ply the eistence of homeomorphic modifictions. Outline. Section 2 introduces definitions from comintoril topology. Section 3 defines oundry; its sic p

Transcription

1 Topology Preserving Edge Contrction Λ Tml K. Dey y, Herert Edelsrunner z, Sumnt Guh nd Dmitry V. Nekhyev Astrct We study edge contrctions in simplicil complees nd locl conditions under which they preserve the topologicl type. The conditions re sed on generlized notion of oundry, which lends itself to defining nested hierrchy of tringulle spces mesuring the distnce to eing mnifold. 1 Introduction This pper studies the opertion of shrinking or contrcting n edge in simplicil comple. The repeted ppliction of this opertion eventully reduces ny connected comple to single verte. During tht process, the comple loses ll non-trivil topologicl properties. We re interested in recognizing edges tht cn e contrcted without chnging the topologicl type. The repeted ppliction of such contrctions simplifies the comple while preserving its type. This mens there is homeomorphism tht connects the originl with the simplified comple, nd we re lso interested in constructing such homeomorphism. Motivtion. Edge contrctions re used in computer grphics to simplify surfces for fst rendering. A surfce consists of tringles in R 3 connected to ech other Λ Reserch y the first two uthors is prtilly supported y the U.S. Deprtment of Energy, under Sucontrct B Reserch y the second uthor is lso supported y NSF under grnt CCR , nd y ARO under grnt DAAG Reserch y the third uthor is lso supported y NSF under grnt CCR y Deprtment of Computer Science nd Engineering, Indin Institute of Technology, Khrgpur , Indi. z Deprtment of Computer Science, Duke University, Durhm, North Crolin 27708, nd Rindrop Geomgic, Reserch Tringle Prk, North Crolin Deprtment of Electricl Engineering nd Computer Science, University of Wisconsin t Milwukee, Milwukee, Wisconsin 532. Rindrop Geomgic, Reserch Tringle Prk, North Crolin long shred edges nd vertices. In mthemticl lnguge it is 2-comple, nd ppers in computer grphics generlly restrict themselves to 2-mnifolds with or without oundry. The pioneering puliction in this contet is Hoppe et l. [5], ut see lso Grlnd nd Heckert [3] for n effective numericl prioritiztion of edge contrctions. A relted topic is prmetriztion, where surfce is to e covered y possily few nd lrge ptches, ech the homeomorphic imge of n open disk in R 2. The first workle solution is descried in recent pper y Lee et l. [6]. The lgorithm simplifies the surfce incrementlly nd mintins piecewise liner homeomorphism. The simplifiction is generted through repeted verte removl, ut the sme lgorithmic ide cn lso e sed on edge contrction [2]. For this to work it is essentil tht ech simplifiction step preserves the topologicl type. Summry of Results. Edge contrctions tht preserve the topologicl type re recognized y locl criteri phrsed in terms of generlized concept of oundry. Roughly ut not ectly, the order of simple ff in simplicil comple is the smllest integer i such tht the underlying spce of the str of ff is homeomorphic to R h X, for some topologicl spce X of dimension i. We could let X equl the str of ff nd let h e zero, so the order is well-defined nd t most the dimension of the str of ff. The j-th oundry of simplicil comple consists of ll simplices of order i j. We introduce the Link Conditions for n edge considered for contrction. They require tht within ech oundry the link of is equl to the intersection of the links of nd of. More precisely, for ech j the reltion etween the links of ; ; is required within the j-th oundry etended y cones connecting the (j +1)-st oundry to dummy verte. For 2-complees we prove tht the Link Conditions chrcterize edge contrctions tht permit homeomorphic modifiction limited to the strs of nd. We prove the sme for 3-mnifolds. For generl 3-complees we only prove tht the Link Conditions im-

2 ply the eistence of homeomorphic modifictions. Outline. Section 2 introduces definitions from comintoril topology. Section 3 defines oundry; its sic properties re estlished in Appendi A. Section 4 introduces edge contrctions nd unfoldings. Sections 5 nd 6 prove the mentioned results for contrctions in 2- complees nd in 3-complees. Section 7 concludes the pper. 2 Bsic Definitions We use concepts nd terminology from comintoril topology nd discuss simplicil complees, topologicl spces, nd mps etween spces. Most ut not ll definitions re stndrd, nd the stndrd ones cn lso e found in tetooks such s Munkres [9]. Simplicil complees. A k-simple, ff, is the conve hull of k ffinely independent points. Its dimension is dim ff = k. A fce of ff is simple, fi, defined y non-empty suset of the k + 1 points, nd fi is proper if the suset is proper. We cll ff cofce of fi nd write fi» ff. The interior, int ff, is the set of points contined in ff ut not in ny proper fce of ff. A simplicil comple, K, is finite collection of simplices so ff 2 K nd fi» ff implies fi 2 K, nd ff;ff 0 2 K implies ff ff 0 is either empty of fce of oth. All complees in this pper re simplicil. The dimension of K is dimk = mfdim ff j ff 2 Kg. A d-comple is simplicil comple of dimension d. A sucomple is simplicil comple L K. A simple in K is principl if it hs no cofce in K. The verte set contins ll 0-simplices: Vert K = fff 2 K j dim ff = 0g. The underlying S spce is the union of simple interiors: j K j = ff2k int ff. Let B e suset of K tht is not necessrily sucomple. The closure of B is the set of ll fces of simplices in B. The str of B is the set of ll cofces of simplices in B. The link of B is the set of ll fces of cofces of simplices in B tht re disjoint from simplices in B. In more forml nottion, B = ffi 2 K j fi» ff 2 Bg; St B = fff 2 K j ff fi 2 Bg; Lk B = St B St B: The closure is the smllest sucomple tht contins B. The link is lwys comple while the str is generlly not comple. The closure of the str is lwys comple nd denoted s St B = St B. The concepts of dimension nd of underlying spce etend immeditely to S susets of comple: dim B = dim B nd j B j = ff2b int ff. We introduce opertions tht crete new simplices nd complees from old ones. The cone from point to simple ff is defined if is not n ffine comintion of the vertices of ff, nd in this cse it is the simple ff = conv( [ ff) of dimension dim ff + 1. A sudivision of K is comple Sd K so j K j = j Sd K j nd every simple in Sd K is contined in simple