Moran
3
Inventiones math. 5, 160--162 (1968) A Short Proof that Compact 2-Manifolds Can Be Triangulated P. H. DOYLE and D. A. MORAN* (East Lansing, Michigan) The result mentioned in the title of this paper was first proved by RADO [1]; a proof can also be found in [2]. The idea for the present proof is that of the first-named author, who discovered it while investi- gating the possibilities of engulfing in low dimensions. It is shorter than previous proofs, and is presented in the interests of economy. We commence by listing a few familiar facts from geometric topology: The Jordan-Schoenflies Theorem. A simple closed curve J in E ~ separates E 2 into two regions. There exists a self-homeomorphism of E 2 under which J is mapped onto a circle. Thickening an Arc. Each arc in the interior of a 2-manifoM lies in the interior of a 2-cell. This 2-cell can be chosen to be disjoint from any preassigned compact set in the complement of the are. Cellularity and Quotients. For our purposes, a cellular set K is one that can be written as the intersection of a sequence of 2-cells oo K= 0 E~, where E~cIntE~_ 1 (i=2,3 .... ). i=l If K is a cellular subset of a 2-manifoM M, then M/K is homeomorphic to M. (The corresponding statement also holds in n dimensions; see [3], for example.) We shall also have need for the following Lemma. Let M be a closed 2-manifold, and let C be a connected subset of M which is the union of n simple closed curves, C= U c i. i=1 Let A be a compact, totally-disconnected subset of C. Then .4 lies in the interior of a closed 2-cell in M. (A totally-disconnected set is characterized by the property that each of its connected subsets consists of at most one point.) Proof. Since A is compact and totally-disconnected, some subarc S of C1 contains no points of A. Thicken C1- S to obtain a closed 2-cell D1 whose interior contains A n C 1. * Work on this paper was supported by National Science Foundation Grant GP-7126.
-
Upload
yasahswi91 -
Category
Documents
-
view
215 -
download
2
description
moran
Transcript of Moran
Inventiones math.5,160--162 (1968) AShortProof thatCompact2-Manifolds CanBeTriangulated P.H. DOYLEandD. A.MORAN*(East Lansing, Mi chi gan) Theresultment i onedint hetitleof thispaper wasfirstpr ovedby RADO[1];apr oof canalsobef oundin[2].Theideaf or t hepresentpr oof ist hat of t hefi rst -namedaut hor , whodi scovereditwhileinvesti- gatingt hepossibilitiesofengulfing inl owdimensions. I t isshor t er t han previ ousproofs, andispresent edint heinterestsofeconomy.We commence by listing afew fami l i ar facts f r om geomet ri c t opol ogy:TheJordan-SchoenfliesTheorem.AsimpleclosedcurveJinE ~ separatesE 2intotworegions.Thereexistsaself-homeomorphismof E 2 underwhichJismappedontoacircle. ThickeninganArc.Eacharcintheinteriorof a2-manifoMliesinthe interiorof a2-cell.This2-cellcanbechosentobedisjointf r omany preassignedcompactsetinthecomplementof theare. CellularityandQuotients.For our purposes, acellularsetKisone t hat canbewri t t enast hei nt ersect i onof asequenceof 2-cells oo K= 0E~,whereE~cI nt E~_ 1( i = 2 , 3. . . .) .i =lI f Kisacellularsubsetof a2-manifoMM, thenM/ Kishomeomorphic toM. (Thecor r espondi ngst at ement alsohol dsinndi mensi ons; see[3], f or exampl e. ) Weshallalsohaveneedf or thefol l owi ng Lemma. Let Mbeaclosed2-manifold,andletCbeaconnected subsetof Mwhichistheunionof nsimpleclosedcurves, C=Uc i.i =1 Let Abeacompact,totally-disconnectedsubsetof C.Then.4liesinthe interiorofaclosed 2-cell inM. (Atotally-disconnectedset is charact eri zed byt hepr oper t yt hat eachofitsconnect edsubsetsconsistsof at mostonepoi nt . ) Proof. SinceAiscompact andt ot al l y-di sconnect ed, somesubarcS ofC1cont ai nsnopoi nt sofA.Thi ckenC1 - St oobt ai naclosed2-cell D1whosei nt eri orcont ai nsA nC 1. *WorkonthispaperwassupportedbyNationalScience FoundationGrant GP-7126. AShortProof that Compact 2-Manifolds CanBe Triangulated161 Next , supposet hat a2-cellDkhasbeenconst ruct edt osatisfy k Ac~UCicI ntD k . i =1 Ck+l--Dkist heuni onofacollectionofcount abl ymanymut ual l y disjointopenarcsX. whoseendpoi nt slieonBdD k.EachX. cont ai ns somesubarcS. f or whichA c~ S.