proc. london math. soc. 1953 gugenheim 29 53

8/13/2019 Proc. London Math. Soc. 1953 Gugenheim 29 53

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PIECEWISE LINEAR ISOTOPY AND EMBEDDINGOF ELEMENTS AND SPH ERES I)By V. K. A. M. GUGENHEIM[Received 15 October 1951.Read 15 November 1951]

I n t r o d u c t i o nTHIS paper deals with the embedding, mostly of elements and spheres, inpolyhedra l manifolds by piecewise linear hom eom orphism s. The questionsstudied are mostly 'local' in character; by this we me an th a t if embedd ingis in an w-manifold, the properties with which we are concerned assertthemselves if we restrict attention to the w-elements of the n-manifoldwhich contain a given point.We use some of the combinatorial results and methods due to M. H. A.Newman (1,2) and developed by J. W. Alexander (3) and J. H. C. White-head (4, 5). Otherwise our methods are geometric. Some similar questionswere studied b yH .Schubert(7),who discovered th e semigroup here deno tedby (1, 3) and developed its algebraic str uc tur e in some deta il. I am g rea tlyindeb ted to his paper. There is some very slight overlap with a paper byW . Graeub (12), a copy of which reached me when this paper w as co mp leted;I have indicated the lemmas in question in the tex t; G raeub also proves t heresu lt [2, 3] = 0 by a method sim ilar to th e proof of Theo rem 8 below.For publication the paper has been divided into two parts: Part II willapp ear late r. The exposition is as follows: The principal results of the paperare stated, with all necessary definitions, in sections 1 and 7, the latterbeing the first section of Part II; the proof of these results occupies therem ainder of P ar ts I an d II respectively. A very short r6sume* of th e ma inresults of Part II is printed at the end of Part I.I t is a pleasure to express my g ratitu de to Professor J. H . C. Whitehea d,who was my supervisor. He suggested the subject, and ha s given me a verylarge amount of help and advice at all stages of work.

1. The main definitions an d results of Pa rt I of the pape r are st ated .1.1. Rqwill stand, throughout, for ^-dimensional metric Euclidean space.By 'simplex' we shall mean 'closed Euclidean simplex ', by 'comp lex', 'recti-linear closed locally finite simplicial complex of some Euclidean spac e'. Le tK be a complex; we denote by \K \ the point set covered by the simplexesoiK; such a po int set will be called a polyhedron, a ndK a partition of thepolyhedron. If K is a homogeneous complex, R. will deno te its mod 2Proc. LondonMath.Soc. 3) 3 1953)


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30 V. K. A. M. GU GE NH EIMboundary; and if P = \K\, we shall write P = \R \ and IP = PP.Clearly the point sets P and IP , the boundary and interior of P, areindependent of the partitions used in their definition.Polyhedra having isomorphic partitions will be said to be equivalent;this clearly is an equivalence relation in the technical sense. Let K, L beisomorphic partitions of polyhedra P, Q. Let : P ->Q be the homeo-morphism obtaine d by map ping each simplex ofK linearly onto its correlatein L. W e call a piecewise linear homeom orphism onto , or PL O. Le tQc R, where R is a polyhedro n. Then the map ip: P ->R defined byI/JX = (f>xforx E P is called a piecewise linear homeomorphism i nto , or P L I .Thus, a PL O is a P L I. The iden tity m ap will, in all conte xts, be denote dby 1. If P is a polyhedron, 1:P ->P is a PLO. If : P -> Qis a PL I andJP is defined, we write for\P.

1.2. By / we shall denote the closed unit interval [0,1]; if P is a poly-hedron, so is the cartesian product Pxl. L e t.P, Q be polyhedra and letbe a PLI such that i{x,t) = (y,t), where x e P, y e Q, and t E I. Sucha P L I will be said to be ^-consistent. Ifx, y, and tare related as above, wewrite y = tz, and have thus defined a map t: P -> Q which is a PLI(cf. 2.33) and is such that

4>j{x,t) = (fax,t).D E F I N I T I O N . In the above situation, the PLI j>j or the family of PLI (f>t,indifferently, are called an into isotopy between the PLI 0and 1} which aresaid to be into isotopic; we write ^ ~ v If j is a PLO or, equiva lently, ifeach t is a PLO, then we refer to an onto isotopy, we say that 0, ^x are ontoisotopic and write 0 vEven if each of cf>0 and (j>x is onto, Q


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P IE CE W IS E L INE AR IS OT OP Y 31A combinatorial g-manifold M is defined as a complex M such thatI S t ^ ^ l is a ^-element for every simplexAc M, cf. (3). Alternatively, ag-manifoldischaracterizedasfollows: Le tabe any ver tex , i.e. any 0-simplex,ofif. Then \LkMA\ is

a (


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32 V. K. A. M. G U G E N H E I MT H E O R E M 3 . T here is a -\-PLO : J f ->>MQ such that

\P = 1, \E\ = Rn; cf. 2.32. The subset % of ^ n is defined as follows:e &Q if and only ife &n and there is an n-element E%c Rn such that\Cl[RnE$\ = 1.9% is a normal subgroup ofe S?nsuch t ha t P = Q.Then we say that P, Q are congruent in Rn and write P = Q in i2n . I nstatements using this and consequential definitions the words 'in Rn> willoften be omitted w hen they are clear from the con text. The inv estigationof congruence relationships is the g eom etry ofRn induced by the group &nin the sense of Klein's Erlanger Programm\ cf. (11). Thus, by Theorem 4,for finite polyhedra the geometries induced by S?nand Qbeagiven PLO. Then thereisE&nsuch that \P =*p.(See4.3.)In particular, equivalent finite g-dimensional polyhedra of i? n are con-gru ent if2q-\-2 ^ n. As can be seen from t he existence of 'kn o ts ' in 3-space,this is the be st possible res ult.1.52. Let E* be a ^-element ofRn. We say tha t E* is flat in R if it iscongruent in Rn to a ^-simplex . Clearly all flat g-elements ofJB"form onecongruence class. It is an immediate consequence of Newman's theoremon homogeneity of manifolds (3.31 below) that all 7i-elements are flat inRn. We shall also prove th at all1-elementsare flat (cf. 6.51), and that allg-elements are flat in R* it 2q-{-l < n ; see 7 .33.


