Annals of Mathematics A Representation of Orientable Combinatorial 3-Manifolds Author(s): W. B. R. Lickorish Source: Annals of Mathematics, Second Series, Vol. 76, No. 3 (Nov., 1962), pp. 531-540 Published by: Annals of Mathematics Stable URL: http://www.jstor.org/stable/1970373 . Accessed: 28/01/2015 05:03 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 of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of 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 of Mathematics. http://www.jstor.org This content downloaded from 210.212.192.130 on Wed, 28 Jan 2015 05:03:47 AM All use subject to JSTOR Terms and Conditions ANNALS OF M1IATHE MATICS Vol. 76, No. 3, November, 1962 Printed in Japan A REPRESENTATION OF ORIENTABLE COMBINATORIAL 3-MANIFOLDS BY W. B. R. LICKORISH (Received December 20, 1961) 1. Introduction The following question has been posed by Bing [1]: "Which compact, connected 3-manifoldscan be obtained fromS3 as follows: Remove a finite collection of mutually exclusive (but perhaps knotted and linking) poly- hedral tori T1,T2, * - *, To fromS3, and sew them back. " This paper answers that question by showing that every closed, connected, orientable, 3-manifold is obtainable from S3 in the above way. Whereas this fact can now be deduced from general theorems of differentialtopology, the combinatorialproof given here is direct and elementary;while, in the proof, a study is made of a certain type of homeomorphismof a two dimensional manifoldthat is of interest in itself. Having obtained the above mentioned result on 3-manifolds, it is then easy to deduce the well known result (Theorem 3) that the combinatorialcobordism group fororientable 3-mani- folds is trivial. 2. Isotopies and homeomorphisms DEFINITION. Let K be a finitesimplicial complex, and let I denote the unit interval. If F0 and F1 are two piecewise linear homeomorphisms mapping K to K, F, is isotopic to F1 if there exists a piecewise linear homeomorphismh: I x K > I x K such that: (i) h(t x K) = t x K, 0 < t < 1; (i.e., for each t there is a piecewise linear homeomorphismht: K K such that h(t, k) = (t, htk)). (ii) ho = Fo, h1 = F1. If K is a simplicial complex, the set of piecewise linear homeomorphisms mapping Kto Kis obviously a group, GK. Let NK be the subset consisting. of all elements of GK that are isotopic to the identity. It is immediately seen that NK is a normal subgroup of GK, and that elements of G .are in the same coset of NK if and only if they are isotopic. DEFINITION. LK is the quotient group GKINK. LK is the group of isotopy classes of piecewise linear homeomorphisms of K to K. 3. Homeomorphisms of 2-manifolds If X is an orientable combinatorial2-manifold, we can 'perform' homeo- 531 This content downloaded from 210.212.192.130 on Wed, 28 Jan 2015 05:03:47 AM All use subject to JSTOR Terms and Conditions 532 W. B. R. LICKORISH morphismson X in the followingway. Let C be a simple closed polygonal path in X. A neighbourhoodof C in X is a cylinder, S1 x I. Cut X along C, twist one of the (now free) ends of the cylinderthrough 2w, and glue together again. We have then 'performed' a homeomorphismof X which leaves X fixed except in a neighbourhoodof C. The process is illustrated in Fig. 1. ZC I \ / Twit SI XI c u GLUE Fig. 1. A piecewise linear homeomorphismof this type will be called a C-homeo- morphism. If p1 and p2 are paths in X, we shall write p1 -, p2 if there exists a sequence of C-homeomorphismsh1, , h. and an element n of Nx, such that nhlh2* hmip=2. Since the inverse of a C-homeomorphismis a C-homeomorphismand N, is normal in Gx, "'- '" is an equivalence relation on the set of paths in X. LEMMA1. If p1 and p2 are simple closed polygonal paths in X, such that p1 and p2 intersect in just one point, then p1 C p2 PROOF. Let h1be a C-homeomorphismusing p2as C. If p1 and p2 inter- sect at point w, hp1 is, in effect,a copy of p1 broken at w with