an -defrmatin retractin f K nt L. Observe that if K n* L is an f-cllapse and if K^r> is an rth derived subdivisin f K, then the cllapse f K t L^T' can be arranged s that it is an we-cllapse, where m depends nly n the diameter f the simplexes f K L and the dimensin f K - L. A plyhedrn X cllapses (e-cllapses) t a plyhedrn Y if there are lcally finite simplicial cmplexes K and L such that K = X, L = Y, and K cllapses (e-cllapses) t L. Suppse that X Si Y and Z is a subplyhedrn f X. Let K and L be chsen as abve s that K "S L. Then there is an rth derived subdivisin /Or' cntaining a subcmplex H such that \H\ = Z (see Chapter 1, Lemma 4 f [16]). Define tr Z = j tr H\ C X. Of curse, tr Z depends upn the triangulatin invlved. It fllws, hwever, that in any event each f the cllapses X \ Y u tr Z and Vu trz \ V is an wze-cllapse (fr sme m as described abve). This can be seen frm ur abve remarks tgether with the prf at Lemma 45 f [l6]. Preliminary lemmas. The apprximatin therem we require t prve Therem 1 is stated as fllws. (See [5], [6] and [l0].) Therem 2. Suppse that M and Q are PL maniflds f dimensins m and a respectively with q m > 3, and that f: M * Q is a tplgical embedding. Then fr each cntinuus e: M * (0, <x>) there exists a PL embedding g: M * Q such that d(f, g) < e.

3 1972] APPROXIMATING EMBEDDINGS OF POLYHEDRA 87 I am deeply indebted t Rbert Edwards, Richard Miller, and the referee fr helping me simplify and crrect many f the prfs appearing in the riginal versin f this paper. Lemma 1. Suppse that f; B» Int Q (q k > 3) is an embedding such that / lntb is PL. Then there exists an extensin f f t an embedding F: Bq > Int 0 such that F\Bq - Bd B k in PL. therem. Prf. This is essentially a sequence f applicatins f Therem 9 f [15]. The next lemma is a straightfrward applicatin f the Tietze extensin Lemma 2. Suppse that f; B Int 0 is an embedding. Let H be a neighbrhd f /(Int Bk) in Q and let e. [Q - /(BdBfe)]»(0, ) be cntinuus. Then there exist a neighbrhd V f /(Int B ) in Q and a cntinuus 8: Int B (0, ) such that, if g: Bk» V u f(bk) is an embedding within 8 f f n Int Bk that agrees with f n BdB, then there exists an e-hmtpy h : V U f(b ) U U f(bk) such that h = identity, h \g (Bk) = identity fr all t I, and h l is a retractin f V U f(bk) nt g(bk). Lemma 3. Suppse that f: B IntO, U and e are as in Lemma 2. Then there exist a neighbrhd V f /(int B ) in Q and cntinuus 8: IntB (0, ) and -q: [Q -/(Bd Bk)]-*(0, ) such that, if g: Bfe-> V Uf(Bk) is a PL embedding within 8 f f n IntB that agrees with f n BdB and if X C V is a plyhedrn that r cllapses t g (Int B ), 7ie?2 there exists an e-hmtpy h : V U /(Bd Bk)» U U /(Bd Bk) such that hq = identity, b \X = identity fr all t I, and h is a retractin f V U f(bk) nt X U g(bk). Sketch f prf. Assume initially that 8 and V ate chsen s as t crrespnd t e/2 and U as in Lemma 2. Suppse X C V 77-cllapses t g (Int B ), where g: B» V is PL n IntB and within 8 f / n Int B. Let TV be a small, secnd-derived neighbrhd f X in V. Let h' be the 1-level f an (e/2)-hmtpy h' (t e 7) given by Lemma 2. Think f N as the tplgical mapping cylinder f a map Bd TV X given by the cllapse TV N» X, and let a: Bd TV x I *Nbe such that a(p, 0) = p, cx Bd TV x [O, 1) is a hmemrphism nt TV - X, a(bd TV x Si!) = X, a(p x /) is small fr p Bd TV (say, less than r (p)). Let ß : X»X (7 I) be a strng rj-defrmatin retractin f X nt g (Int B ), with ßQ = 1 and ß,: X g (Int B ) a retractin. Fr 3 and 77 sufficiently small, the maps ' Bd TV: Bd /V-»g(Int Bk) and ß xa( -, 1): Bd TV -* g (int Bk) will be (c/2)-hmtpic via a hmtpy y : Bd TV» g (Int B ) (t I) with yq(p) = ß^(p, 1) and yap) = h'(p). Define h" : TV -» X by

