b--._ < Topology and its Applications 11 (1980) 17-29 Company @ North-Holland Publishing
METRIC RETRACTION SPACES AND THE HOMOTOPY
Laurence BOXER PA 18104' USA Department of Mathemarics, Muhlenberg College' Allentown'
Received 8 JulY 1978 Revised 8 December 1978 and 20 July 1979
of retractions and compactu m.Let gl(x).and "t1x)be the spaces Let X be a finite-dimensional (=sup-metric) topology' rerractions oi x respectively,.with the compact-open non_deformation of X with topology induced by the 2f be the space of totputt ANR subs"ts Let .non-ttp'v AN R's in X rhat are retracts of x ..,ri.. r_., n r u. ,i.'i"irprlJoi 2 f; .onsisring of the il;;l";; that if X is an ANR and Ao e Rf, is simply-connected for ln t 1. w-. rh;* we show that J[(s-) almost all j" rerraction ro of X onto Ar) there are, for ,nJi,"n,,__ i, = ," , f i, ,"d only if tor every : 16 in 9?(X)"Y'/e show that if X is an ANR' then retractions 1 of X onto Ai such that limi-o | i, ;plittth^*inf wtprovethat fit(M)isloc.allv connectedif M thelocalconnectednessof ;(Xl when is not show how some of our results weaken X is a closed surface. We give examples to assumed to be an ANR.
AMS (MOS) Subj. Class. (19?0): Primarv 54820' 54C351 S..ona*r 54C1i, 54F40, 55D99, 57A05, 57A10' 57A15' FANR metric f undamental retraction deformation retraction homotopy sup-metrlc Hausdorft metric movable finite-dimensionalcompactum homotopydomination
1. Introduction
space El(x) of retractions of a Among the ideas introduced by Borsuk are the [4] (:sup-metric' if X is a compactum) topological space X with the compact-open ANR subsets of a finite- topology and the space 2f [3] of non-empty compact induced by the homotopy metric' In this dimensional compactum X with topology gt(X) and 2f' these results were paper we examine some relations between - in to study a component of 2f motivated by the observation that methods used [8] canbeemployedtoobtaincorrespondingtheoremsforacomponentof4(S.)(see Section 3). t7 18 L. Boxer / Relraction spaces and the homotopy metric
2. Preliminaries
All ANR's considered in this paper will be compact. By map we will mean 3 continuous function. If / is a map, Im(/) will mean the image of /. If e ) 0, an s-map (X will mean amap f whose domain and range are subsets of a metric space d) such that d(x,f(x))
d'(f, g) = sup{d(/(Y), s(Y)) lY e Y}. positive Let (X, d)be afinite-dimensional compactum, A e2{, and let 6 and e be numbers. We will use s(A, 5, e) to abbreviate "every subset of A with diameter less than 6 is contractible to a point in a subset of A with diameter less than e'" The topology of.2{ is induced by the homotopy metric dnand may be described as follows:
2.1. Theorem [3, p. 197]. Limr- * dn(A,,Ao) = 0 if and only if (a) lim;-- dg(A,,Ao) :0 (where dg is the well-known Hausdorff metric), and e)' (b) for euery €>O there is c 6 > O such that for all i, s(Ai,6,
We will use the following:
Let Then thert 2.2. Theorem [3, Theorem on p. 196 and Lemma on p. 188]. A e2{. isaneighborhood0llofAin2{suchthatforeueryt>0thereisaS>OsuchthatYe0l' implies there is a retraction of Nt(Y) onto Y that is an E-map'
with th, 2.3. Theorem [3, p. 2OO]. Let A e2I.-. Then the collecrion of members of 2{ same homotopy type as A is open (and closed) in 2{'
3. Global Properties of 'M(S-)
path-connected for > 1. In thi wagner has shown [10, 3.9, p. 42] that "^r(s- ) is 'n section we prove that "M(S-) is simply-connected for m>2'
3.1. Lemma. Let f:I-.M(S^) be a map, m>0. Let ues^\Im(/(0)), Dl L. Boxer I Retraction spaces and the homotopy metric l9
S-\Im(/(l)). Then there is a map g:(1,0,1)+(S-,u,u) such that for all t€1, g(t)e Im(/(t)).
