3 Simplicial Complexes
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1 CS 468 Wintr 2004 Simpliil Complxs Afr Zomoroin Comintoris is th slums of topology. J. H. C. Whith (ttr.) 3 Simpliil Complxs In th first ltur, w look t onpts from point st topology, th rnh of topology tht stuis ontinuity from n nlytil point of viw. This viw os not hv omputtionl ntur: w nnot rprsnt infinit point sts or thir ssoit infinit opn sts on omputr. Strting with this ltur, w will look t onpts from nothr mjor rnh of topology: omintoril topology. This rnh lso stuis onntivity, ut os so y xmining onstruting omplit ojts out of simpl loks, n uing th proprtis of th onstrut ojts from th loks. Whil our viw of th worl our ontology will mostly omintoril in ntur, w will s onpts from point st topology rmrging unr isguis, n w will rful to xpos thm! In this ltur, w gin y lrning out simpl uiling loks from whih w my onstrut omplit sps. Simpliil omplxs r omintoril ojts tht rprsnt topologil sps. With simpliil omplxs, w sprt th topology of sp from its gomtry, muh lik th sprtion of syntx n smntis in logi. Givn th finit omintoril sription of sp, w r l to ount, n th mirl of omintoril topology is tht ounting lon nls us to mk sttmnts out th onntivity of sp. W shll xprin first instn of this mrvlous thory in th Eulr hrtristi. This topologil invrint givs simpl onstrutiv prour for lssifying 2-mnifols, omplting our trtmnt from th lst ltur. 3.1 Gomtri Dfinition W gin with finition of simpliil omplxs tht sms to mix gomtry n topology. Comintions llow us to rprsnt rgions of sp with vry fw points. In othr wors, llow us to sri simpl lls whih om our uiling loks ltr. Dfinition 3.1 (omintions) Lt S = {p 0, p 1,..., p k } R. A linr omintion is x = k i=0 λ ip i, for som λ i R. An ffin omintion is linr omintion with k i=0 λ i = 1. A onvx omintion is n ffin omintion with λ i 0, for ll i. Th st of ll onvx omintions is th onvx hull. You my hv sn th onpt of inpnn in stuying linr lgr. Dfinition 3.2 (inpnn) A st S is linrly (ffinly) inpnnt if no point in S is linr (ffin) omintion of th othr points in S. Figur 1 shows th linr, ffin, n onvx omintions of thr ffinly inpnnt points in R 3. W my now fin our si uiling lok. Dfinition 3.3 (k-simplx) A k-simplx is th onvx hull of k + 1 ffinly inpnnt points S = {v 0, v 1,..., v k }. Th points in S r th vrtis of th simplx. A k-simplx is k-imnsionl susp of R, im σ = k. W show low-imnsionl simplis with thir nms in Figur 2. Sin ll th points fining simplx r ffinly inpnnt, so is ny sust of thm. This uss th simplx to hv n intrsting strutur: it is ompos of simplis of lowr-imnsion, or its fs. Figur 1. Comintions. Th linr omintions of thr ffinly inpnnt points in R 3 ovrs th whol sp. Th ffin omintions fill th pln fin y th thr points. Th onvx hull is th tringl fin y th thr points. 1
2 CS 468 Wintr 2004 Simpliil Complxs Afr Zomoroin vrtx g [, ] tringl ttrhron [,, ] [,,, ] Figur 2. k-simplis, for h 0 k 3. Dfinition 3.4 (f, of) Lt σ k-simplx fin y S = {v 0, v 1,..., v k }. A simplx τ fin y T S is f of σ n hs σ s of. Th rltionship is not with σ τ n τ σ. Not tht σ σ n σ σ. Not tht simplx is lwys f of itslf y this finition. W tth simplis togthr to rprsnt sps. This tthing is vry muh lik using lgo loks to uil stls: w n only tth lgo loks on th spil intrfs. Similrly, w my only tth simplis long thir spil intrfs: thir fs. Th following finition formlly fins our struturs, whih w ll omplxs. All tht th following rypti finition stts is tht if simplx is prt of th omplx, so r ll its fs; n if two simplis intrst, th intrstion is prt of th omplx. It is goo to s this forml finition, howvr, s w will nountr similr ons in ring urrnt rsrh in omputtionl topology, n w shoul los our fr of thm! Dfinition 3.5 (simpliil omplx) A simpliil omplx K is finit