4 88 J.L.BRYANT [August Íy1_2t(a(p, 0)), 0<t<y2 ß2_2t(a(p, 1)), la<t<l (p Bd N), and h; V > X by h(p) p V - N, p N. Then h is a retractin f V nt X, and if 8 and r ate sufficiently small, then h will be the 1-level f the apprpriate hmtpy h (t I) (extended t /(Bd B ) via the identity, f curse). Lemma 4. Suppse that f: B» Int 0 is an embedding (q - k > 4). Let U be a neighbrhd f /(Int Bk) in Q and let e: [Q - /(Bd Bk)] > (0, <*>) be cntinuus. Then there exist a neighbrhd V f /(int B ) in Q and cntinuus functins 8: Int Bk»(0, <*>) and r : [0 - /(Bd Bk)]»(0, ) satisfying the fllwing cnditins: If g: B V Cl f(b ) is an embedding that is PL and within 8 f f n Int B and agrees with f n Bd B, if Y is a plyhedrn in V (embedded as a clsed PL subset f Q /(Bd S )) with q - dim Y > 4, if W is a neighbrhd f g (Int S ) in Q, and if X is a plyhedrn in W such that a - dim X > 3 and XN XÍ1 g (int B ), then there exists an istpy h (t I) f 0 such that (i) h = identity, (ii) h = identity n g(b ) U X and utside f U, (iii) hx(w) 3 Y, (iv) d(h identity) < e n Q - /(Bd Bk) fr each t I. This lemma is prved by using hmtpies btained frm Lemma 2 tgether with standard engulfing arguments. The reader is referred particularly t [l], [ll], and [14] fr the engulfing techniques required here. Lemma 5. Suppse that f:b Int 0 is an embedding be a neighbrhd (q k > 4). Let U f /(Int Bk) in 0 and let e: [0 - /(Bd Bk)}»(0, ) be cntinuus. Then there exist a neighbrhd V f /(int B ) and cntinuus functins i}- [Q - /(Bd Bk)]» (0, ) and 8: Int Bk» (0, ) satisfying the fllwing cnditins : If g: B» V is a PL embedding within 8 f f that is PL n Int B agrees with f n Bd B, if X is a plyhedrn in V (q dim X > 3) that cllapses t g (int B ) via an -q-cllapse, and if Y is a plyhedrn in V (q dim Y > 4), then there exists a plyhedrn Z in U such that and

5 1972] APPROXIMATING EMBEDDINGS OF POLYHEDRA 89 (i) Z D X U Y, (ii) Z N g (Int Bk), and (iii) dim(z -X) < dim Y + 1. Prf. Suppse that /: Bk-* Int Q, U D /(Int Bk), and <r: [0 - /(Bd Bfe)]~» (0, ) are given as in the hypthesis f the lemma. Chse V and 5 crrespnding t U and e/3 as in Lemma 3, and let g: B» Int g be a -apprximatin f /. Let 72 = e/3- Suppse that X and Y satisfy the cnditins in the lemma. Let W be a small, secnd-derived neighbrhd f X in V such that every rth derived subdivisin f W 77-cllapses t X. Frm the cnclusin f Lemma 3 we can btain an istpy h (t I) that is fixed n g (B ) U X and utside U such that h AW) D Y and d(h{, identity) < e/3 n Q - /(Bd Bk) fr each t I. Let Yj = h~l(y), and let Zj = tr(yju X) under the cllapse W >. X >. g (Int Bk). Then Z, = X u tr Y1 and Z jn g (Int Bfe) is a 2r/-cllapse. Let Z = AjiZj) = X UÄjdx Yj) (since h ^X = identity). Then Z ^ g (int B^) and dim(z- X) <dimy + 1. Main lemmas. Suppse that / is a cmplex and that ff is a simplex f /. Let K = St(ff; /), L = Bd a * LK(er; ]), K = \K\ - \L\, and ff = Inter. Dente by ff the barycenter f cr. There is a triangulatin K. f K - Bdcr such that K.