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PI EC EW IS E LIN EAR ISOTOPY 33Let Mq be a ^-manifold and Eq a ^-element such that

Mn Eq = Mq n $q = ^a-1 ,a (g1)-element. We say, cf. (1), that Mq and Eq have regular contact inEq~x. Then, cf. (1) or (3) , Mq and ifa UEq are equivalent . If we tr y togeneralize this to the case of embedded manifolds and congruence, we areled to the following definition:

Let Mq c Rn be a g-manifold and eqcMq a g-element such thatMq n eqDe91,

a (g1)-element. Let Z9c Rn have regular contact withMq in e91and letthe ^-element Eq U e be flat. Then we call Eq a flat attachment to Mq.T H E O E E M 6. If Eq is a flat attachment to the q-manifold Mq, then

Mq = MqU Eq. (See 6.2.)The next theorem is similar. Let Mq c Mn be a ^-manifold, and letEq+1 c Rn be a (g-fl)-element such that E 1 nMq = q+1 n if = ^,a ^-element; and let ^+1be flat in Rn.THEOREM 7. Jf = Cl[Mqu J&fl+i_^]. (^ee 6.3.)The next theorem shows that failure to be flat is, for elements, a 'local'

phenomenon; for spheres, of course, the situation is quite different, thereare 'knots'.T H E O R E M 8. Let Eq c Rn be a q-element and let K be any given partition

of Eq. Then Eq is flat in Rn if and only if IStg-al is flat for every vertex aofK. (SeeGA.)

The last theorem in this section is a generalization of Theorem 3. LetMq cRn be an orientable manifold, let Eq (i= 1,2) be flat ^-elements inMq, and let ef cIEq be g-elements. Let PcRne?e\ be a polyhedronwhich does not disconnect Mq\ and let ip:ef ->e\be a given PLO, positivei n i / 3 .

T H E O R E M 9. There is e&nsuch thatMq= Mq, \P= 1 \el= f See6.53.)

A similar result will be proved concerning regions on the boundary of amanifold, Lemma 6.54.

2 . The fundamental lemmas in the theory of piecewise linear maps andisotopies are proved. See 1.1 and 1.2 for definitions and notation.

2.1. DEFINITION:The complex K' is called a partition of the complex K if(i) \K'\ = \K\, i.e. if K' is a partition of the polyhedron \K\, and if

(ii) every simplex of K' is a subset of some simplex of K.5388.3.3 D


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34 V. K.A. M.G U G E N H E IM2.2. The following lemmas arewell kn ow n; seeforinstance(4) and7of (6).2.21. IfK, Lare complexes such th at |1JL|= \L\,the n th ere isacomplexM whichisapart i t ion bothofKand ofL.2.22. IfK, Lare complexes, there are partitionsK', UofK, L respec-tively which intersectinacommon subcomplex.2.23. IfPv P2 ,..., Pkarepolyhedra, then so arePxUP2U... uPk andPjnP2n ... nPk.2.24. IfXcRnisanysubsetofEuclide an space, we deno teby0\Xi tsclosure.L E M M A . If P, Qare polyhedra, soisC1{PQ).Proof.By2.22P, Qhave partitions K, L which intersect in a commonsubcomplex, Msay.Now,

Q$P-Q) = C\(\K\-\L$= C l ( | X | - | J f | )= Cl(uL4) = DC\(IA) = uA,

where U denotes the union overall theopen simplexes IA ofKwhicharenot inM.2.31. Notation. If the PLO:P->Q is induced by the isomorphicpartitions K,LofP , Qrespectively, we shall writeK= L\this notationthen implies three statements:

(i) 4\K\= \L\,(ii) K,Lareisomorphic complexes,(iii) the map maps each simplex ofK linearly onto itscorrelate inL.2.32. LEMMA. LetP,Q,Rbepolyhedra, andeach of:P->Q,i(i:Q->RaPLI (PLO). Then # : P-*R isaPLI (PLO).Proof.Let\K\= P,\L\= \L2\ = Q,\M\ = Rbe partitions such th atKis asubcomplex ofLxandL2is asubcomplex ofM. Then,by 2.21,there is apart i t ion L3ofQwhich is a common partition both ofLxandof L2. LetL*bethesubcomplex ofLs which is apart i t ion ofK. LetK* = ~xL*. By2.22 there isapart i t ionM*ofMwhich co ntains I/JL*asa subcom plex. Clearly th e m ap iftis inducedbythe isomorphic partitio nsK* and ifiL*.2.33. L E M M A . Let P, Qbe polyhedra and: P- Qagiven PLI. Let

Poc P beasubpolyhedronandletQo= PQ. Then(i) Qoisa polyhedron,(ii) cf>\P0:P 0->Q0isaPLO.Proof.Thishas infact already been proved in2.32ifwe remember th atthe identity injection Po->PisaPLI.