Proof. Since f is compact, there exist 0:/o(/r("'1tp=1, points u; and open sets Br c S- \Im(/(ri)) such that u1e 81 fuo= Lt, Ltk: u), and connected neighbor- hoods Ui of 4 in l with (J1a(J1*r*0fori p; e A B e A a B be such that the subarcs ulpi and q1t1 1 of Ai lie in B Let i a 1, e i i 1 + r - 1 and B;+r, respectively. The map g is taken to be any map satisfying: glU,, cil maps ([ti, cif, ti, c) onto tu1pi, u1 p); Sllcj, di) maps ([c;, d1l, c1 d1) onto (pfii, pi, ei)| glldl t1*Jmaps ([d;, ti*t7, di' /i*r) onto @tuftr, q1 u1*). Since I{ is connected, our choices of c; and p1 imply that if r e [4, c;], then sU)eIm(f(t)). Similarly, g(t)elm(/(t)) if. teldl, r;*rl' our choices of Ti, c1, and d1 imply g(r)eIm(/(r)) if re[ci, d]. Thus g:(I,0,1)+(S-, u, u) satisfies: for all red g(.t)eIm(f(t)). This completes the proof. For the remainder of this section, xo will be a basepoint of .S-, and ,\ : S- - lf(S-) will be the embedding sending x € ,S^ to the retraction of S- whose image is {'}. 3.2. Corollary. Let (P,p) be a pointed one-dimensional compact polyhedron. Let f:(P,p)-'("f(S-), I(xd)beamap.Thenthereisamapg:(P,p)- (S^, -xo) suchthat for all y e P, g(y) e Im(/(y)). Proof. We may assume P is connected and that p is a vertex of P.Let S( p) : -xo. We proceed inductively on the one-simplexes of P. Let o be a one-simplex of P with endpoints a and b such that ,: g(.a) has been defined. If g(b) has not already been defined, choose u : C@) € S- \ Im(/(r)). Apply 3.1 to obtain glo. Thus we obtain g:(P, p) - (.S', -xo) such that for all y e P, S(y) e Im(/(y1;. 3.3. Theorem. Let f :(P,p)+("4f(S-),4(ru)) be a nap, where P is a one' dintensional compact polyhedron. Then f is homotopic rel p to a map whose image /ies ln A (S- ). Proof. By 3.2 there is a map g:(P,p)-(,S^,'-xo) such that g(y)eIm(/(y11 for all yeP. 20 L. Boxer I Retraction spaces and the homotopy metric For all x € S- let h': (S- \{x})x 1+ S- be given by (l tx h'\z,tj=;;--- t)z - ll(r-t)z-txll' h'(2,0)= z and Since the denominator is never 0, f is well-defined. observe < < 1' h'(2,l)= -; for all z eS-, and /ri is an embedding for 0 I For all (y, t)e P x I we define . tftl(') if o Hence F(y,t)e "'tl(S-) for all (y,t)ePxL is clear that F is continuous in Observe F(y,0)=/(y) and r('' t):'t(-g(y))'It lm[lrf(t)] lies in an (y, r) for 0 denote the collection of For pointed spaces (A,a) and (B,b),let [(A, a),(B,b)] (B' have: pointed homotopy classes of maps from (A' a) to b)' We 3.4. Corollary. If (P,p) is as aboue, then the function p)' ('/f(S-)' l (ro))] f x : [(P, p), (S-, xo)]- [(P, suriectiue' giuen by *([/]) : [A " .f] is Froof. This is an immediate consequence of 3'3' Hence Jf(S-) is simply- 3.5. Theorem. A*:I1r('ll(S-),i(xo)) is a surjection' connected for m>I' Proof. Take P: 51 in 3'4' 27 L. Boxer I Retraction spaces and the homotopy metric 4. Convergence in fit(X) and R{ d., charac- we begin this section with results concerning the metric of continuity, (for all terized in [1] by: d"(A,B) 4.1.Theorem[1,p.t9ol.LetXbeafinite-dimensionalcompactumandletA' B e2{. Then ds(A, B)< d.(A, B)< dn(A, B). 