in K. Sudivisions cn e creted y vrious opertions. One such opertion is strring from point 2 j K j: remove ll simplices tht contin nd dd together with the cones from to the fces of the removed simplices tht do not contin. Topologicl spces. A d-dimensionl point is d- tuple of rel numers. The norm of point = ( 1 ; 2 ;::: ; d ) is kk = ( P 2 i )1=2. The d-dimensionl Eucliden spce, R d, is the set of d-dimensionl points together with the Eucliden distnce function tht mps ech pir of points ; y to the non-negtive rel k yk. In ddition to R d we need nmes for three other stndrd topologicl spces: the (d 1)-sphere, the d-ll, nd the d-hlfspce: S d 1 = f 2 R d j kk = 1g; B d = f 2 R d j kk» 1g; H d = f 2 R d j 1 0g: Most spces in this pper re underlying spces of complees K. The spce j K j is suset of some Eucliden spce R e, nd it is equipped with the suspce topology inherited from the Eucliden topology of R e. A d-mnifold is non-empty topologicl spce, M, so every point 2 M hs n open neighorhood homeomorphic to R d. For complees it suffices to check the defining condition t the vertices: j K j is d-mnifold iff the underlying spce of every verte str is homeomorphic to R d. A d-mnifold with oundry is non-empty topologicl spce, N, so every 2 N hs n open neighorhood homeomorphic to R d or to H d. The oundry of N is the set of points with neighorhoods homeomorphic to H d. The oundry is either empty or (d 1)- mnifold. For complees it gin suffices to check the defining condition t the vertices: j K j is d-mnifold with oundry iff the underlying spce of every verte str is homeomorphic to R d or to H d. Note tht every d- mnifold is d-mnifold with oundry, nmely empty oundry, ut d-mnifold with non-empty oundry is not d-mnifold. Mps. A homeomorphism etween two topologicl spces X nd Y is ijection h : X! Y so h nd h 1 re oth continuous. If such n h eists then X nd Y re homeomorphic, denoted s X ß Y, nd they re sid to hve the sme topologicl type. A tringultion of X is simplicil comple K with X ß j K j. X is tringulle if it hs tringultion. 2

3 We need some definitions to introduce the comintoril counterprt of homeomorphism. A verte mp for two complees K nd L is function f : Vert K! Vert L so the vertices of simple in K re mpped to the vertices of simple in L. The rycentric coordintes of point 2 ff, ff 2 K, re the unique rels u (), u 2 Vert K, so u () 6= 0 only if u» ff nd = 1 = X X u2vert K u2vert K u () u; u (): We use rycentric coordintes to etend f in piecewise liner fshion. The simplicil mp ffi : j K j! j Lj is defined y ffi() = X u2vert K u () f(u) for every 2 j K j. The mp ffi is continuous y construction, ut it is neither necessrily injective nor necessrily surjective. It is homeomorphism iff f is ijective nd f 1 is lso verte mp. In this cse ffi is n isomorphism nd K nd L re isomorphic, which is denoted s K ο L. K nd L re comintorilly equivlent if they hve isomorphic sudivisions, which is denoted s K ' L. We need the concept of comintoril equivlence lso for susets of complees: B ' C if there is n isomorphism j B j! j C j tht mps j B j to j C j. We comment tht there is sutle difference etween the piecewise liner nd the topologicl ctegories. This ws first discovered y Milnor [8] who ehiited two homeomorphic tringultions tht re not comintorilly equivlent. To void relted difficulties we sty within the piecewise liner ctegory y sing further definitions on the notion of comintoril equivlence. All pplictions of comintoril equivlence in this pper re to complees nd to sets of simplices whose complement in the closure re complees. We therefore do not need comintoril theory of non-compct spces. 3 Order nd Boundry The results of this pper rest on the fundmentl concept of strtifiction defined s nested sequence of oundries. For 2-complees these oundries hve een defined erlier y Whittlesey [13]. We egin y defining the order of simple nd then proceed to introduce the strtifiction. Order of simple. Let ff e simple in comple K, nd let k e the dimension of the str: k = dim St ff. The order of ff is the smllest integer i = ord ff for which there is (k i)-simple with comintorilly equivlent str: St ff ' St. We ssume elongs to some suitle other comple so its str is defined. Since int is homeomorphic to R k i the str of ff is homeomorphic to R k i X, for some topologicl spce X of dimension i. Recll tht i is chosen s smll s possile. The order cnnot eceed the difference etween the dimension of ff nd the dimension of its str: Order Bound. ord ff» dim St ff dim ff. Proof. For i = k dim ff we hve dim = k i = dim ff nd cn therefore choose = ff. The strs of nd ff re the sme nd therefore certinly comintorilly equivlent. Shrk-fin emple. The shrk-fin comple in Figure 1 illustrtes some of the definitions. It is constructed y gluing two closed disks (tringultions of B 2 ) long simple pth. Tht pth is contiguous piece of the oundry of one disk (the fin) nd it lies in the interior of the other disk. The dimension of the shrk-fin Figure 1: The shrk-fin comple hs dimension 2 nd vertices of ll orders: 0, 1, 2. comple is 2 so every tringle hs order 0. Ech edge elongs to one, two, or three tringles nd we cll this numer the degree of the edge. All degree-2 edges hve order 0 nd the others hve order 1. The degree-3 edges re witnesses of the fct tht the shrk-fin comple is not mnifold with oundry. The violtion of the mnifold property y degree-3 edges is less severe thn the violtion tht cn e found t the two endpoints of their pth. The str of n order-0 verte is disk, nd tht of n order-1 verte is cycle of hlf-disks glued long pth of two edges. There re two order-2 vertices, nd their strs re more complicted thn disks or glued hlf-disks. Boundry of comple. The j-th oundry of simplicil comple K is the set of simplices with order no less thn j: Bd j K = fff 2 K j ord ff jg: By the Order Bound, the j-th oundry contins only simplices of dimension dim K j or less. Consider gin 0 2 3