>» H., where 77, C K. and \H. = <j. Thus we have tr Z defined fr a subplyhedrn Z f K.. Set k = dim K and assume q - k > 3 thrughtut. Given /, ff, K, and L as abve, let us cnsider \K\ as the cne a * \L\. If x e /C, let [zj, x] dente the segment jining a and x in K. Similarly, let us cnsider Bq as the cne 0 * Bd Bq, with subcne 0 * Bd B' (where j = dim a), and let [0, y] dente the segment jining y Bq t 0 in this cne structure. Let g: a» B7 be a fixed PL hmemrphism with g (a) = 0. We use this ntatin in the fllwing lemmas. Lemma 6. Suppse that f: a *Q is an embedding and e: a * (0, 00) is cntinuus. Then there exists a cntinuus 8: a * (0, 00) such that x, y a and d(f(x), f(y)) < 8(x) imply d(f(z), f(d(z))) < e(z) fr each z [3, x], where 6: [ff, *1 * [ff, y] is the linear hmemrphism satisfying 8(a) = a and 0(x) = y. The prf f Lemma 6 is btained by applying the cntinuity f / and /"" and shall be left fr the reader. The next tw lemmas are successive generalizatins f Lemma 6. Lemma 7. Given f: a~*q and e: *(0, <x) as in Lemma 6, there exist 8, T]-. *(Q, 00) such that if gq, g a» O are mappings with d(g.(x), f(x)) < tj(x) fr each x a, then x, y a and d(g (x), g Ay)) < 8(x) imply d(g0(z), ga^(z))) <e(z) fr each z [5, x].

6 90 J. L. BRYANT [August Lemma 8. Suppse that A and B are cnes ver a each cntaining a as a subcne, that f: *Q is an embedding, and that e: [A Bda] * (0, ) is cntinuus. Then there exist 8: [A - Bd» (0, ) and -q: a >(0, ) satisfying the fllwing prperties: If G0: A» 0 and G.; B» O are mappings with d(g.\a, f\a) < r] (i = 0, 1), then there exist neighbrhds U f a in A and V f in B such that x U, y V, and d(g Q(x), G {(y)) < 8(x) imply d(gq(z), G x($(z))) <((z) fr each z [d,x\. Lemma 9. Suppse f: K~* Q is a clsed embedding with f\k - a PL. Then fr each e: K» (0, ) and each neighbrhd N f a in K, there exists a PL map G; K-* Q such that (9.1) G\K-N = f\ K-N, (9.2) G " is a PL embedding, (9.3) d(f, G)<e, (9.4) S- CAÍ, and (9.5) G is in general psitin. Prf. This is a straightfrward applicatin f Therem 2 (as applied t the embedding f\a: a *Q) and general psitin. Lemma 10. Suppse that j: K 0 is a clsed embedding with f\k a PL. Given e: 0 * (0, ) there exist ~~ a neighbrhd N f a in K and cntinuus functins n: 0 (0, ) and 8: K *(0, ) satisfying the fllwing cnditins: If G: K-* 0 is a PL map satisfying (9.1)-(9.5) f Lemma 9 with \8, N\ replacing \e, N\ and if X. and Z are plyhedra such that (10.1) SGu CX1 = trxj C N, (10.2) Yj = G(Xj) CZj -r/n, G (a), and (10.3) dim[(zj- Yj) ng(k)]<k-j (j > 4), then there exist plyhedra X? and Z? such that (10.4) Xj CX2 = trx2 C K, (10.5) Y2 = G(X2) C Z2 ^. G la), and (10.6) dim[(z2 - Y2) n G( K)] < k - j - 1. Prf. Given e: 0 + (0, ), let U = N (f(a)), and chse 8, ri, and V as in ~ c Lemma 5. Chse N t be a regular neighbrhd f a in K such that N = tt N and f(n) C V. Assume that G: K > Q satisfies (9.1) (9.5) f Lemma 9 with 5 replacing the e f Lemma 9 and that G(N) C V. Suppse that XjCN and Z^CQ satisfy (10.1) (10.3)- Let X2 = tt(g-l(zx)), and let Y 2 = G(X2). Then dim(x2 - X j) = dim(y2 - Y J <k-j+ 1