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P IE CE W IS E L INE AR IS OT OP Y 352.34. L E M M A . Let eachof: P - > Q , ':P'->Q' bePLO, P, P', Q, Q'being

polyhedra. LetPnP' = 'Pn P' = Qn Qfand\Pn P' = '\Pn P \Then if>: P UP' ->QUQfdefined byif>\P= , ifj\Pr = ' is a PLO, and so

Proof.Let \K\= P, \K'\ = P', \L\= Q, \L'\= Q'bepartit ions sucht h a t K= L, 'K' = U. By 2.22there arepartit ions Klt K[of K, K'respectively w hich intersectin a common subcomplex whichis apart i t ionof P nP'. If we nowdefine Lx = ^Kj andL\ = 'K'v then Lv L[areclearly partitions of Q, Qr respectively which intersect in a commonpartit ionofQnQ'. Then L UL' = $(KUK')and thelem ma follows.2.35. LEMMA. Let P, Qbepolyhedraand let P cQ. Let:P ->.P beaPLO such that\C$Q P)flP= 1. Then f:Q-*Q, definedby

isa PLO.Proof.In 2.34take P' = C1(QP) and f = l . Then by 2.24P' is apolyhedron,andhence 'aP L O,and thelemm a follows.2.4. Let P, Q-$Q polyhedra in disjoint Euclidean spaces Mpand Rq.Then it is well known that the cartesian product P XQis a polyhedronin i?p+= RPXi?. Let each of :P ->P\ 0:Q->Q'be a PLI (PLO),whereP , P ' cR*, Q,Q'c i2. The n

defined by^X*p(x, y)= (x,ifry),wherex e P, yeQ,is a PLI(PLO). Weomitthesimple proofofthis fact.2.5. LetX, Y c Rn be anysubsetsofRn. WedenotebyXY = YX thejoinofX andY,i.e. the set ofpoints

tx-\-{lt)y wherexe X, y eY and 0 t< 1.Note tha twe do not require this jointo benon-singular.IfP ,Qc Rnarepolyhedra,so isPQ. For a proof,see7of(6).

2.61. Let P beapolyhedron, an dKapartit ionofP . Asimple p artitionof PX/ isobtained as follows: Le tA be any simplex ofK, andgits centroid.Assuming inductively thatall thesetsA'xl have alreadygot partit ionswhen dimA'


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36 V. K. A. M.GUG ENHEIMare clearly ^-consistent partitionsofPX/, Qxl, and they inducea PLOj: Pxl^-Qxl s u ch t h a t

.The proofsforthe symm etric law and the transitive lawof are extremelysimple,and are omitted.

2.62. L E M M A . Let (f>0,fa: P -+Q, d: R-> P, fa Q-> S each be a PLO,P, Q, R, 8being polyhedra, andlet0 fa with isotopy t. Then (f>06 & fadwith isotopy t9and tfj(f>0 tpfa with isotopy ifjt.Proof.Wet rea tthecase with tp.Letpt= ifjt, whichis aPLOby 2.32.Asin2.61wefinda PLO fa-.QxI Sxl such th at

A/(y>0 = (#>0 where yeQ,teI.Then,forx e P,wedefine pI{x,t) = {Ptx,t)

Nowjis a PLO byhypothesis andhenceplis a PLO by 2.32, and thelemma follows.2.63. LEMMA. Let cf>0, x\P-+Q, if/0,0X:Q->R be PLO, P, Q, R beingpolyhedra, and let0 1} tp0 ijjv Then ifjo


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P I E C E W I S E L I N E A R I S O T O P Y 37q = 0, A = xi arep o in t s ofICn, andAis th e ma pXxx= x2. In th is casewe sha l l wr i te (tjj,xv x2, C

n) for (A,t/j),w h e r e0: C

n ->C

n is a g ivenPLO.3.12. L E M M A . Let Ev E2be elements and I/J: SX -+$2 agiven PLO. Thenthere is a PLO : Ex-> E2 such that = ip.

Proof.SinceEx, E2areelementsofthe same dim ensionality,we can findPLO 0E a PLO such that ^ 1 ,with isotopy ip t, where fa = , IJJ0= 1. Then & I with isotopy t, where


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38 V. K . A. M. GUGENHEIMProof. Since P does not. disconnect Mq and is disjoint from E\ U E%we

can find a partition of Mq containing a sequence of successively incidentsimplexes A j _ ^ A J - I , . . . , ^ ^ Ag - i a l l m m_psuch th at A? n E\ = A? n # = A?"1,

A%_xnE = A J U nJS|= A JT1an d [#f U A? U ...U A?]nA ?+ 1= A ? 1 (*= 1, 2, . . . , k - 2 ) ,[E l uAfu ... uA _ J nEl = A g 1 ,where the last two conditions can clearly always be realized by taking adouble subdivision of any given partitio n. Then El UAfU ...UAj{_1UElis a ^-element disjoint from P, and the lemma follows from 3.21.3.23. LEMMA.Let E%~1(i= 1,2)bed isjoint (ql)-elements on the boundaryof a q-manifold M q,bothdisjoint from apolyhedronP cMqwhich disconnectsneither M q nor Mq.^ Thenthereis aq-elementFq c Mq, d isjoint from P, andsuch that FqnMq = $qr\Mq = Fq~\a (ql)-element,.and E^1 U El~x c IF*-1.

Proof. Find first a (g-1)-element F 1 cMqP such that^f-'1 UEl-1 c IFlr1,by 3.22. Then we find Fq as a suitable 'regular neighbourhood' of F^~xin Mq, appealing to the same theorems as in the proof of 3.21.3.3. We now give an adde ndum to a theorem due to M. H . A. New man (2)and sharpened by J. H. C. Whitehead (4), namely:

3.31. Let El, El be two g-elements interior to a ^-manifold Mq, and letPcM qElEl be a polyhedron which does no t disconnect Mq. Thenthere is a PLO : Mq -+Mq such that El = El and \P = 1.3.32. Addendum. In 3.31, can be chosen so that 1.Proof.The adden du m will follow ifwecan find a g-element Fq c MqPsuch that El U El c IFq. Fo r the n we replace Mq by Fq and P by J^ in3.31,and we find fa: Fq-+Fq such that fa El = El and fa 1. But thenfa 1 with isotopy fat such that fat = 1, by 3.13. Now put\Fq= fa, \Mq-Fq=l,

and$53 1 follows easily, using 2.35.The required elementFqcertainly exists in two cases, nam ely (i)El c El,by applying 3.21 to El; (ii)El, Eq2 disjoint, by 3.22.In general, let El and El overlap with out being identical. Then there isa g-simplex Asuch tha t Ac El andAcMqE%.Then, by cases (i) and(ii) above, we can find v 2:Mq ->Mq such thatfaEl = Aq 2^q==El,both being isotopic to 1. Then = xfasatisfies t h e addendum,

t Whichweassumeto beconnected.