4 the shrk-fin comple in Figure 1. The 1-st oundry consists of two circles. Even though the 1-st oundry of the two circles is empty, the 2-nd oundry of the shrk-fin is non-empty nd consists of the two endpoints of the pth long which the two disks re glued, see lso Property 5 in Appendi A. Note tht oth oundries of the shrk-fin re complees. Property 3 in Appendi A shows tht this is not coincidence nd tht every oundry of comple is gin comple. Property 2 sserts tht the j-th oundry is topologicl concept nd does not depend on the tringultion. More precisely, the restriction of simplicil homeomorphism to the j-th oundries of two complees is gin simplicil homeomorphism. The j-th oundry contins the (j + 1)-st oundry. Hence, if the j-th oundry is empty then ll lter oundries re lso empty. Underlying spces of complees with empty 1-st oundry re mnifolds, ut the reverse is not true. Underlying spces of complees with empty 2-nd oundry cn e mnifolds with oundry ut cn lso e different. For emple, the 2-nd oundry of 2-comple tht tringultes sphere with equtor disk is empty, ut the comple is not mnifold with oundry. Hierrchy of complees. The oundry concept cn e used to define hierrchy of progressively more complicted complees. Let M j e the clss of simplicil complees with empty oundries eyond inde j. Since succeeding oundries re contined in preceding ones, we hve M j = fk j Bd j+1 K = ;g: The only memer of M 1 is the empty comple. The clsses form nested hierrchy: f;g = M 1 ρ M 0 ρ M 1 ρ M 2 ρ ::: ; nd ll inclusions re proper. For comple we use the minimum inde i with K 2 M i s mesure of how complicted it cn get loclly. It is plusile ut lso true tht the i-th oundry is t lest i clsses simpler thn the originl set. Nesting Lemm. If K 2 M j then Bd i K 2 M j i. Proof. K 2 M j iff Bd j+1 K = ;. By Property 5 in Appendi A we hve Bd` Bd i K Bd i+` K for every inde ` 0. This implies Bd j i+1 Bd i K = ;, which is equivlent to Bd i K 2 M j i. Note, however, tht Bd i K cn e more thn i clsses simpler thn K. Consider for emple two tetrhedr tht meet t common verte, u, nd let K e the 2- comple of proper fces. Verte u hs order 2 nd ll other simplices hve order 0. It follows tht K 2 M 2 nd Bd 1 K = fug 2 M 0, which is one inde stronger thn climed y the Nesting Lemm. 4 Edge Contrction The motivtion for defining j-th oundry is its role in chrcterizing edge contrctions tht preserve the topologicl type. We egin y defining edge contrctions nd then proceed to discussing conditions under which they permit the construction of isomorphic sudivisions. Edge contrctions. The contrction of n edge in comple K replces St = St [ St y the str of new verte, St c. Let E = St nd C = St c e the closures of the two strs. E nd C connect to the rest of K t the common link of nd c, which is X = E St = C St c. We cn think of the contrction s surjective simplicil mp ' : j K j! j Lj defined y the surjective verte mp f(u) = ρ u c if u 2 Vert K f; g; if u 2 f; g: Outside j E j, ' is the identity, ut in the interior it is not even injective. We re interested in wys to mke edge contrctions homeomorphic. An unfolding of ' is simplicil homeomorphism : j K j! j Lj. It is locl if differs from ' only inside j E j, nd it is reled if differs from ' only inside j St E j. Clerly every locl unfolding is lso reled, ut not every reled unfolding is locl. Isomorphic sudivisions. Ech unfolding of ' corresponds to pir of isomorphic sudivisions of K nd L. If the sudivisions ffect only St nd St c then is locl, nd if they only ffect St E nd St C then is reled. Sudivisions of oth kinds cn e generted from isomorphic sudivisions Sd E of E nd Sd C of C. Sudivisions tht eploit symmetry need to e voided since they cnnot e comined with the identity. We therefore sy the isomorphism : j Sd E j! j Sd C j preserves the connection if () 2 ff for every point 2 ff 2 X, where X = E C s efore. We cll Sd E trnsprent if X Sd E, nd similr for C. The restriction to j X j of ny connection preserving isomorphism defined y trnsprent sudivisions is necessrily the identity. Isomorphic Sudivision Lemm. If E nd C hve sudivisions Sd E nd Sd C dmitting connection preserving isomorphism then ' hs reled unfolding. If furthermore Sd E nd Sd C re trnsprent then ' hs locl unfolding. Proof. We first show the second clim. Since SdE is trnsprent, we cn replce E y Sd E nd get sudivision of the entire comple, nd similr for C. Let these sudivisions e K 0 = (K E) [ Sd E nd L 0 = (L C) [ Sd C. To see tht K 0 nd L 0 re isomorphic note tht they shre X = E C y ssumption 4

5 of trnsprency. On one side of X we hve n isomorphism j Sd E j! j Sd C j whose restriction to j X j is the identity. On the other side of X we hve the identity ecuse K 0 Sd E = L 0 Sd C. To show the first clim we form K 0 nd L 0 s efore, ut ecuse X my hve een sudivided, K 0 nd L 0 my not e complees. Let ff 2 K E e simple with fce fi 2 E tht hs een sudivided in Sd E. In this cse fi 62 K 0, nd to loclly repir the comple property we only need to sudivide ff y strring from n interior point. The strring is done inductively in the order of non-decresing dimension, nd it effects only simplices in St E E. Whenever ff is sudivided within K 0 it is similrly sudivided within L 0. The result re isomorphic sudivisions of K 0 nd L 0 defining reled unfolding of '. By definition, if ' hs locl or reled unfolding then K ' L. We will see in Section 5 tht the reverse is not true: there re edge contrctions with unfoldings tht re necessrily glol. Link conditions. We formulte generl condition, which we show implies edge contrctions with locl nd reled unfoldings in some cses. For ech i we etend the i-th oundry y dding dummy verte,!, nd cones from! to ll simplices in the (i + 1)-st oundry: Bd! i K = Bd i K [! Bd i+1 K: If Bd i+1 K = ; then Bd! i K = Bd i K. We re only interested in the topology of the etended comple nd do not worry out the loction of! nd the geometric shpe of the cones. For simple ff 2 Bd! i K we denote the link within Bd! i K s Lk! i ff. Link Conditions. Lk! i Lk! i = Lk! i, 8i 0. Refer to the two portions of surfce tringultion in Figure 2 s emples. In oth cses only the links within K re relevnt, tht is, we only consider the cse i = 0. y Figure 2: To the left we hve Lk Lk = f; yg = Lk, nd the contrction of hs locl unfolding. To the right we hve Lk Lk = f; y; z; zg 6= Lk, nd the contrction of hs no unfolding. z y Redundncy. Tle 1 unwinds the Link Conditions for d-comple in M j. The conditions simplify for lrge vlues of i nd j. For i = j in the digonl of Tle 1 the etension with! is redundnt. For i = j = d the i-th oundry is 0-comple so ll links re empty nd the condition is void. For i = j = d 1 1 the condi- M 0 M 1 ::: M d 1 M d 0 Lk 0 Lk! 0 ::: Lk! 0 Lk! 0 1 Lk 1 ::: Lk! 1 Lk! d 1 ; Lk! d 1 d ; Tle 1: For comple in M j there re j + 1 conditions, some my e void (i = j = d) nd some my e susumed y others (i = j = d 1). tion is susumed y the condition for d 2. To see this note tht Bd d 1 K hs dimension t most 1. The condition thus simplifies to Lk d 1 Lk d 1 = ;, which is violted iff nd elong to cycle of three edges. Let e the third verte. Then the edge! 2 Lk! d 2 ecuse elongs to the (d 1)-st oundry nd thus! 2 Bd! K. Similrly, d 2! 