7 1972] APPROXIMATING EMBEDDINGS OF POLYHEDRA 91 ( <q - 6). Observe that if 5 and 27 are sufficiently small, then G~ (Z ) C N, and s Y2 C V. Nw apply Lemma 5 with Z replacing X and Cl (Y - Y j) replacing Y. This gives us a plyhedrn Z.CO satisfying (i) Zj uci(y2 - y,)cz2, (ii) Z2 f\. G(ff), and (iii) dim(z2-zj) <dim(y2- Yj) + 1 < * Putting Z in general psitin with respect t G (K) (keeping Z \j Y - fixed), we have (since Zj H G( K) C Y2) dim[(z2 - Y2) n G(K)] <dim[(z2 - Zj) O G(K)] < (k - j + 2) + k - q < k '- ; 1,., Lemma 11. Suppse that f: K» Q is a clsed embedding with f\k a PL. Given e: O *(0, 00) and a neighbrhd TV f a in K, there exists 8: K (0, 00) such that, if G: K > Q is a PL map satisfying ÍS, N\ replacing \e, TV}, then there exist plyhedra (11.1) ff USGCX = trx CN, (11.2) G(X) = ZD G(K), and (11.3) Z N, G(a). Prf. Apply Lemma 10 inductively. (9.1) (9-5) f Lemma 9 with X C TV and Z CO satisfying The next lemma is the key lemma f this paper. Its prf depends upn a result f J. Cbb annunced in [7]. The specific frm f Cbb's therem we wish t cnsider is the fllwing. Therem 3 (Cbb [7]). Suppse that L is a subcmplex f a finite k-dimensinal cmplex K and that f is an embedding f \K\ int a PL q-manifld Q (q - k > 3) such that f\ \L\ and f\ \K\ - \L\ are PL. Then fr each e > 0 there exists an istpy h; 0 x I Q x / such that (i) h. = identity, (ii) hj: \K\->Q is PL, (iii) hj\ \L\ =f\ \L\,and (iv) 73 (Ox/)-(/( L )x!0!) is PL. Lemma 12. Suppse that f: \K\» 0 is an embedding with f\k - a PL. Then O ~ fr each e: K» (O, 00) and each neighbrhd TV f a in K, there exists an embedding /': K Q such that (i)/' jl U( K - TV) =/ \L\ u( K -/V), (ii) f'\k is PL, and (iii) d(f(x), f'(x)) <e(x) fr x K. Prf. Given : K» (0, 00), chse 8: K * (0, 00) and 77: a»(0, 00) as in

8 92 h L. BRYANT [August Lemma 8 crrespnding t e =. Let G: K» (Q - f(l)) be a PL map btained frm Lemma 11 with d(f, G) < r n a and let X and Z be the assciated ply hedra having the prperty that a \J Sr C X = trx C N, G (X) = Z Cl G(K), and Z\t,G(). Triangulate K and Q-/( L ) s that G: K > Q - f(\l\) is simplicial, and let U and V be small, secnd-derived neighbrhds f X in K and Z in 0 - f(\l\), respectively, such that G(U') = V' D G( K). Assume that 5 is chsen s that G extends t a map f K int Q (which we shall still call G) that agrees with /n L. Let H: Bq 0 be an embedding H\(Bq-BàBj) H(0) = G (a), and Hg = G. is PL (/ = dim-), such that Let U be the clsure (in K ) f a small, secnd-derived neighbrhd f a in K lying in the clsure f N in K, and let V be the clsure in B f a small, secnd-derived neighbrhd f Int B1 in Bq Bd B', bth chsen s as t satisfy the cnclusin f Lemma 8 with Ç, replacing e. We shall make the further assumptin that U is starlike in K frm a, V is starlike in Bq frm 0, and H(V) O G( K ) CG(A/u a). that Since X = tr X N a, there exists a small hmemrphism F.: \K\ \K\ such Fj K is PL, F j L U ( K - N) u a = identity, Fj(l/)n K= U'. It r is sufficiently small and if X is suitably chsen, then we can guarantee that d(g(x), GFj(x)) < S(x)/2 fr x K. Similarly, there exists a small hmemrphism F2: 0 0 such that F2 /( L ) U G(a) = identity, and F2\Q-f(\L\) is PL, F2H(V)-f(\L\) = V', and d(f2g(x), G(x)) < C(x)/3 fr x K. Thus, GFj(Bd 17) = G( K ) OF2/i(Bd V) is hmemrphic t L. Let L' = H'^-HGilKDn F2H(Bd V)) C Bd V, and let K' = L' * 0 C V C Bq. Observe that U = a * Bd U and GF j(bd U) = F HlL'). Define F3: (7-» V by Fj(x) = H~^F~ lgf%1x) if x e Bd (7 and F [5, x] = Ö: [â, x]» [0, F Ax)] is the linear hmemrphism described previusly. Nw define G': K -> Q by G'(x) = F2HFA,x) it x U, G'(x) = GFj(x) if* $17.