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PIECE W ISE LINEAR ISOTOPY 39Note that if M9 is orientable, i s a + P LO .3.41. Proof of Theorem 1. Let S = S9, a g-sphere. W e use an ind uctio non q. For q= 0 the theorem is trivial, the only + P L O being 1. Le tE9 cS9 be a g-element. By 3.3 there is a + P L O ip:S9->S9 such th at0f = E9 = El, say, and such th at 0 1. Now 00 -1 #? = #? , andhence, if we define 0 2 = 00~1|2f, fa is a -f-PLO and so, therefore, is if/vHence, by the inductive hypothesis, fa 1. Henc e, by 3.13, fa 1, wi thisotopyfatsay,aQ= 1,fax= 0 rLet El = C\(S-El). T he n # " 1 | ^ = 0 0 - 1 |#? = 0X. Hence, by 3.13,0 0 - 1 | | i 1, with 'boundary isotopy' fat. Hence, by a simple applicationof 2.34, 00 -* 1. B ut 0 1. H ence , by 2.62, 0 1.3.42. Proof of Theorem2. 0 1, by Theo rem 1. Hen ce 0 1, by 3.13.3.43. Proof of Theorem3 (see 1.4). By 3.3 the re is a + P L O fa: M*^M*such that fa 1, .0 i| P = 1 and faEl = ^f. Th en 0 0 fx | ^ is a PLOE% ->JEJ which is positive inMQ. It thus suffices to prove the theorem forthe case E\= E% = E*say. Let 0: E -+ E*,then, be a + P L O . By 3.21there is a g-element F*cM*P such that E* c IF9.Let A9, B9 be g-simplexes similarly situated with respect to a centre of

similitude in IA9 and such that A9 c IB9. The n it is easy to define in anatural manner a PLO a: Cl[B9A9] ->A9xI such th atax = (x,0) when x e A9,ax = (/te, 1) when x 6 i8,

wherefi: B9 ->A9 isa, PLO (obtained by projection).Let d^.F9^ B9 be a PLO. Then 0X#c IB9 is a g-element and hence,by 3.31, there is a PLO 02:B9 ->B9 such tha t 020XE9 = 4. Then, putting6 = e2d1} 0 is a PL O such th at OF9= , 6E9 = ^4.Now r 0 = 000~1 |^9: ^4->^43 is a + P L O . Hence by Theorem 1 thereis a +P L O rI:A 9xI-^A9xI such tha t

TJ(X,t)= (rtx, t) where x e A9, te I,where also rt: A9 -A9 is a PLO and r1= 1.Now define 0:M9 -> M9 as follows:

0|Cpr-#] = 1,

Our theorem now follows from 2.34 if we can verify that this definitionis consistent on fi9 and Fq.


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40 V. K. A. M. GU GE NH EIMLet x e 9\ O-ârâB x = Bôt-h-jidx,0)

x,o)= e-1-1^, o= e îfjx = 4>x

Let XE/ ; e-ôc-^rjoidx = dôc^r^dx, 1)= fl-1*" 1^*, 1)= e isx= x

3.44. Proof of Theorem 4 (see1.51). Let En c Rn be an n-element sucht h a t P c 2n; such an element exists sinceP is bounded . Le t Fn = En.Then Qc i^n. Let Gnc Rn be an rc-element such that En U Fn c /(?.Then, by Theorem 3, there is a PLO 0: Gn-> Qn such that ^ = 1 andt/j\En = \En. We define ^0 b y f> \G = 0

Note. An othe r proof of Theorem 4, no t depending on Theo rem 3, isgiven in (12), 3, Satz I I .3.51. Let MQ, Nq be orientable g-manifolds and let Z#(i = 1,2,...,k) beorientable g-dimensional submanifolds of Mq\ we say tha t a set of P L Iifsf L\-> Nq is orientation-consistent if for a given orientation of Mq theyinduce the same orientation of N9.

L E M M A. Let M9, N9 be equivalent orientable q-manifolds, let E\ c IM9,F9 c IN9 besets ofq-elements,the elements ofeach set beingmutually disjoint.Let ipf E-i-^F-l be a set of PLO which are orientation-consistent whenregardedas PLI into N9. Then, if q > 2,there is a PLO : M9 ->N9 suchthat\E% = ifjifor every i (i = 1,2,..., k) .When q = 1, th e lemm a continues to be true provided th at we ma ke anadditio nal hypo thesis : A given orientation ofM1 implies an ordering of thepoin ts and of disjoint connected subsets ofM1; let this ordering be denotedby < . Le t < also deno te the ordering which is implied by this orderingand the orientation-consistent PLI ijjiin N1. Choose the not atio n sucht h a t El < E9


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PIECE WISE LINEAR ISOTOPY 41hypothesis on ordering assures tha t E%*andF%are in the same comp onent

Note. The special hypothesis on ordering for the caseq= 1 is automati-cally satisfied w henk==1;and also whenk= 2,in the case tha tN1 is a circle.3.52. Let El, F? be pairs of elements having regular contact in e?"1(t = 1, 2). Le t Gi = El U F*.LEMMA. There is a PLO 6: G\-> G\ such that 6E\ = E% and 6F\ = F%.Addendum 1. Given a PLO ip: E\-+ E\ such that \pe\ ~x = ef"1, 6 can beso chosen that 9\E\ = ip.Addendum 2.Given, further, q1elements/ ? - 1c i ^ " 1 whicharedisjointfrom e?""1and a PLO a: f\~1->f%~1 which,regardedas a PLI into$%t isorientation-consistentwith \p\e\~1,we canchoose 6such thatalsoQ\f%~x = a.Proof. It is clearly sufficient to prove the lemma with both addenda.First find a PLO : F\ ->$% such tha t ^e?" 1 = i/He?"1and ^ l / ? "1 = aThis is possible by 3.51. Then e xtend this to a PL O F% by 3.12.Now define 0 by