2 Lk! d 2, ut! 62 Lk! d 2 ecuse the etended (d 2)-nd oundry is 2-comple nd thus contins no tetrhedr. 1-complees. It is instructive to consider the firly strightforwrd cse of 1-comple or grph G. The contrction of n edge 2 G chnges the topologicl type iff nd hve common neighor or oth hve degree different from 2, see Figure 3. The two cses re Figure 3: The contrction of removes loop to the left nd verte of the 1-st oundry to the right. cptured y the Link Conditions for d = j = 1. Indeed, Lk! 0 Lk! 0 = ; iff is different from the two cses illustrted in Figure 3. Suppose now tht stisfies the link condition nd ssume without loss of generlity tht hs order 0. In this cse we get locl unfolding y sudividing c into u; uc, where 6= is the other neighor of. If violtes the link condition then there is no unfolding, not even non-locl one. Theorem A. If G 2 M 1 is 1-comple then the following sttements re equivlent: (i) Lk! 0 Lk! 0 = ;. (ii) ' hs locl unfolding. (iii) ' hs n unfolding. 5

6 5 2-Complees This section proves tht for 2-complees the Link Conditions chrcterize edge contrctions with locl unfoldings. This result is strengthened for 2-mnifolds where the Link Conditions chrcterize edge contrctions tht hve ny unfolding t ll. Strs nd hlf-strs. To prove tht n edge contrction ' hs locl unfolding we estlish trnsprent sudivisions of E = St nd C = St c tht permit connection preserving isomorphism. This tsk is simplified y ssuming ord = ord nd choosing the new verte c equl to. The contrction cn then e visulized y sliding towrds nd eventully merging into, see Figure 4. The opertion only ffects simplices in A = St nd leves simplices in E A unchnged. Cll R = Lk = A St the rim of A. Of the simplices in A the ones in St dispper or merge into the rim, nd the others remin ut ssume different geometric shpe nd position. We cll A 0 = A St the hlf-str of nd R 0 = A 0 (St St ) the rim of A 0. The imge of A 0 under the contrction is n isomorphic sucomple C 0 of C. To estlish trnsprent sudivisions of E nd C tht permit connection preserving isomorphism it suffices to construct such sudivisions of A nd C 0. This is equivlent to constructing isomorphic sudivisions of A nd A 0 tht permit n isomorphism whose restriction to the intersection of the two rims is the identity. It is therefore essentil tht R ο R 0, which will lwys e the cse when we pply the construction. Figure 4: The overly of two sudivisions of regulr k- gon. To the left the sudivisions re otined y strring from the center nd from verte. To the right they re otined y strring from the midpoint nd from n endpoint of n edge. Figure 4 illustrtes the construction of isomorphic trnsprent sudivisions. Both A nd A 0 re mpped isomorphiclly to sudivisions of the sme regulr k- gon. To the left, A nd A 0 re disks of k nd k 2 tringles. To the right, A nd A 0 re hlf-disks of k 1 nd k 2 tringles. Generl 2-complees. Let K e 2-comple nd 2 K. Recll tht the contrction of is simplicil mp ' : j K j! j Lj. There re three Link Conditions nd Tle 1 indictes tht the lst one is void. Theorem B. If K 2 M 2 is 2-comple then the following sttements re equivlent: (i) stisfies the Link Conditions for j = 2: (i.0) Lk! 0 Lk! 0 = Lk! 0, nd (i.1) Lk! 1 Lk! 1 = ;. (ii) ' hs locl unfolding Cse 1 Cse 2 Cse Figure 5: From left to right, is principl, is not principl nd does not elong to the 1-st oundry, elongs to the 1-st oundry. In ech cse ord = ord nd in the upper row hs the sme nd in the lower row it hs higher order. Proof. (i) =) (ii). The rgument for sufficiency of the Link Conditions distinguishes three cses ll of which re illustrted in Figure 5. We note tht the order of n endpoint of edge is t lest s lrge s the order of, see Property 3 of Appendi A. The Link Conditions imply tht not oth endpoints cn eceed the order of, nd we ssume without loss of generlity tht ord = ord. Cse 1. is principl. Thus ord = ord = 0, nd elongs to ectly two edges: nd. L is otined from K y removing ;; nd dding. The isomorphic trnsprent sudivisions of f; ; g nd fg re ovious. Cse 2. is not principl nd ord = ord = 0. Mp A = St isomorphiclly to sudivision of the regulr k-gon s in Figure 4 to the left, where k is the numer of tringles in A. Similrly, mp the hlf-str A 0 = A St isomorphiclly to sudivision of the regulr k-gon into k 2 tringles. The two rims mp isomorphiclly to the oundry of the k-gon nd the mps gree t R R 0. We cn therefore form common sudivision tht leves the oundry of the k-gon unchnged. This sudivision mps ck to trnsprent sudivisions Sd A, 1 0 6

7 Sd A 0, nd Sd C 0. We dd the remining simplices of E to Sd A nd the remining simplices of C to Sd C 0 nd get trnsprent sudivisions of E nd C tht permit n isomorphism whose restriction to the underlying spce of X = E C is the identity. In other words, the isomorphism preserves the connection. The Isomorphic Sudivision Lemm implies (ii). Cse 3. ord = ord = 1. Let e the other edge with order 1 in the str of. A is cycle of hlfdisks, nd ech hlf-disk is fn of tringles strting t nd ending t. Let D e the closure of such hlf-disk nd let k 1 e the numer of tringles in D. Mp D isomorphiclly to sudivision of the regulr k-gon with the imge of t the midpoint of k-gon edge, s in Figure 4 to the right. A 0 is nother cycle of hlf-disks, this time round. Let D 0 e the closure of the hlf-disk tht corresponds to D. Mp D 0 isomorphiclly to sudivision of the regulr k-gon, with the imge of t the verte tht is the imge of under the erlier mp. We form common sudivision tht leves the k-gon oundry unchnged, ecept for one edge which is cut into two y the imge of under the first mp. We denote this edge s yz. Tht sudivision mps ck to isomorphic sudivisions of D nd D 0. Both sudivisions re trnsprent ecept t the preimges of yz. The sudivisions of the hlf-disks of A meet long ; nd together they form Sd A, which is trnsprent. Similrly, the sudivisions of the hlf-disks of A 0 meet long. The imges of these sudivisions together form Sd C 0, which is lso trnsprent. By construction Sd A nd Sd C 0 re isomorphic. After dding the remining simplices of E to Sd A nd those of C to Sd C 0 we hve trnsprent sudivisions of E nd