9 1972] APPROXIMATING EMBEDDINGS OF POLYHEDRA 93 Then, assuming r <,, G is an embedding that clsely apprximates /. Let H V * Bq be a hmemrphism such that H Aß1 = identity and Hr\Bq - Bd By is PL. Then G ^ U -» Bq, defined by G, = H1H-1F~1G', prper embedding f the cne â * Bd 77 int Bq = 0 * Bd Bq such that G. c7 is PL and G X\U - a is PL. Cnsider GjBd U: Bd U -» Bd Bq. GjBd Bj is PL and GjBd U - Bd Bj is PL. Hence, by Therem 3, there exists a small istpy k (t I) f Bd Bq suchthat zeq = identity, jgjíbd Í7) is a subplyhedrn f BdBq, &1G, Bda = GjBd is a a, and (Bd Bq x /) - (Bd B> x \0\) is PL. Cpy this istpy in a small PL cllar W f Bd Bq in B«. Let U, be (7 minus a small cllar f Bd 77 in (7 and btain a PL embedding G. U. * Bq - W by cning ver k jg, (Bd U) C Bd W. Extend G2:U1^Bq-W t G : (7 -» Bq by sending (7 - U r t &(Bd 77 x /) C W in a natural way. Then G,: U * Bq is a prper embedding, G j 77 - Bd ff is PL, G, Bd 77 = G, Bd 77, and G, is a clsed apprximatin t Gj. Thus if G is sufficiently clse t G,, then the embedding / : \K\ > 0 defined by /'(*)= F2HH~lGAx) if x 77, /'U) = G'(x) if x ^ 7/ is PL n K, is an apprximatin f / n K, and agrees with /n L u ( K - TV). Prf f Therem 1. Suppse that /: P Q is a tplgical embedding. Let / be a triangulatin f P. Apply Therem 2 t the pen p-dimensinal simplexes f / t btain an embedding /.: P 0 that apprximates / and satisfies the tw prperties: /J \Jp~l\ = f\ \JP~ l\ and /1 ( / /p-1 ) is PL. Nw prceed inductively using Lemma 12. Appendix. Observe that in the prf f Lemma 12 we applied Therem 3 in dimensin q 1 t a cmpact plyhedrn t get Lemma 12 in dimensin q. In view f the fact that a prf f Cbb's therem [7] (Therem 3) has nt appeared as yet, we shall utline a prf f Therem 3 in dimensin q, assuming Therem 1 in dimensin q. (The idea appears t be essentially the same as that indicated in [7l.) We shall require an istpy therem and an engulfing therem due t Edwards [9] and Bryant-Seebeck [3], respectively. Istpy Therem (Edwards [9]). Suppse that L is a subcmplex f a finite k-dimensinal cmplex K and that f: \K\ 0 (q k > 3) is a tplgical embedding such that f\ \L\ is PL. Then, given e > 0, there exists 8 > 0 such that if g.: \K\ * 0 (i' = 1, 2) is a PL embedding such that g \ \L\ = f\ \L\ and dig-, f) < 8, then g. and g2 are PL e-ambient istpic in 0 via an istpy that is fixed utside TV (/( K - L )).