4 . Theorem 5 and some related lemmas are proved.The lemma s 4.1 and 4.2 occur in (12), 3, Satz I.Let K be a complex and aif for some indexing set of i's, its vertices; let6t- be a set of points in some Euclidean space indexed by the same set.The map which maps every simplex aio...a iq ofK barycentrically onto thesingular or non-singular simplex bio...b it is called 'the map of the correla-tion at: / .If x, y 6Rn, we denote byd(x, y) the Euclidean distance between them.Let K, L be finite isomorphic complexes of Rn, and let the given iso-morphism induce the PLO if/: \K \ -> \L\, such that ipK= L; see 2.31.Let ai be the vertices ofK, and bt = \\sav Then we write

(#,, .) = max[d(a 0 with the following proper ty:If L = L0U K x is a complex and ip:\K\ -> \L\ is a PLO such that tpK = L

(see2.31), 0 | 1^1 = 1 and D(K0,L 0,if>\ \K0\) Rn such that\\K\ = ip.Proof. Let a{ (i = 1,..., k) be the vertices ofKQ. By Theorem 5 in (4)there is a partition M of Rn which contains K as a subcomplex. Let bjdenote the vertices ofM other than those ofKQ. Let


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42 V. K. A. M. GU GE NH EIMcorrelation a{:ifjai} bj:bj. In general this will be singular; it will be a -fPL Oif the vertices at can be moved to the points fa t without any one of acertain finite set of determinants which are polynomials in the coordinatesof these poin ts passing through zero, being non-zero initially; from th is a ndthe continuity of these determinants, the lemma is immediate.

4.2. Let Kt = KQIDL be a family of isomorphic complexes of Rn(0 ^ t ^ 1),the subcomplexesK being finite and the PLOtptr: \Kt\->\Kr\induced by the isomorphism being such that tjjlt,Kot= K^, ifjtl>\ \L\ = 1;also, if a is any vertex ofKo, le t iffota depend continuously on t.L E M M A. There is a -\-PLO (f>01: Rn ->Rn such thatProof. Let t0G/ . Then, since the vertices of K depend continuouslyon t,by 4.1 there is a (relatively) open neighbourhood oft0, O(t0)say, in /such that for t, t' eO{t0) there is a +PLO l>v: Rn ->Rn such that

4HX\131= fc-The sets O(tQ) for toe I form an open covering of /; but I is compact,hen ce the re is a finite subcovering. Le tAbe the Lebesgue number of thissubcovering, let N be an integer gr eater t ha n I/A and let e = 1/N; le ttn = ne. Then 01 = tx-i,v< >o,h clearly satisfies the lemma.4.3. Proof of Theorem5 (see1.51). A set of points in a Euclidean spacewill be said to be in ^-general position if for r Q' which can eachbe extended to a -f PLO Rn ->Rn if the displacements are small enough,by 4.1. Hence it is sufficient to prove the theorem for the case when theverticesav...,bk are in w-general position; this, the n, we assum e.Let Sbe the point set which is the union of all the r-spaces Rr (r 2q-\-2,this set doesnot disconnect Rn; henceak andbkcan be joined by a continuous curve inRn S (for exa mple, a 1-element). More precisely, we can find a con tinuo us1-parameter family of points xt (te /) such th at x0 = ak, x = bk, and


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PIECE W ISE LINEAR ISOTOPY 43xt e RnS. Then , for any t e /, the point setav..., ak_v %i is in (2q-{-l)-general position, and the m ap of the correlationai:a i{i = 1,...,k1), ak:x tis a PLOfa. K-+Kt of the kind discussed in4.2. Hence0Xcan be extendedto a +PL Ofa: Rn - Rn. faKisa, complex isomorphic toKwith verticesiv 5 ak-v bkBy the same process we now 'move' the vertex ak_ t to 6&_i, leaving allother vertices of faK fixed; and continuing in this way we obtain therequired +PLO in k steps.4.4. Let P, QcRno be finite polyhedra, o being a point of Rn suchthat every rayop (pe P) (see 5.0 for the notation) meets Qin exactly one>point # = ippand every rayoq(q eQ)meets P in exactly one po inty = ip'tyin other words, the mapiff is 1-1 and onto.

LEMMA, (i) 0: P -+ Q is a PLO.(ii) There is a -{-PLO : Rn -> Rn such that \P = I/J.Proof, (i) Let K, L be given partitions of P, Q respectively an d letA nc RnPQ be an n-simplex such th a t o e /A * (see 5.21). If Ais an ysimplex ofK, it is clear that all the simplexesoAform a complex oK, say,

which is a partition of oP. Let R = An noP = An noQ. By 2.22 thereare partitions of oK, An which intersect in a common subcomplex whichis a partition ofR; let this partition beK '. Similarly, we define a part itio nU of R with respect to L. By 2.21 there is a partitio n M' of R which isa partition both ofK' andL'. LetMQ,MXbe the pa rtitions ofP, Qobtainedby projecting th e simplexes ofM'ontoP,Qrespective ly fromo.ThenMQ,MXare partitions of K, L respectively, and the isomorphism between theminduces the maptp.(ii) If p GP, write pt = (1t)p-\-t*Jjp in th e usual vectorial no tatio n;

p0 = p, Px = tftp. If S c P denote by St the point set of all pointspt forpe 8. Then, ifMQ,Mxm ean th e same as in (i) and A is any simplex of Mo,then A, is a simplex, and the set of all these simplexes is an isomorphiccomplex Mt,as is easily seen. Our result now follows from 4.2.Let ^r*:oP->oQ be the map defined by+*{tp+(l-t)o} = tifip+(l-t)o (peP),

then the following result also is immediate:LEMMA, (i) tp*:oP -+oQ is a PLO .(ii) There is a +PL0 *:Rn->Rn such that*\oP =tff*.5 . Some lemmas on the geometry ofRn and some simple map-extensionsare established.5.0. Notation. All po int sets of this section are subsets of a given Rn.