4.2. Proposilion. Let X be a metric space, Iimi-ari:roin Ot(X) such that Ai= Im(ri) is compact for i :0, 1' 2, . . ' . Then limi-* d"(Ai, Ad=0' > proof. It is easily seen that given e > 0, there is an n such that i n implies r,lArr:Au-Ai and rolAi:Ai+Ao ar€ e-maps' 4.3. Corollary. (Jnderthehypothesesof 4.2,if inadditionAoeANR thenforalmost all i, Ai dominates Ao homotopically. Proof. This is a direct consequence of 4.2 and [9, 3'14, p' 852]' our For the rest of this paper, we assume X is a finite-dimensional compactum' main result relating fit(X) and Rf is the following: 4.4. Theorem. Let Ao e Rf. Consider the following statementsi (a) I-imi-- dn(Ai, Ad=0. (b) There is a positirte integer N such that i>N implies e'eR,x' and for euery that retraction r,r of X onto A,r there are retractions ri Q>N) of X onto Ai such (b) limi-*ri : rs in nq).Then (a) implies (b). If X e ANR, then implies \a)' Proof.Suppose(a).Lete>0.LetrobearetractionofXontoAn.Wemustshow 16) { e' that for almost all i there are retractions 11 of X onto Ai such that d'(r;, regard-X as Since dimX<€, there is a positive integer m such that we may in 2{-. By embedded in the m-cube .I^. Hence we may regard {A,}Ln as a sequence ki of 2.2,thereexist a positive integer No, a positive 6 with 5 Hence d'(ri, ro) < e. This completes the proof that (a) implies (b). Let X €ANR. To show (b) implies (a), suppose for i =0 and i >N there are retractions ri of X onto Ai such that limi-- d"(ri, ri = 0. Given e > 0, there is an N,>N such that for i >N1 and for all x€As and all leAi, d(x, t(x)) = d(ro?), rr(x)) < e and d(y, ruOD: d(ri!), ru(y)) < e. Hence limi-- ds(At, Au) :0. Let e)0. Let 4>0 be such that if {x,y}cX and d(x,y)<4 then d(ro?), ro(y))< *e. Since X is an ANR, there is a 6 > 0 such that s(X 6, 4)' Let n be such that i > rz implies d"(ri, ri <*e +*e +1, :lr, f.or i> n. Let BcA; such that diam B<6. There is a contraction lr: BxI ->X of B to a point such that diam h(B x I)< 4' Therefore ri o h : B x I + X is a contraction of B to a point with image in Ai such that diam(r, o h(B x I)) 4.5. Corollary. Rf is open in 2{. Proof. Let Aenf;. By 4.4,any member of 2f sufficiently close to A must be in Rf' The assertion follows. 4.6. Corolla ry. If X e ANR, then the function p : {t (X) -> R{ giuen by p (r) = Im(r) is a conlinuous open surjection. 0lI gl(X), Proof . It is clear from 4.4 that p is a continuous surjection. Let be open in let A e p(,U), and let {Ai}3r c nf; be a sequence such that lim'-- Ai = A.It is clear from 4.4 that for almost all i, Aiep(0ll).It follows that p(oll) is open in Rf. 4.7. Corollary. If Xisan ANRandri -'>i-qriln(X),thenforalmostalli,lm(ri)has the homoropy type of lm(r). Proof. This is an immediate consequence of.4.4 and2-3. We present a series of results to show how "(b) implies (a)" of 4.4 is weakened if X is not an ANR. Recall that a compact subset Y of the Hilbert cube Q is a L. Boxer Retraction / spaces and the homotopy metric 23 fundamenal absolute neighborhoad retract (FANR) if and only if there is a compact ANRZ in y O that contains and a fundamental retraction f ={f,,, Z, y}q.() VIII(1.4), p.25al. f6, 4.8. Theorem. Let YeFANR and let limi-- r;=r11 i1 gt(y), with Ai:Im(4)e ANR i 0, 1, /or = 2'. . . . Then for armost ail i, A1 and As haue the same homotopy type. Proof. we will show that for armost ar i, r,lArand relA; are homotopy inverses. From the proof of 4. 1, limr-- d" (Ida., rs. r,l Ao1= o. Sin"" Ao e ANR, it follows that for almost all i, Ida,, - r11o r;1A6. we may assume Y c and Q that Z and/ are as above. Since A; e ANR, for alr i there is a neighborhood y Ll of in e such that 4 extends to a map ri: (J, -+ l,.There is a positive integer k; such that f1,,(Z)c. U,. Since close maps into an ANR are homotopic, for armost ail i there is a map Hi: Ai x I '+ Z such that HiG,0): r and Hie, l): rio r11(x) for all x e A;.Thus, for almost all i we can --> define Gi:A;x J Ai by Gie, t) : rlo fk o Hi(x, t). It is easily seen that for re{0, 1}, G1(x, t) = Hi(x,r). The p.ooi is' complete. Thus under the hypotheses of 4.8, limi-- d"(Ai, Ao): 0 (by 4.2) andfor almost all i, Ai and Ae have the same homotopy type. Even for so nice a finite-dimensional FANR Y as a contractible compactum, we cannot improve this to convergence of {Ar}Et to As in the homotopy metric, as the following shows: 4.9. Example. Let y : 0 < {(r,d) I r t l, 0 = 0 or 0 : r / n for n = l, 2,3, . . .