10 94 J. L. BRYANT [August Engulfing Therem (Bryant-Seebeck [3]). Suppse that X is a cmpact k- dimensinal ANR in Q (a k > 3, q > 5) such that Q - X is 1-ULC (unifrmly lcally simply cnnected) and A C X is clsed in X. Then, given e > 0, there exists 8 > 0 such that if g: X * 0 is an embedding with d(g, 1) < 8 and g\a = 1 and if U is an pen set in Q cntaining g(x), then there exists a PL e-istpy h (t I) f 0 that is fixed utside N (X - A) such that hq = 1 and h A\V) 3 X. Mrever, if X is a plyhedrn and g: X * 0 is PL then ht (t I) can be chsen s as t fix g (X). (This statement is slightly strnger than the statement f Therem 2.1 f [3I, but its prf is cntained implicitly therein. Indeed, it has been pinted ut t the authr that the abve refinement f Therem 2.1 f [3] is actually needed t prve sme f the therems f [3]. The "mrever" part was btained by Cernavskir in [5]-) Prf f Therem 3 in dimensin a, assuming Therem 1 in dimensin a. Suppse that L is a subcmplex f a finite ^-cmplex K and that /: \K\» 0 (a - k > 3) is an embedding such that f\ \L\ is PL and f\ \K\ - \L\ is PL. (We shall assume that q > 5, since bth Therem 1 and Therem 3 are well knwn in dimensin a = 4 [4].) Let N, N-, be a sequence f PL neighbrhds f \L\ in \K\ such that A'.,, C Int/V. and 0 -, N.= \L\, and let P.= \L\ UC1( K -N), i= 1, 2,.... Nw prceed just as in the prf f Therem 2 f [3]. Start with a PL apprximatin /': K» 0 f f that agrees with / n P. Use the Istpy Therem and Engulfing Therem t cnstruct sequences f small PL pushes {G.P and G;.! =j f (0, f(\k\)) satisfying (1) G = lim. G. ' G, and G = lim. G. * G, are small pushes f 0, (2) Gf = G'f, (3) g...gj/ p. = g;...g;/' p., (4) the PL istpy f G. +, (respectively, G^+j) t the identity is fixed n G. G j/( X - N.) (respectively, G! G'.f'(\K\ N.)). The hmemrphism G'G~ is istpic t the identity via the apprpriate istpy f Q. REFERENCES 1. R. H. Bing, Radial engulfing, Cnference n the Tplgy f Maniflds (Michigan State Univ., E. Lansing, Mich., 1967), Prindle, Weber and Schmidt, Bstn, Mass., 1968, pp MR 38 # H. W. Berkwitz, Piecewise linear apprximatins f hmemrphisms (preprint).

11 1972] APPROXIMATING EMBEDDINGS OF POLYHEDRA J. L. Bryant and C. L. Seebeck III, Lcally nice embeddings f plyhedra, Quart. J. Math. Oxfrd Ser. (2) 19 (1968), MR 38 # J. C. Cantrell, n-frames in euclidean k-space, Prc. Amer. Math. Sc. 15 (1964), MR 29 # A. V. Cernavskii, Tplgical embeddings f maniflds, Dkl. Akad. Nauk SSSR 187 (1969), = Sviet Math. Dkl. 10 (1969), MR 41 # , Piecewise linear apprximatins f embeddings f cells and spheres in cdimensins higher than tw, Mat. Sb. 80 (122) (1969), = Math. USSR Sb. 9 (1969), MR 40 # J. I. Cbb, Taming almst PL embeddings, Ntices Amer. Math. Sc. 15 (1968), 371. Abstract #68T R. Cnnelly, Unkntting clse embeddings f plyhedra in cdimensin greater than tw, Ph. D. Dissertatin, University f Michigan, Ann Arbr, Mich., R. D. Edwards, The equivalence f clse piecewise-linear embeddings (t appear). 10. R. J. Miller, Apprximating cdimensin three embeddings, Ann. f Math, (t appear). 11. C. L. Seebeck III, Cllaring an (n l)-manifld in an n-manifld, Trans. Amer. Math. Sc. 148 (1970), MR 41 # A. Sctt, Infinite regular neighburhds, J. Lndn Math. Sc. 42 (1967), MR 35 # C. Weber, Plngements de plyèdres dans le dmaine métastable, Cmment Math. Helv. 42 (1967), MR 38 # P. Wright, Radial engulfing in cdimensin three, Duke Math. J. 38 (1971), E. C. Zeeman, Unkntting cmbinatrial balls, Ann. f Math. (2) 78 (1963), MR 28 # , Seminar n cmbinatrial tplgy, Inst. Hautes Etudes Sei., Paris, 1963 (mimegraph ntes) DEPARTMENT OF MATHEMATICS, FLORIDA STATE UNIVERSITY, TALLAHASSEE, FLORIDA