=0, sot hat t het woarcscompri si ng X~- S. canbet hi ckenedt of or mclosed2-cellsE.1andE. 2which (a)avoi deachot her, (b)avoi dX~f or 2 # p , and(c)meet Dkinclosed 2-cells.Then Dk+, =Ok uU(E.1 uE.2) / tisaclosed2-cell whosei nt eri orcont ai ns k+lAc ~UCi.[ ]i =1 Theorem. Anyclosed2-rnanifoMMcanbetriangulated. Proof.Cover Mi rreduci bl ybyafinitecollectionofdos eddisks {B1,B 2 . . . . . B.},andput C~=BdB~. Let Abet hesingularsetof n C= UC/ ,i =1 i.e.Aconsistsoft hosepoi nt sof Cwhi chdonot have1-dimensional eucl i deannei ghbor hoodsinC.Aiscompact andt ot al l y-di sconnect ed,and, sincet hecover{Bi}wasirreducible, nopr oper subsetof {B~} coversM; t husCisconnect ed, sobyt hel emma, Aliesint hei nt eri or ofaclosed2-cellDinM. Not et hat M- Cist heuni onofacount abl e col l ect i onofmut ual l ydisjointopen2-cells,eachofwhoseclosuresin Misaclosed2-cell;thisisaneasyconsequenceoft heJor dan- Schoen-flies Theor em. Not ealsot hat C- Dconsistsof count abl y many mut ual l y disjointopenarcswhoseendpoi nt salllieonBdD.Now D cUc M, whereUi s an open 2-cell, as can be seen by t hi ckeni ng Bd Dslightly;henceDiscellular(againbyt heJordan-Schoenfl i es Theor em) , andsoM1 =MIDist opol ogi cal l yequi val ent t oM. M1is t hecopyofMwi t hwhichweshallwor kf r omthispoi nt on.Let R cM1bet hei mageof ~under t hequot i ent map; t husR istheone- poi nt - uni onofacount abl ecollectionofsimpleclosedcurves, andislocallyeucl i deanexcept at onepoi nt p(theimageofDundert hequot i ent map). Mor eover , M1 - Rist opol ogi cal l yequi val ent t o M- ( C uD).Fur t her mor e, any2-cellnei ghbor hoodVof pwillcont ai n allbut afinitenumber oft hesimpleclosedcurveswhi chcompri seR;thisfollowsf r omthecompact nessofC- D, henceofR.Ifeachofthe 162P. H. DOYLE andD. A. MORAN: Compact 2-Manifolds si mpl edos e dcurveslyingwi t hi nVisspannedbyt hedi skitbounds,acellularsetT,consi st i ngofaone- poi nt - uni onofcloseddisks,is obt ai ned. ( Toseet hat Tiscellular,l et Nbeanynei ghbor hoodofT. ThenNcont ai nsa2-cellnei ghbor hoodV' ofp, whi chmus t int ur n cont ai nallbut finitelyma nyoft hesi mpl ecl osedcurvescompr i si ngR.Thi ckeni ngt hearcsof Rwhi chlie out si deV' andappendi ngt heresul t i ng cellstoT u V' yieldsacellQ,whereTcI nt Q cQ=N. ) Therefore, pass- i ngt oM1/ T ifnecessary, wel osenogeneral i t yinassumi ngt hat Risan r-l eafedrosewhosecompl ement inM1iscompos edoffinitelyma ny component s t hat areopen2-cells. Fi nal l y, enclosepinasmal l closed2-cellEmeet i ngeachsi mpl e cl osedcurveinRint wopoi nt sofitsboundar y, BdE. E u R ist hena di skwi t hafinitenumbe r ofmut ual l ydi sj oi nt closedarcsA1,- 4 2 . . . . ,- 4 rmeet i ngBd Eint hei rendpoi nt s. Each-4ima ybeencl osed(except f oritsend-poi nt s)inthei nt eri orof acloseddi skmeet i ngEinapai r ofarcs onitsboundar y. Byselectingthesediskst obedi sj oi nt inpai rsone obt ai nsat ri angul abl e2- mani f ol dNwi t hboundar y. Thet r i angul at i on of Nis nowext endedt ot hecl osureof eachc ompone nt ofM 1 - -N.[ ]Cor ol l ar y1.Anycompact2-manifoMcanbetriangulated. Proof.I ncl udet hebounda r ycurvesint hesetCoft hepr oof oft he t heor em; mi nor modi f i cat i onsoftheabovear gument willconver t it i nt oapr oof oft hecorol l ary. [ ]Corol l ary 2. Anycompact2-manifoldwithnon-voidboundaryisem- beddableinE 3. Proof.Gi vensuchamani f ol dM, Mcanbeembeddedi naclosed mani f ol dM 1.Let M 1 = E 2 u R,ast andar ddecomposi t i on[4],whereR isanr-l eafedrosedi sj oi nt f r omt heopen2-cellE2.ThenacopyofM liesineverynei ghbor hoodofR, f or anyembeddi ngofRinE3.[ ]References 1.RAI)6, T.:~berdenBegriffderRiemannschenFl/iche.Acts.Litt.Sci. Szeged. 2,101--121(1925). 2.AHLFORS,L. V.,andL. SAmo: Riemannsurfaces. Princeton:PrincetonUniversity Press1960. 3.DOUADY,A. : Plongementsdes spheres.S6minaire Bourbaki13, expos6 205(1960). 4.DOYLE,P.H.,andJ. G. HOCKING: Adecompositiontheoremforn-dimensional manifolds.Proc.Amer.Math.Soc.13, 469--471(1962). Dr. P. H. DOYLE andDr. D. A.MORAN MichiganStateUniversity Department of Mathematics East Lansing,Michigan48823, USA (Received December 8, 1967)