} in polar coordinates. There is a conuergent sequence ti + i-qro in g(y) with Ai: Im(ri) e ANR for i =0, 7,2,.. . such that Asllimi-* Ai in 2{. Proof. Let rg(r, 0') = (r,0); and for i > 0 let n/i) if 0>r/i: ri1, 0) - l(r', ( (r, 0) if. 0 The notion of mouability generalizes that of FANR: an FANR must be movable, )ut not conversely [6]' we give an example to show that the conclusion of 4.g fails if ;he assumption that y is an FANR is weakened so that y is only assumed movable. Let E be the "Hawaiian earring" LJl, c,, where c is the circle in the plane R2 vith center (l i" and / U radius 1/ i. Letp d"not" the origin in R2. Since E is a plane 24 L. Boxer I Retraction spaces and the homotopy melric compactum, it is movable [6, V6.1, p. 160], but since Rt\E has infinitely many components, ,Ee FANR [6, VIII9.2, p.267). 4.10. Example. There is a conuergent sequence ti + i,ero in A@) such that Im(ri) e ANR /or i = 0, L, 2,. . ., but i * A implies Im(r) does not hane the homotopy type of Im(ro). Proof. It is clear that the following maps satisfy the claim: r6(y):p forallxeE; andfori>0, (P if -teE\C; r,(x) = {' [x if xeCi. 5. Local connectedness We continue to assume X is a finite-dimensional compactum. A question of Borsuk14,2.S,p.lg8lsuggeststheproblemof findingforwhichXthespace 9?.(X)is locally connected. Ball and Ford have asked [1., 6.5, p. 491:f.or which X is 2f;locally connected? We explore some relations between these questions. 5.1. Theore m. Let X be an ANR. ry A(X) is locally connected, then R{ is locally connected. Proof. Let A eRf, let 1tt be a neighborhood of A in Rf, and let p be as in 4.6. Let r e p-t(olt) such that p(r)= A.lf Et(X) is locally connected, then there is a connected neighborhood T of. r such that Tcp-l(lt). Since p is an open map, p(7) is a connected neighborhood of A that is contained in 41. In the following, M will denote a closed surface, i.e., a compact metric 2-manifold without boundary. It is known l7 ,4.4lthat 2f is an absolute neighborhood retract for the class of metrizable spaces. We use some methods from the proof of that theorem to show the following: 5.2. Theorem. A(M) is locally connected. Proof. Let rse El(M). Let % be a neighborhood of 16 in nM). It follows from 4.7 that IdM is an isolated point of n(M), so we may assume ro*ldu. Let A: Im(ro). Let e ) 0 be such that {r eA(M)ld'(r, ro)< e}c 6lt. By 2.Z,there is a positive 6 < 1s and a neighborho od Y of A in R f such that Y e Y implies there is a (}e)-retraction of A/r(y) onto Y. L. Boxer / Retraction spaces and the homotopy metric Zs A well-known consequence of [5, IV3.1, p. 86] is that close maps into an ANR may joined be by a small homotopy: thus there is a positive 4 < 6 such that if /, g: M -+ M are maps y satisfying d"(f, s) < 3 a and /l = sly for some closed y in M,then there is a map K: M x I -+ M such that Ko= f, Kt= g,and for all t e I, K|y :11y and d'(K,, /)<4. p gl(M)ld"(r, Let be as in 4.6. Then 7/1: p(r e ro1< 4)) is a neighborhood of A in yt, nf. BV L7 , 3.2 and 4.21, there exist N e a neighborh ood V2 of A in Rf with 7/zc 'l/ n V1, and a sequence of positive numbers s; Such that l[, e; .