C tht permit connection preserving isomorphism. The Isomorphic Sudivision Lemm implies (ii). :(i) =) :(ii). The rgument for necessity of the Link Conditions distinguishes etween the violtion of (i.0) nd tht of (i.1). We will show tht in either cse the topologicl type of the link of t lest one simple in Lk chnges. Since locl unfolding is the identity outside St, this contrdicts its eistence. Note tht the link of n edge is contined in the link of its endpoints nd define L 0 = (Lk! 0 Lk! ) 0 Lk! 0 nd L 1 = Lk! 1 Lk!. 1 Cse 1. (i.0) is violted, or equivlently L 0 6= ;. First suppose tht L 0 contins n edge y. The link of y is finite set of vertices tht contins oth nd. The contrction of decreses the crdinlity of Lky nd thus chnges its type. Net suppose tht L 0 contins no edge ut it contins verte. Lk is 1- or 0-comple tht contins oth nd s vertices. If nd elong to different components then the contrction of merges these components. Otherwise, every pth from to in Lk hs t lest three edges, for y; y 2 Lk implies y; y 2 K nd therefore y 2 L 0, which contrdicts the ssumption. The contrction of thus forms one or more cycles. In oth cses the topologicl type of Lk chnges. Cse 2. (i.0) is stisfied nd (i.1) is violted, or equivlently L 0 = ; nd L 1 6= ;. Then ; 2 Bd 1 K for else L 1 would e empty, nd 2 Bd 1 K for else! 2 L 0 would violte (i.0). Let e verte in L 1. If =! then ; 2 Bd 2 K. St thus contins two vertices of order 2, while St c contins t most one such verte, nmely c. This contrdicts E ' C. Finlly suppose 6=!. We hve ; ; 2 Lk, nd ecuse ; 2 Bd 1 K the degrees of nd in Lk re oth different from 2. In other words, nd oth elong to Bd 1 Lk, which is set of vertices. The contrction of removes verte from the 1-st oundry nd thus chnges the type of Lk. Non-locl isomorphism. An edge contrction with locl unfolding preserves the topologicl type, ut there re type preserving edge contrctions tht hve neither locl nor reled unfolding. An emple is the folding chir comple illustrted in Figure 6. Before the contrction of the comple consists of 5 tringles in the str of nd 4 disks U; V; Y; Z glued to the link of s shown. Vertices nd elong to the 1-st oundry, ut does not. The dummy verte! thus violtes the Link Condition for i = 0 nd so does. After the con- U Y V Z Figure 6: The 2-dimensionl folding chir comple. Ft edges elong to three tringles ech. trction there is one less tringle in the str of, U loses two tringles, nd V; Y; Z re unchnged. The contrction echnges left nd right in the symmetry of the comple. We cn find homeomorphism j K j! j Lj tht cts like mirror, mpping U to V, V to U, Y to Z, Z to Y. Indeed, every homeomorphism must ct this wy nd differ from the identity lmost everywhere. U Y c V Z 7

8 2-mnifolds. The 2-comple K elongs to M 0 iff j K j is 2-mnifold. In this cse condition (i.1) in Theorem B is void nd condition (i.0) simplifies ecuse the etension with! is redundnt. We strengthen the result implied y Theorem B y proving tht the violtion of the Link Condition contrdicts the eistence of ny unfolding, whether locl or not. Theorem B 0. If K 2 M 0 is 2-comple then the following sttements re equivlent: (i) Lk Lk = Lk. (ii) ' hs locl unfolding. (iii) ' hs n unfolding. Proof. (i) =) (ii) follows from Theorem B nd (ii) =) (iii) from the definitions. To prove :(i) =) :(iii) we distinguish two cses depending on the dimension of the violting simple. Let L 0 = (Lk Lk ) Lk. :(i) is equivlent to L 0 6= ;. Cse 1. L 0 contins n edge y. Then y nd y re replced y single tringle cy. Hence y elongs to only one tringle in L, which contrdicts L 2 M 0. Cse 2. L 0 contins no edge ut it contins verte. Then nd re edges in K. Ech elongs to two tringles: p 6= q, r 6= s. The four tringles re pirwise different ecuse 62 K. The four vertices p; q; r;s re pirwise different ecuse L 0 contins no edge. Hence c elongs to four tringles in L, which contrdicts L 2 M Complees This section etends Theorems B nd B 0 to complees of dimension 3. We egin with the min geometric tool, which is Steinitz' clssicl theorem for conve 3- polytopes [11]. Steinitz' theorem. A conve 3-polytope is the conve hull of finitely mny points in R 3 tht do not ll lie in common plne. Its oundry is comple of vertices, edges, nd (2-dimensionl) fcets. If the points re in generl position then ll fcets re tringles. The 1-skeleton is the sucomple of ll vertices nd edges. A grph G is 1-comple. It is connected if for every prtition of Vert G into two non-empty sets, G contins n edge with one endpoint in ech set. A connected grph is three-connected if the deletion of ny two vertices together with their edges leves the grph connected. G is plnr if it is isomorphic to 1-comple in R 2. For emple, the 1-skeleton of every conve 3-polytope is plnr nd three-connected. A fundmentl result y Steinitz sserts tht these 1-skeletons ehust ll threeconnected plnr grphs [11]. Steinitz' Theorem. For every three-connected plnr grph there is conve 3-polytope with n isomorphic 1-skeleton. Plnr grphs tht re not three-connected rise y removing edges nd vertices. Let X e tringultion of S 2. Let 2 X e verte nd consider the grph G consisting of ll edges nd vertices in X other thn the ones in the str of. A drwing of G in R 2 hs one k-gon nd otherwise only tringles. The oundry of the k-gon is the link of within X, nd we denote it y U. In Figure 7 we hve k = 8 nd the k-gon is the unounded outer region of the drwing. G is three-connected iff no edge y Figure 7: A tringultion of the 2-sphere fter removing verte of degree k = 8. There re three edges y tht re cut into two ech to restore three-connectedness. y 2 G U hs oth endpoints in U. If there is such n edge y we repir three-connectedness y cutting y t n interior point z nd connecting z to the opposite vertices of the two djcent tringles. This is done in ny opportune sequence over ll such edges y, s in Figure 7. The opertion corresponds to sudividing X y strring from the points z in the sme sequence. Generl 3-complees. Let K e 3-comple nd let 2 K. There re four Link Conditions, nd Tle 1 indictes tht the lst one is void. Theorem C. If K 2 M 3 is 3-comple then the first sttement implies the second: (i) stisfies the Link Conditions for j = 3: (i.0) Lk! 0 Lk! 0 = Lk!, 0 (i.1) Lk! 1 Lk! 1 = Lk! 1, nd (i.2) Lk! 2 Lk! 2 = ;. (ii) ' hs reled unfolding. Proof. We distinguish four cses, the first of which hs een treted in the proof of Theorem B. Recll tht the order of n endpoint of is t lest s lrge s the order of. The Link Conditions imply tht not oth endpoints cn eceed the order of, nd we ssume ord = ord. y 8