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44 V. K. A. M. GU GE NH EIMIf x, y are distinct points, xy will denote the R1 through x and y, xy willdenote the closed half-line onxy with boundary-pointx; xywill be calledthe ray from x through y.

5.1. L E M M A . / / P, Q areconvex polyhedra, so is PQ.This is immediate from 2.5 and the easily verified fact that the join ofconvex sets is convex.5.21. L E M M A . / / A8 is a q-simplex ofRnX, where X is a closed subsetof Rn, there is an n-simplex Anc RnX such thatA9c /A n .This is easily proved.5.22. LetA, B be simplexes and letA n B = P = A nB,a closed convexpolyhedron on their bound aries. LetAobe the face of least dimension of Asuch that P cA0.L E M M A. There is a convex n-element Cn such that

A Ao CICn, Ao cCn, BnCnc AQ.Proof. The case dimA = 0 is trivia l, and can be dismissed. L etdim A > 0.P contains points of IA0; for otherwise P c Ao and either P is in some

proper face ofAQ, against hypothesis, or P contains pointsx, y which lieon no common face ofAo which is proper; then P contains xy and hencepoint of IA0. Thus, letp e IA0 n P.L et S be the set of all rays aob for every a0e Ao and b e B (b# a0).Clearly 8 is a closed set, and B c S. Also,8 ft A c Ao. For, otherwise, letxc 8 nA Ao. Then there are points aoeA o and b e B such that

x e aobao, b not lying in the subspace of Rn defined by Ao, for x is notin this subspace. First assumep = a0. Thenpb = aob = aox = pxand px npbcA D B is a segment of P no tin Ao, against hypothesis {pxc A, pbc B).Now assum e ) ^ a0. Thenpa0cAo and hencedim^40 ^ 1 {pe /-40)-Hence there is a point a*eAo such thatp GIaoa*. Then A = xaoa* c -4 is a triangle ,and confining attention to the R2 of A,

pb c B lies on the same side of aoa* as x.Hence pb n A c P contains points not in Ao1- aga inst hyp othes is. See Fig. 1.Now since P contains interior points of Ao and since dimA ^ 1, wehave dim^40


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PIECEWISE LINEAR ISOTOPY 45join. Then S andAx aredisjoint by wha t wehave proved. Hence,by5.21,the re is an w-simplexA

ndisjoint fromSsuch tha tAx cIA

n. C

n= AoA

nisa convex polyhedron, by 5.1,and isw-dimensional,for it containsAn.By considering neighbourhoods of pointsitis easytosee tha tAA oc ICn,A-ocCn. Also, BnCnc Ao. For, let y e BnCnAo. Then therearea0e AoandxEAnsuch tha ty eaQx a0. Bu t thenaQxca^y c S,foryEB.HencexE8, against hypothesis. This completestheproofof the lemma.

5.23. LEMMA. LetA = A0Axbe asimplex,Ao andA1beingproper facesof A. Let P be apolyhedron such that PnAcAo. Then thereis an n-dimensional convex elementCnsuch thatA-Ao c ICn, AocCn, P(\Cnc Ao.

Proof.LetKbeapartit ionofP,letLbethe set ofsimplexes ofK whichhave points in common withA, andM the set ofsimplexesofK disjointfrom A. Sinceourcomplexes are locally finite, L contains only a finitenumberofsimplexes. LetA1}..., A&beitssimplexes. Le tAoi be thefaceoflowest dimensionofA containingall thepointsofA nAf. ThenAoi c Ao,by hypothesis. By 5.22thereis a convex w-element C?such t ha tA~Aoi c IC2, Aoi= O , A


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46 V. K. A. M. GUGENHEIMWe write H\ = H UR -1, Hn_=Hn_U Rn~l, and C% = Cnn E%,Cn_= CnnHn_. Then C= C\V Cn_and .4 = C^ n Cn_= C%n #L .

5.3. /Some map-extensions5.31. L E M M A . 2/e i?6ea PLO. Then

there is a +PLO : Rn-> Rn such that \R9= tp.Proof. We regard Rn as the cartesian product RQxRn~Q. Then put0= tf/X1 (see 2.4). If0isa-{-PLO,putJ?n- isareflectionin ahyperplaneofi2n~.5.32. LEMMA. Let EqcR9c Rn (q .n) be a q-element, and P c Rn a

polyhedronsuch thatP n ^ c ifc. Letipq:E-+ E*beagiven+PLO suchthatipq= 1. Then we can find ann-element En cRn such thatIE*cIEn, & cfin, Pn Enc tln

and a +PLO n :En -+En such that n\E9 = if>q, iftn= 1.Proof. Forq = n, t//n= ff, -/n= EQ trivially satisfies thelemma. ForqRn such tha t dE -1 = A -1, an ( n - 1 ) -

simplex. Then OPC\An - 1cA71-1. Henceby5.24, 5.25we can find convex7i-elements Cn, C%,Cl such thatCn = C%uCL, A -1= C\ n C = (? n Cn_, P n Cnc6?71

and An - 1c


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PIECEWISE LINEAR ISOTOPY 47LetPj = P nRq. ThenPxisdisjoin t from IE*-1, hence0PXfrom A9A""1,thus 6P1nA9cA9"1. Hence,by5.23, thereis aconvex g-element C9cR9such that A 9A9"1cIC9, A9"1c C9, ^PxnC9cC9. Hence C= A u D whereD9= C1[C9A9]is a g-element such that