a copy of C inserted at the break. Thus hpi is as shown in Fig. 2(b). Now take C2 as a simple closed polygonal path in a neighbourhood of p1 (see Fig. 2(b)). If h2is a C-homeomorphismusing C2 as C, then h2h1p,is as shown pi ~ ~ ~ ~ C r____>_ I ht Pa (a~) (b) (c) Fig. 2. This content downloaded from 210.212.192.130 on Wed, 28 Jan 2015 05:03:47 AM All use subject to JSTOR Terms and Conditions COMBINATORIAL3-MANIFOLDS 533 in Fig. 2(c). There is then an isotopy of X sending h2hlp,to p2; i.e., there is an n e Nx such that n h2hlp, A2.p COROLLARY. If PA,P2, ..., Pr are simple closed polygonalpaths in X, such that pi intersects pi,, in one point only (i = 1, 2, , r - 1.), then PA c Pr. This follows immediately from the lemma and the fact that "c'" is an equivalence relation. LEMMA2. Let p and q be simple closed polygonal paths in X, and Y be any neighbourhoodof q in X. Then there exists a path p* such that P -c p*, m*n (X- Y)czp n (X- Y), and p* either does not meet q, or meets q precisely twice with zero algebraic intersection. (This last statement is significant,as X is orientable). PROOF. The proofis by inductionon the numberof points ofintersection, r say, of p and q. If r = 1, by Lemma 1, p -c q, and there is an element n e Nx such that nq c Y, but that nq does not meet q. We then take p* as nq. If r = 2, and p can be oriented so that p has mutually different directions,with respect to an orientationof q, at the two points of p n q, then the lemma is trivial. Thus suppose the lemma is true for all paths p and q with r < k, and suppose we are given a p and q with r = k. Orient p and q: Case 1. Suppose that at two points a and , of p n q, which are adjacent on q, p is oriented in the same direction at a as it is at 3 with respect to the orientationof q. Let p1be a closed polygonal path starting at a point A in a neighbourhood of a (A e Y), proceeding along near p (without intersectingp) to a point B in a neighbourhoodof 3 (B e Y) and returning to A so that BA is contained in Y, and BA meets p and q at one point each (see Fig. 3). By Lemma 1, p -, p1, and p1 n q contains less than k points. There is an n E Nx suchthat npln (x-Y) c pn (x-Y) and that np, n q has less than k points. The inductionhypothesis now implies the existence of a p* such that p* -c np, -c p, p* having the required properties. $ P j, I Pt P Fig. 3. This content downloaded from 210.212.192.130 on Wed, 28 Jan 2015 05:03:47 AM All use subject to JSTOR Terms and Conditions 534 W. B. R. LICKORISH Case 2. Suppose that at three consecutive points on q of p n q, (a, , and ' say) p is oriented in alternating directionswith respect to the direction of q. One of the p segments aoror -a does not contain F. Suppose the nota- tion be chosen so that this segment is -a. Now take for C the closed (polyg- onal) curve shown in Fig. 4. C begins at a point B C Y in a neighbourhood of ai, continues in a neighbourhoodof p (without meeting p) to A e Y, near or,and then returns to B. AB is contained in Y, and AB cuts p twice and q once. , ~~~P I p , ~~~~P I <C A Fig. 4. If h is a C-homeomorphismusing this C, hp c X is as shown in Fig. 5. There is then an n e N, such that nhp is as shown in Fig. 6. n is chosen so that nhpn (x- Y) c p n (x-Y) but so that nhp intersectsq twiceless than p intersects q. Now p - nhp and, by the induction hypothesis, nhp p , where p. is a path with the required properties. C/ y~~~~~/ Fig. 5. Fig. 6. COROLLARY. If p, qj , q21 *O q, are simple closed polygonal paths in X with qj not meeting qj if i j, there exists a path p,* such that: For This content downloaded from 210.212.192.130 on Wed, 28 Jan 2015 05:03:47 AM All use subject to JSTOR Terms and Conditions COMBINATORIAL 3-MANIFOLDS 535 i = 1, 2, ***, r either p* does not meet qi, or p* meets qi twice, with zero algebraic intersection. Also p - c, PROOF. Take neighbourhoods Yi of qi so that Yi n Yj is empty if i # j.