9 Cse 1. dim St» 2. The Order Bound implies ord = ord» 1. The presence of tetrhedron in St would imply one in St, hence dim St» 2. In other words, the neighorhood of is s in Theorem B. Let K 0 e the 2-comple otined from K y removing ll simplices ff tht stisfy dim St ff = 3 nd ord ff = 0. For simple ff 2 K 0 the order in K 0 is either the sme or less thn the order in K. Specificlly, if dim St ff» 2 in K then the str remins unchnged nd so does ord ff. If dim St ff = 3 in K then the dimension of the str drops nd so does ord ff. We will rgue shortly tht the 3-dimensionl Link Conditions for K imply the 2-dimensionl Link Conditions for K 0. Theorem B therefore pplies nd we get locl unfolding ' 0 : j K0 j! j L 0 j which differs from the identity only inside j St j. A locl unfolding of ' is otined y etending ' 0 with the identity inside ll simplices in K K 0. We now rgue tht if stisfies (i) then 2 K 0 stisfies the Link Conditions for j = 2. Since dim St» 2 in K, the str of is the sme in K nd in K 0. It follows tht Lk! 0 is the sme in K nd K 0. In generl, strs nd oundries cnnot increse from K to K 0. If follows tht Lk! 0 nd Lk! 0 do not increse. Since the link of n edge is lwys contined in the links of its vertices, Lk! 0 Lk! 0 = Lk! 0 in K implies the sme in K 0. The second Link Condition pplies only if the order of oth nd is t lest 1 in K 0. Then ord 1 ecuse of (i.0). But is principl in Bd 1 K so its link is empty. The links of nd cn gin not increse from Bd! 1 K to Bd! 1 K0. Hence Lk! 1 Lk! 1 = ; in K implies the sme in K0. Cse 2. dim St = 3 nd ord = ord = 0. Define A = St s usul. Using Steinitz' Theorem we mp A isomorphiclly to sudivision S of conve 3- polytope P, with the imge of in the interior, the imge of t verte v of P, nd the rim R = Lk isomorphic to the oundry comple of P. Similrly, we mp the hlf-str A 0 = A St isomorphiclly to nother sudivision S 0 of P, with the imge of t v, nd the rim R 0 isomorphic to the oundry comple of P. We construct common sudivision T of S nd S 0 tht keeps the oundry comple unchnged. T mps ck to Sd A, Sd A 0, nd Sd C 0, ll trnsprent nd isomorphic. We dd the remining simplices of E to Sd A nd the remining simplices of C to Sd C 0 nd otin trnsprent sudivisions of E nd C. By construction there is n isomorphism tht preserves the connection. The Isomorphic Sudivision Lemm implies tht ' hs locl unfolding. Cse 3. dim St = 3 nd ord = ord = 1. By definition there is tringle with 3-dimensionl str St ' St consisting of one, three, or more tetrhedr tht shre. The numer of tetrhedr shring is not two, else we would hve ord = 0. Let U e the set of simplices with order 1 in St. It corresponds to in St nd therefore forms n open disk tht decomposes St into one, three, or more components. A component B 1 of St U hs only order 0 simplices. The closure A 1 = B 1 is tringultion of B 3, nd the oundry X 1 = Bd 1 A 1 is tringultion of S 2. We use Steinitz' Theorem to mp A 1 to sudivision S 1 of conve 3-polytope P 1. Ecept for ll vertices of A 1 mp to vertices of P 1, nd mps to point in the interior of fcet of P 1. To ccomplish this proceed s descried erlier: construct the 1-comple of edges nd vertices in X 1, remove nd its edges, cut nd dd edges to restore three-connectedness, nd let G 1 e the isomorphic 1-skeleton of conve 3-polytope P 1. Finlly, sudivide P 1 y strring from the imge of. Becuse edges were cut nd dded, the oundry comple of P 1 is isomorphic to sudivision of X 1 ut not necessrily to X 1 itself. Similrly, S 1 is isomorphic to sudivision of A 1 ut not necessrily to A 1 itself. The imge of is verte of the k-gon fcet. We form second sudivision S 0 1 of P 1 y strring from this verte. S 0 1 is isomorphic to sudivision of the hlf-str A 0 1 = A 1 St. The right picture in Figure 4 illustrtes S 1 nd S 0 1 in the 2-dimensionl cse where P 1 is conve polygon. The common sudivision T 1 of S 1 nd S 0 1 is otined s usul, y intersection nd strring. T 1 mps ck to sudivision of A 1 nd sudivision of A 0 1. These sudivisions re not necessrily trnsprent. To finish the rgument we repet the construction for ll other components B` of St U. The sudivisions of the A` re glued long the preimge of U, which is sudivided s result of mpping ck the sudivision of the k-gon fcet, see left picture in Figure 4. Here it is importnt tht the sudivisions of the fcets e isomorphic, ut this cn esily e chieved ecuse the imge of cn e freely chosen nywhere in the interior of the fcet. The result is sudivision Sd A of A. We dd the remining simplices of E, possily fter sudivision ecuse Sd A is not necessrily trnsprent, nd otin Sd E. Similrly, the sudivisions of the A 0` re glued long the preimge of U nd mpped to Sd C 0. We dd the remining simplices of C, possily fter sudivision, nd otin Sd C. Since the oundries of Sd A nd Sd A 0 re isomorphic, the sudivision of the remining simplices in C cn e done such tht Sd E ο Sd C. By construction, Sd E nd Sd C permit n isomorphism tht pre- 9

10 serves the connection. The Isomorphic Sudivision Lemm implies (ii). Cse 4. ord = ord = 2. By the Order Bound, the dimension of the strs is dim St = dim St = 3. It follows tht elongs to ectly two edges of order 2, nd. The rgument is similr to Cse 3. The disk U is replced y ring U of hlfdisks glued long ;. Agin, U decomposes St into one or more components. The closure A` of ech such component B` is tringultion of B 3. The oundry X` = Bd 1 A` is tringultion of S 2, ; re edges of X`, nd the closed str of within X` is disk consisting of two hlf-disks in the ring U. The use of Steinitz' Theorem is similr to Cse 3 ecept tht now we mp to the interior point of n edge. To ccomplish this, we modify the construction of the grph y dding the edge fter removing nd its edges. The drwing in the plne hs k-gon djcent to n m-gon nd otherwise only tringles. Three-connectedness is recovered y cutting nd dding edges tht neither elong to the k-gon nor to the m-gon. Let G` e the isomorphic 1-skeleton of conve 3-polytope P`. A sudivision S` of P` is otined y strring from the imge of. Another sudivision S 0` of P` is otined y strring from the imge of, which is n endpoint of the edge common to the k-gon nd the m-gon. T` is gin common sudivision of S` nd S 0` nd is mpped ck to isomorphic sudivisions of A` nd A 0`. The sudivisions of the A` re glued to form Sd A nd the sudivision of the A 0` re glued nd mpped to form Sd C 0. Finlly, the remining simplices of E nd C re dded, possily fter sudivision, to otin sudivisions Sd E nd Sd C with connection preserving isomorphism. The Isomorphic Sudivision Lemm implies (ii). 