A9nD9= A9nD9= aA-J.Hence, putting F9= 0-1 )9and (?= UJ = fl^C9, 9is a g-elementof R9 such tha t (?n?jC (X Nowdefine0':G9->G9by

0'|-#fl = F* bea givenPLO (positivein Rv if q= n). Then thereis a PLO :Rn ->Rnsuch that4>\E*=ifj.Proof. Since E*andF*are flat,we can f ind +PLOs Rn. Then $ = fe Vo^isatisfies thelemma, for whena;e i/9,


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48 V. K. A. M. GU G EN H EI M6.15. L E M M A . If Eqc Fqc Rn areq-elements, and if Fq is flat, so is Eq.Proof. Immediate from 3.31 and 5.31.6.2. Proof of Theorem 6 (see 1.52).6.21. LEMMA. Let M be a combinatorialmanifold, and E a subcomplexof M which is thepartition of aq-elementand such that M D De,the parti-tion of a (q1)-element. Then C\[ME]contains no interior vertex of e.Proof. We use the notation of (3) for the proof of this purely combina-

torial lemm a. Let a be an interior vertex of e. ThenM = aE0+P,

e = aS+R,where Eo, Ex are combinatorial [q1)-elements and S a combinatorial(g-2) -sphere . & = aNow , e cM n j. Hence S c i0n v B ut no prope r subse t of a sphere isa sphere. Hence S = i0= Is^. B ut Ec Eo] hence Eo = E^K, whereit is empty; hence a simple argument based on the strong connectivity ofEQshows tha t K is em pty. ThusEQ= Ev Th us the re is no g-simplex ofMhavin g a as a verte x w hich is no t a simplex ofE, and hence C\[ME] doesnot contain a.

6.22. COROLLARY. Let Mq bea q-manifold and Eq cMq a q-element suchthat M q n q Deq~x, a (q1)-element. ThenProof. For, if C\[MqEq] contains a point of Ieq~x, we merely have totake a partition having this point as a vertex to get a contradiction of 6.21.6.23. We now come to the main part of the proof. See 1.52 for then o t a t i o n . e'q= eq~x Ugq~ l, w h e r e g9'1 = C\[e'qeq~x]is a {q1)-elementsuch that eq~x= $*-x. By 6.22 C\[Mqeq]neq~xceq~x. Also, clearly,Cl[iHfffe3] contains no interior points ofeQ. Hence

C\[Mqeq] n e c gq~x.Now letAq = aAq~x be a ^-simplex ofRn, Aq~x being one of its(ql)-faces,and letb e IAq. Then Aq = Bq UHq, where

Bq = bAq~x and Hq = C\[Aq-B q]are ^-elements having regular contact inbAq~x. The n, sinceeqU Fq is flat,we can by 3.52 and 6.11 find a +PLO x: Rn -+Rn such that

cf>igq-x = Aq ~ x .


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P I E C E W I S E L I N E A R I S O T O P Y 49e]n c A*-1. Hence, by5.23,thereisa convex rc-elementCn c Rn such that

A*-A*-1 CICn, A*-1c Cn, ^ Cp f e ] n C c ( XL et 2\Rn^-Rn bedefined by

< 2 | O = ( i , a , 6 , Cn) (see3.11), 2\R n-C n = I.Then we have ^ a[Mq_eQ ] = l f

^ 2 ^ a = B.Now let 0 = x: Rn -> Rnsuch thatNow Cl[iWaE*\ n EZ+1= Cl[itfJ&].n ^ c #. Hence

^ C l [ J f - ^ ] nA**1c A*.Hence, by 5.24,there is a convex ^-element Cn c Rn such that

Az+tA* c ICn, A c Cn, xQ\[Ma J0]n C71c (Let 6G/ J .3 . Define the +PLO 2: Rn ->i2ra by

Then ^ 2 ^ a = aA* =^ 2 | ^ C 1 [ ^ -

Now, write = a n c^ we have

from which the theorem is immediate.6.4. Proof ofTheorem8 (see1.52). Let E*,F*be g-elements ofRn whichhave partit ions K, L respectively, such that K UA = L, where A is oneof the g-simplexes ofL and has regular contac t with \K \ in an elementEQ~X5388.3.3 E


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50 V. K . A . M. G U G E N H E IMconsisting of (ql)-faces ofAq. W ith Newman, we write L-> K. Then Lcontains at least one (ql)-face ofA, Ao say. Then A = aA0. The only(q l)-face of A not containing a is AQ and this, being on L, is incidentw ith one g-simplex ofL, namelyA, only. HenceAo is not a simplex of K,and hence all (q l)-simplexes of Eq~x contain a, and

E^c \S tKa\c \K\.B ut | S t^ a | is a ^-element which is a subset of the ^-element | S t za|. He nce,if |St z a| is flat in En, then A is a flat attachment to \K\ = Eq.Now by (5), Theorem 29, corollary, there is a partition K' .of K suchthat there is a sequence

2 * 2 2 2 2whereA is a simplex. Le tKi+ 1 be obtained fromK{by rem oving a g-simplexA^ Now if all th e stars of vertices are flat inK, the same is tru e inK', by6.15. Hence, by wh at we just showed, Ai is a flat attachment to Ki+ 1.Hence, by Theorem 6,\Ki+1\= |-K^| in Rn. From this and the transitivityof = , \K \ = A, and the theorem is proved.

6.5. Proof of Theorem9 (see1.52).6.51. A class of flat elements. Let Aq~2 be a {q2)-simplex of Rn,a0, ax,..., ak points of Rn such that AQ = A9-^^^ (i= 1, 2,..., k) areg-simplexes, and such that

El = Al\J...uAl (i=l,...,k)is a ^-element, having regular contact in Aq~x = -4s~2a i with A9+1 fori = 1,.., k1. The case q = 1, w ith Aq~2 the empty simplex, is notexcluded.