-l,satisfying: Y e7/2 implies there is a sequence {A,}3, in Z with Ar : N, y: OE, Ai, and for all c,Int r; Ai*r A; there are finitely many components of Ai \Ai*r, and if p is the closure in M ofsuch a component then there is a homeomorphism h: p -Sr x / with diam &-1({z} x 1) < er for all z e Sr. 6112= gt(M)ld"(r, Let p-l(7/) o{r e ro)< 4}. clearly %2is aneighborhood of 16 in gt(M) with %zc %. Fix r e 6il2. Let Y = Im(r). Let {Ar}Er be a sequence for Y as described above. Fix i and let p and h:P->S1x1 be as described above. We may assume lr-t(St x {1}) c Bd Ai, h-t(S'x {0}) c Bd Ai+r. Let a1:S1X7+Sr,az:StxI->I be projection maps. For teI, define h,:P-+Pby h,(x):[:-'r,. h(x), 1-t) if a2oft(y];>t-t; if a2o ft(2i1< 1 - t. Bychoiceof ft,eachft,isans;-ffiop.Clearly h,egl(p),ho=Idp,and/rrretractspto /r-115t x {0}) c Bd Ai*r. Apply the construction to each component of A; \ A,*r and extend for all r e r via the identity map on Int Ai*r to obtain a continuous family of e;-retractions {ellr e /} of Ai such that Qb is the identity map and Im(ei A,*r. : ) = Let 0 ts1 t11 tz{. . ., limi*- ti = 1. For 0< t< 1, define/,:N-N by,fo= Oi = Idrv;,f, = Q|o f,,_,if i>t and /i-1< ts fi, where ,:(t-t,_r)/(t,-Ii_r). Since Qi of,,_,=f,,:Ob*t "f,, fq is well-defined, and it is easily seen that the family{tl0 By choice of N, there exists F e q.(M) with Im(F) : N and d"(rs, F) < 4" Define G:I-A@[)by if 0 It is easily seen that G is well-defined and continuous, with Im(G(0)) : Im(gr " F): A, Im(G(i)) : Im(/' " F) = Y, and d"(G(t),ro) d"(G(l), r) < d'(G(1) , rn)+ d"(rs, r) <2n + n :3n. Bychoiceof 4,therearemaps H:M xJ ->M andK tM xI -+Msuchthat.F/o: ro, I/r : G(0), Ko: G(1), Kr= r, and for all t e I, H,lA: Ida, Kl Y =ldy, d'(ro, H,)< 5, and d'(r, K) < 6. By choice of 6, there are (le)-retractions a:No(A)+A and €:Ns(Y)-+Y. Observe that for all t e I, q o H, and B o K, areretractions of M whose images are A and Y, respectively. Further, we have a" Ho= to, d" Hr: G(0), Po Ko: G(1), and FoKt:r. We define T:I ->91(M)by (oo Hr, if 0 We will be done when we show d"(r6, < e for all t e .L For 0< r<1, let u = 3t.We have "(t)) d'(ro, T(t))= d'(ro, ao Hu\< d"(ro, H,)+ d"(H,, a" Hu)< 6 +le < e" For 4< r For 3< /< 1, let w :3t -2. We have d'(ro, T(t)): d"(ro, B' K*) < d'(to, r) + d"(r, K*) + d'(K*' I " K*) <4+6+|e This completes the proof. The proof of 5.2 suggests the following: 5.3. Question. Is the conuerse of 5.1 true? .,1 L. Boxer / Retraction spaces and the homotopy metric 6. Local connectedness: X not an ANR We continue to assume X is a finite-dimensional compactum, but we will not assume in this section that X is an ANR. We obtain fairly mild necessity conditions for 2( and CIt(X) to be locally connected. We give an example to show the importance of the assumption that X is an ANR in 5.3. We willuse the following: 6.1. Lemma 16,2.1). If U is open in X, then 1V ez{lY c U} is open in 2{. 6.2. Theorem. (a) tf 2{ is locally connected then X is tocalty connected. (b) If g1(X) is locally connected and X is connected, then X is tocally connected. Proof. Recall that in a complete metric space, local connectedness and local arcwise connectedness are equivalent. That 2f is complete was shown in [3, Coroll ary 4,p. 1981. Since Xx is complete and fr.