3-mnifolds. Steinitz' theorem cn e pplied to the verte links of 3-mnifold K to prove K 2 M 0. For 3-mnifolds, the Link Conditions consolidte to single reltion. We strengthen the result implied y Theorem C in two respects: we construct locl unfoldings, nd we show the Link Condition is equivlent to the eistence of n unfolding. Theorem C 0. If K 2 M 0 is 3-comple then the following sttements re equivlent: (i) Lk Lk = Lk. (ii) ' hs locl unfolding. (iii) ' hs n unfolding. Proof. Only Cse 2 of the proof of Theorem C rises for 3-mnifolds. In this cse the conclusion is tht ' hs locl unfolding, which shows (i) =) (ii). (ii) =) (iii) follows from the definitions. To prove :(i) =) :(iii) we distinguish three cses depending on the dimension of the violting simple. Let L 0 = (Lk Lk ) Lk. :(i) is equivlent to L 0 6= ;. Cse 1. L 0 contins tringle yz. Then yz; yz re replced y single tetrhedron czy. If follows tht yz elongs to only one tetrhedron in the comple L otined from K y contrcting. This contrdicts L 2 M 0. Cse 2. L 0 contins no tringle ut it contins n edge y. Then y nd y re tringles in K. Ech elongs to two tetrhedr: py 6= qy, ry 6= sy. The four tetrhedr re pirwise different ecuse y 62 K, which follows from y 62 Lk. The four vertices p; q; r;s re pirwise different ecuse L 0 contins no tringle. Hence, cy elongs to four tetrhedr in L, which contrdicts L 2 M 0. Cse 3. L 0 contins no edge nd no tringle, ut it contins verte. Then nd re edges in K. Their links re two circles. These circles re disjoint ecuse y 2 Lk Lk would imply tht L 0 contins n edge, nmely y. We hve 62 Lk ecuse else 2 K nd hence 62 L 0. Similrly, we hve 62 Lk. After the contrction of to c oth circles elong to Lk c, which contrdicts L 2 M 0. 7 Discussion This section concludes the pper with n open prolem nd comment on simplifying mnifolds using edge contrctions. Link conditions. The most importnt remining prolem is the etension of the link condition results to complees of dimension eyond 3. At this time the limittion of Steinitz' Theorem to conve polytopes of dimension t most 3 is n ostcle in etending the proofs of this pper. Do Theorems B 0 nd C 0 etend to comintoril d-mnifolds for d 4? Specificlly, is it true tht for every K 2 M 0 the contrction of n edge 2 K hs n unfolding iff Lk Lk = Lk? Is there generl result tht reltes the Link Conditions with topology preserving edge contrctions for simplicil complees of ny fied dimension? Specificlly, do the Link Conditions for d nd j imply tht the contrction of in d-comple K 2 M j hs n unfolding? Irreducile tringultions. A simplicil comple is irreducile if it hs no edge whose contrction preserves 10

11 the topologicl type. It is not difficult to prove tht the oundry comple of the tetrhedron is the only irreducile tringultion of the 2-sphere. It is lso known tht every compct 2-mnifold hs only finitely mny irreducile tringultions [1]. In other words, topology preserving edge contrctions cn e used to quickly clssify tringulted 2-mnifold. The clssifiction prolem for 3-mnifolds is considerly more difficult [4], nd for 4-mnifolds it is known to e undecidle [7]. Algorithms tht recognize the 3- sphere hve een found only recently [10, 12]. In view of the pprent difficulties, it is not surprising tht even the 3-sphere hs infinitely mny irreducile tringultions. To construct n infinite fmily we use knots mde up of only three edges ech. Tke 3-cue decomposed into n 3 little cues, nd let n e lrge enough so we cn drill one cue wide tunnel in the form of non-trivil knot. Insted of completing the drilling we leve the lst cue of the tunnel so the comple is still homeomorphic to B 3. Let e one of the edges of the retined lst cue tht connects the end of the incomplete tunnel with the outside. A tringultion of B 3 is formed y decomposing ech little cue into tetrhedr, which is done without dding new vertices. Finlly, tringultion of S 3 is otined y dding the cone from new verte,, over the oundry comple of the tringultion of B 3. The cycle of three edges ; ; forms knot of the type of the tunnel. The tringle is not prt of the tringultion ecuse is not prt of the oundry comple. In fct, this tringle cnnot e emedded in S 3 ecuse ; ; is non-trivil knot. Hence none of the three edges cn e contrcted without chnging the topologicl type of the tringultion. We get n infinite fmily y drilling tunnels of different knot types. Indeed, if we hd finite set of irreducile tringultions we would hve only finitely mny cycles of three edges nd thus only finitely mny knot types. This contrdicts the eistence of infinitely mny different knot types. Acknowledgements The second uthor thnks Wolfgng Hken nd Min Yn for interesting discussions nd Günter Ziegler for suggesting the knot construction in the tringultion of the 3-sphere mentioned in Section 7. References [1] D. W. Brnette nd A. L. Edelson. All 2-mnifolds hve finitely mny miniml tringultions. Isrel J. Mth. 67 (1989), [2] H. Edelsrunner, M. A. Fcello nd D. V. Nekhyev. Surfce remeshing nd prmetriztion. Report rgi-tech-98-9, Rindrop Geomgic, Chmpign, Illinois, [3] M. Grlnd nd P. S. Heckert. Surfce simplifiction using qudrtic error metrics. Computer Grphics, Proc. siggrph 1997, [4] W. Hken. Üer ds Homöomorphie Prolem der 3- Mnnigfltigkeiten I. Mth. Z. 80 (1962), [5] H. Hoppe, T. DeRose, T. Duchmp, J. McDonld nd W. Stützle. Mesh optimiztion. Computer Grphics, Proc. siggrph 1993, [6] A. W. F. Lee, W. Sweldens, P. Schröder, L. Cowsr nd D. P. Dokin. MAPS: multiresolution dptive prmeteriztion of surfces. Computer Grphics, Proc. siggrph 1998, [7] A. A. Mrkov. Insoluility of the prolem of homeomorphy. In Proc. Internt. Congress Mth., 1960", Cmridge Univ. Press. [8] J. Milnor. Two complees which re homeomorphic ut comintorilly distinct. Ann. of Mth. 74 (1961), [9] J. R. Munkres. Elements of Algeric Topology. Addison-Wesley, Redwood City, [10] J. H. Ruinstein. An lgorithm to recognise S 3. In Proc. Internt. Congress Mth., 1994", Zürich, Switzerlnd. [11] E. Steinitz. Polyeder und Rumeinteilungen. Enzykl. mth. Wiss. 3 (Geometrie), Prt 3AB12 (1922), [12] A. Thompson. Thin position nd the recognition prolem for S 3. Mth. Res. Lett. 1 (1994), [13] E. F. Whittlesey. Finite surfces: study of finite 2-complees. Mth. Mg. 34 (1960), nd Appendi A This ppendi proves sic properties of oundry s defined in Section 3. All properties re intuitively cler ut the proofs re somewht technicl. Invrince of order. Isomorphic sudivisions re constructed y just one method lso descried in the proof of the Isomorphic Sudivisions Lemm: mp the complees to common underlying spce, intersect simplices, nd sudivide y strring. We use this construction to estlish tht simplices with comintorilly equivlent strs indeed hve the sme order. Property 1. If St fi ' St ff then ord fi = ord ff. 11