L E M M A . E% is flat in Rn.Proof.For k = 1 this is trivia l. If the lemma is tru e for k = 2, thenAq+1 is a flat atta ch m en t to Eq, and the lemma follows from Theorem 6.It remains to prove the case k = 2.Either A\ and A\ lie in the same Bq,and the lemma follows from 3.31 and5.31,or a2does not lie in th eRq of A\.Then a2A\ is a (g+ l)-sim plex containing E9.on its boundary, and th elemma follows from 6.12.

The caseq= 1 shows th at all 1-elements are flat.6.52. Let M be a combinatorial g'-manifold, and Aq~2 a (q2)-simplexofM. Then |StM^48~2| is an element of the typ e treat ed in 6.51, and henc eflat. In partic ular , the star s of vertices on the boundary of a 2-manifoldare flat.If A9'1 is a (q l)-simplex of M, then ISt^^8 " 1 is an element of th esame type, with k 1 or 2, and hence flat.


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PIECE W ISE LINEAR ISOTOPY 51Now let M9 = \M\, and let

EQ c [M9-M

9~

2] U [M

q-M

9~

3]be a g-element, whereM{ denotes the polyhedron covered by the *-sectionofM. Then every star of a ve rtex ofE9 is subset of a flat element, andhence flat by 6.15. Hence, by Theorem 8, E9 is flat.6.53. W e now come to the proof of Theorem 9; see 1.52 for the no tatio n,(i) First we find a +PLO 0: Rn -Rn such that Qel = eg. The proofis almost the same as that given in 3.32, replacing the P of that section

now by P' = Pu Cl[M9U M*-ZE{EQin th e notatio n of 6.52. This clearly does no t disconnect M9. The elementF9 (in the notation of 3.32) is flat, because all the stars of its vertices areeither in M9M9M9~2 or in El or E%which are flat by hyp othesis.Hence the PLO *: F9 ->F9 such that * = 1 can be extended to a+ P L O 0: Rn -> Rn s u c h t h a t 0\C\[M9F9] = 1, b y 6.13.(ii) The rest of the proof consists in noting that any +PLO eg->egca nbe extended to one E9,->E 9, which is 1 on jj[, by Theorem 3; this in turncan be extended to a +PLO Rn ->Rn which is 1 on Cl[ikfJ^f], by 6.13.

6.54. We now prove an analogous result concerning regions on thebo un dar y of. a manifold. Le t Mq c Rn be an orientable (/-manifold, E9(i= I, 2) flat g-elements of M9 such t h a t jf nM9 DE*-1, which are(q l)-elements. Le t ef"1 cIE*-1 be (q1)-elements, and 0: ef"1 -> eg"1a PLO which is positive in M9. We assume M? (as well as M9) to beconnected.L E M M A . There is a +PLO : Rn ->Rn such thatM9 = M9, ^lef 1 = 0 .Proof, (i) Let e\~x, eg"1be disjoint. The n, by 3.23, the re is a g-element

F9 cM9 such thatJ n J i r = | 4 n l s = F*-\ a (ql)-element,ef"1 Uef"1 c IF*-1

and such that Fq is disjoint fromin the nota tion of 6.52. Then

FQ c [M9M 9~2] u [MQM9~3] UIE\ U IE9.and hence is flat. By Theorem 3 the given PLO can be extended to onej f l - i .+ F9-1 which is 1 on Jfa-1; this is extended to a -j-PLO Jh ->$9 bytaking 1 on C\[#9P9'1]; th is + P L O is extended to a + P L O Fq-> F9by 3.12; this is finally suitably extended to a +PLO Rn ->Rn by 5.34.(ii) Let eg"1c ef "1 . Then let Fq = E\, and the argument in (i) applies.


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52 V . K . A . M. G U G EN H EI M(iii) In the general case, we proceed as in the last part of 3.32, using (i)and (ii) above.6.6. The following two lemmas will be used in section 9.6.61 . L E M M A . Let Mq c Rn (i = 1, 2) be two orientable q-manifolds, let

El cM% be flat q-elements, and let eqc IE \ n I El be a common q-element.Let : Rn -> Rn be a -\-PLO such that Ml = Ml, and such that \eq ispositive in Ml Then there is a +PL O '\Rn^ Rn such that 'Ml = M%an d '\eq = 1.

Proof. L e t eq = e%,E\ = E%. Th e n , b y Th e o r e m 9 , t h e r e is a + P L O^:R^Rn such tha t ^Ml = M%

^ | 4 = ^\e%.T h e n l e t '= , a n d f o r x e eQ , x= % x= ^ f a = x.

6.62. L E M M A . Let Mq c Rn (i = 1, 2) be two orientable q-manifolds, letEqcM\t\ Ml be a comm on flat q-element such that-\ M\ n Ml n # D E**-1,a (ql)-element} and let e3 " 1c IE*-1 be a (q-l)-element. Let \ Rn^- Rnbe a +PLO such that M\ = Ml and such that \eq-x is positive in Ml.Then there is a +PLO '\Rn-+ Rn such that 'M\ = Ml and file*-1 = 1.

The proof is exactly like that of 6.61, using 6.54 instead of Theorem 9.Resume of Part IIIfMq is an o rientab le ^-manifold, we deno te byMQthe oriented manifoldobtained by giving to M9 one of th e two possible orien tations. If M a, Nare both oriented g-manifolds and if : MQ->NQ is a PLO we shall writef: M fl-> N9ifand th e orientation of Mflinduce inNQthe orientation ofN.If P, 0 are oriented polyhedra ofRn we say that P, 0 are congruent in

Rn, P = 0 in J?n, if there is G&n such that P = Q.For q


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proc. london math. soc. 1953 gugenheim 29 53

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