(X) is closed in Xx, fr,(X) is complete. Thus we concern ourselves with local arcwise connectedness. If X is not locally connected at p e X,there is a neighborhood V of p in X and a sequence xi + i-ap in V such that for all i, p and.r1 or€ in disjoint open sets V; and lI1 respectively, where V = Vi v Wi. By 6.1, the following are open in 2f: V : {y ez{ly c VI; "t/i = {y eZ{ly c. Vi}) Wi = {Y ez{ly c W}. Also, the following sets are open in {?,(X): @ : {r e A6) ltm(r). yi gi : {r e gt(X) lIm1rl . y,1t : 9i {r e 9t (X) | Imlr;. 1y1. Suppose there is a map f :I-->V such that/(O):{p} and/(1):{x'}. Since (by 2.3) each /(t) must be connected, Im(/)c Tiv"l,lri. Since {p}eTi and {xi}e"Il/1, we conclude Im(/) is disconnected. This is impossible. Since every neighborhood of {p} in 2f contains some {r,},2{ is not locally connected at {p}. Let rs, rteq(X) satisfy Im(ro) = {p}, Im(ri)= {x;}. Suppose there is a map g: I - Q such that S(0) = ro, g(1): ft, and suppose X is connected. Then each Im(g(r)) is connected, so Im(g) c 9i v g;. Since roe Qi and r; e 9;, we conclude Im(S) is dis- connected. This is impossible. Since every neighborhood of re in Q(X) contains some ri, &(X) is not locally connected at rs. In the following, E is the Hawaiian earring and {C}L,, {rr}Lo, and p are as in 4.10. The next two theorems give a negative answer to 5.3 when the assumption that X is an ANR is omitted. 6.3. Theorem. A@) is not locally connected. 28 L. Boxer / Retraction spaces and th. homo,op\ m.ni Proof. In view of 4. 1 0, it suffices to show that for i > 0 rhere is no parh in fr (E) from r1 to ro. Suppose forsome i >0 there were a map f:I - j-rEr such rhat/(0)=ri and f(1)=re. Then the map h:cixI +c; defined by. Irrr. tt=.t,./rrr(.r) would be a homotopy from Idc, to a constant map. Since rhe idenrirl map on a l-sphere is not nullhomotopic, we have a contradiction, and the asserrion follo*s. 6.4. Theorem. 2tn is locally connected. Proof. By 12,2.4, p.2lll, it suffices to show the open and crosed subspace g ot 2F consisting of the connected members of 2f; is localll conn€r-r€d. Let A e G. If. p e A, then for some i, 6. 1 implies oll : Iy e € i l, C \ { p}} is a neighborhood 6lt = of A. Since c ARF, it follows from [7, 2.9] that f is locally- connecred at A. Suppose p e A and let % bea neighborhood of A in 6. 81. [], i. l, p. 43] there is a neighborhood Tr of A andan 6 > 0 such that y e l-1 irnplies dhtr', A) < t l2n and if. q:Y -> E is an e-embedding then q(Y)eott. There is a neighborhood y r2of A such that if ye r 3, then for ail i, e c if and only if Ci c A. There is a positive integer N and a neighborhood l'3 of A such that if Y e 1/3then Y ci e T t.There is a neighborhood that it Y e "t/+ ^ U:r I n of A such and i References [1] B.J. Ball and J. Ford, Spaces of ANR's, Fund. Math. 77 (tg72l 3349. [2] B.J. Ball and J. Ford, Spaces of ANR'S. II, Fund. Math. 78 (1973) 209-216. [3] K. Borsuk, On some metrizations of the hyperspace of compacr sets, Fund. Math.41 (1954) 168-202. [4] K. Borsuk, Concerning the set of retractions, Colloquium Mathemaricum lg (1967) lg7-201. [5] K. Borsuk, Theory of Retracts (Polish Scientific Publ., Warsaw 1967). [6] K. Borsuk, Theory of Shape (Polish Scientific Publ., Warsaw 1975). _ L. Boxer Retruction / spaces and the homotopy metric Zg ANR's or a closed surrace, pacific f]l i. 33Xii. :fr}"".."".t r. Math. 7e 1s78)47_68. ;J 1" 1 H::T::i'_::t"T il: lt *fi : : IH ill{i**:li; r:,. p. , i1 .1 iru '""m . n S c i or the space or retractions or the ::. ] i!1'I:tffl:itfil,:L:tfftres two-sphere and the annurus, ph.D. or retractions or the 2-sphere n#fT,X:ll;.'0"" and the annurus, rrans. Amer. tt":]l] Math. Soc. 158 with apprications to the toporosv il;H:3:::#i,T;Xl':t;::!:1" or 2-manirords, rrans. Amer f