12 Proof. Let ` = ord ff nd k = dim St ff. By definition of order there is (k `)-simple in some hypotheticl comple with St ff ' St. Construct sudivision L of St ff y comining the sudivision isomorphic to Sd St fi with the one isomorphic to Sd St. Mp L to new nd finer sudivisions of Stfi nd St using the simplicil homeomorphisms ffi : j St ff j! j St fi j nd ψ : j St ff j! j St j. By definition of comintoril equivlence for susets of complees we hve ffi(j St ff j) = j St fi j nd ψ(j St ff j) = j St j. Then ψ ffi ffi 1 is n isomorphism etween these finer sudivisions nd ψ ffi ffi 1 (j St fi j) = j St j. This implies St fi ' St nd ord fi» dim St fi dim = ` = ord ff. By the symmetric rgument we get ord ff» ord fi. Boundry commutes with sudivision. We show tht the oundry of sudivided simplicil comple K is the sme s the oundry of K sudivided. This is intuitively wht one epects s the sudivision opertion does not chnge the geometric neighorhood of ny point in the underlying spce of K. Property 2. Bd i Sd K = Sd Bd i K. Proof. For ech fi 2 Sd K there is unique simple ff 2 K with int fi int ff. We prove tht fi nd ff hve the sme order y showing tht their closed strs re simplicilly equivlent. Choose point 2 int fi int ff. Let C e 1 e the oundry of cue with center t, where e is the dimension of the mient Eucliden spce. Sudivide oth St fi nd St ff y strring from. In the first sudivision the closed str of hs the sme underlying spce s St fi nd the link of is the difference etween the closed str nd the str. Similrly, in the second sudivision the closed str of hs the sme underlying spce s St ff nd the link is the difference etween the closed str nd the str: Lk 1 = St fi St fi; Lk 2 = St ff St ff: Let X e the centrl projection of j Lk 1 j to C e 1. Since the sudivision opertion does not chnge the geometric neighorhoods of 2 j K j = j Sd K j, X is lso the centrl projection of j Lk 2 j to C e 1. We construct common sudivision of the two projected links. By projecting this sudivision ck to the two links nd sudividing the two strs ccordingly we get isomorphic sudivisions of St fi nd Stff. Property 1 implies ord fi = ord ff. In words, the sudivision opertion preserves orders so the i-th oundries of K nd Sd K hve the sme underlying spce. It follows tht the i-th oundry of Sd K is sudivision of the i-th oundry of K. Boundry is closed. A firly strightforwrd consequence of Property 2 is tht fces of simple in the oundry lso elong to the oundry. Together with the Order Bound this implies tht the i-th oundry of d-comple is comple of dimension t most d i. Property 3. Bd i K is simplicil comple. Proof. It suffices to show tht the order of simple cnnot eceed tht of its fces: fi» ff =) ord ff» ord fi: Let ` = ord fi nd k = dim St fi. By definition, there is (k `)-simple with St fi ' St. Let Sd St fi nd Sd St e isomorphic sudivisions of the two closed strs so the defined simplicil homeomorphism mps j St fi j to j St j. Let ff 0 e highest-dimensionl simple in Sd St fi with int ff 0 int ff, nd let ο 0 2 Sd St e the isomorphic imge of ff 0. Finlly, let ο e the simple in St with int ο 0 int ο. Clerly, dim ff = dim ff 0 = dim ο 0» dim ο. Using Property 2, Property 1, Property 2, nd the Order Bound, in this sequence, we get ord ff = ord ff 0 = ord ο 0 = ord ο» dim St ο dim ο: The result follows ecuse dim St ο» k nd dim ο dim = k `, so ord ff» ` = ord fi. Boundry decreses order. It is intuitively cler tht in the i-th oundry the order of simple is t lest i less thn in the originl comple. The reson is tht the i-th oundry reduces the dimension of the str of ny simple y i. We write ord i ff for the order of ff in Bd i K. Property 4. ord i ff» ord ff i for ll simplices ff 2 Bd i K. Proof. We hve ff 2 Bd i K iff ` = ord ff i. Let k = dim St ff. By definition of order there is (k `)- simple in some comple Y with St ff ' St. Let Sd St ff nd Sd St e isomorphic sudivisions of the two closed strs so the defined simplicil homeomorphism mps j St ff j to j St j. By Property 1, the orders of corresponding simplices in the two sudivisions re the sme. We formlly rephrse this oservtion in the first line elow nd derive the second using Property 2: Bd i Sd St ff ο Bd i Sd St ; Sd Bd i St ff ο Sd Bd i St : 12

13 When we reverse the order of the oundry nd the closure opertions we get the sme results within the strs. This is ecuse the strs of ll simplices fi 2 St ff re the sme within K nd within Stff. Hence fi hs the sme order in K nd in St ff nd elongs to the i-th oundry of K iff it elongs to the i-th oundry of St ff. The sme is true for simplices in the str of. In other words, the str of ff in Bd i K nd the str of in Bd i Y re comintorilly equivlent. Using Property 1 gin we conclude tht the order of ff in the i-th oundry of K is the sme s the order of in the i-th oundry of Y. By the Order Bound, the ltter is ounded from ove y dim Bd i Y dim. Becuse dim Bd i Y» k i nd dim = k `, the order of ff in the i-th oundry is ounded from ove y ` i, nd the clim follows. Tking oundry simplifies. Tking 1-st oundries i times does not necessrily produce the sme result s tking the i-th oundry once. Indeed, the former opertion tends to produce smller complees thn the ltter. The intuitive reson is tht tking oundry elimintes contet nd lurs the topologicl properties of the remining neighorhood. We prove slightly stronger result. Property 5. Bd` Bd i K Bd i+` K. Proof. We hve Bd i+` K Bd i K. A simple ff elongs to Bd i K Bd i+` K iff i» ord ff < i + `. By Property 4, the order of ff in Bd i K is t lest i less thn in K, which implies 0» ord i ff < `, or equivlently ff 2 Bd i K Bd` Bd i K. In words, if ff in the i-th oundry does not elong to the (i + `)-th oundry then it lso does not elong to the `-th oundry of the i-th oundry. 13