st of simplis suh tht 1. σ K, τ σ τ K, 2. σ, σ K σ σ σ, σ or σ σ =. Th imnsion of K is im K = mx{im σ σ K}. Th vrtis of K r th zro-simplis in K. A simplx is prinipl if it hs no propr of in K. Hr, propr hs th sm finition s for sts. So, simpliil omplx is olltion of simplis tht fit togthr nily, s shown in Figur 3 (), s oppos to simplis in (). () Th mil tringl shrs n g with th tringl on th lft, n vrtx with th tringl on th right. () In th mil, th tringl is missing n g. Th simplis on th lft n right intrst, ut not long shr simplis. Figur 3. A simpliil omplx () n isllow olltions of simplis (). 3.2 Siz of Simplx As lry mntion, omintoril topology rivs its powr from ounting. Now tht w hv finit sription of sp, w n ount sily. So, lt s us Figur 2 to ount th numr of fs of simplx. For xmpl, n g hs two vrtis n n g s its fs (rll tht simplx is f of itslf.) A ttrhron hs four vrtis, six gs, four tringls, n ttrhron s fs. Ths ounts r summriz in Tl 1. Wht shoul th numrs for 4-simplx? Th numrs in th tl my look rlly fmilir to you. If w 1 to th lft of h row, w gt Psl s tringl, s shown in Figur 4. Rll tht Psl s tringl nos th inomil offiints: th 2
3 CS 468 Wintr 2004 Simpliil Complxs Afr Zomoroin k/l ???? Tl 1. Numr of l-simplis in h k-simplx Figur 4. If w 1 to th lft si of h row in Tl 1, w gt Psl s tringl. numr of iffrnt omintions of l ojts out of k ojts or ( k l). Hr, w hv k + 1 points rprsnting n k-simplx, ny l + 1 of whih fins l-simplx. To mk th rltionship omplt, w fin th mpty st s th ( 1)-simplx. This simplx is prt of vry simplx n llows us to olumn of 1 s to th lft si of Tl 1 to gt Psl s tringl. It lso llows us to limint th unrlin prt of Dfinition 3.5, s th mpty st of prt of oth simplis for non-intrsting simplis. To gt th totl siz of simplx, w sum h row of Psl s tringl. A k-simplx hs ( k+1 l+1) fs of imnsion l n k l= 1 ( ) k + 1 = 2 k+1 l + 1 fs in totl. A simplx, thrfor, is vry lrg ojt. Mthmtiins oftn o not fin it pproprit for omputtion, whn omputtion is ing on y hn. Simplis r vry uniform n simpl in strutur, howvr, n thrfor provi n il omputtionl ggt for omputrs. 3.3 Astrt Dfinition Our isussion on th siz of simplx shows tht w n viw simplx s st long n its powr st (th olltion of ll its susts. This viw of simplx is ttrtiv us it vois rfrns to gomtry in fining simpliil omplx. It lso shoul giv you ri flings of éjà vu, s it mths th finition of topology Dfinition 3.6 (strt simpliil omplx) An strt simpliil omplx is st K, togthr with olltion S of susts of K ll (strt) simplis suh tht: 1. For ll v K, {v} S. W ll th sts {v} th vrtis of K. 2. If τ σ S, thn τ S. Whn it is lr from ontxt wht S is, w rfr to K s omplx. W sy σ is k-simplx of imnsion k if σ = k + 1. If τ σ, τ is f of σ, n σ is of of τ. Not tht th finition utomtilly llows for s ( 1)-simplx. W will oftn us nottion n rfr to S s th omplx. Th strt finition ffirms th notion tht topology only rs out how th simplis r onnt, n not how thy r pl within sp. W now rlt this strt st-thorti finition to th gomtri on y xtrting th omintoril strutur of (gomtri) simpliil omplx. 3
4 CS 468 Wintr 2004 Simpliil Complxs Afr Zomoroin Dfinition 3.7 (vrtx shm) Lt K simpliil omplx with vrtis V n lt S th olltion of ll susts {v 0, v 1,..., v k } of V suh tht th vrtis v 0, v 1,..., v k spn simplx of K. Th olltion S is ll th vrtx shm of K. Clrly, th st K n th th olltion S togthr form n strt simpliil omplx. It llows us to ompr simpliil omplxs sily, using isomorphisms twn sts. Dfinition 3.8 (isomorphism) Lt K 1, K 2 strt simpliil omplxs with vrtis V 1, V 2 n sust olltions S 1, S 2, rsptivly. An isomorphism twn K 1, K 2 is ijtion ϕ : V 1 V 2, suh tht th sts in S 1 n S 2 r th sm unr th rnming of th vrtis y ϕ n its invrs. Thorm 3.1 Evry strt omplx S is isomorphi to th vrtx shm of som simpliil omplx K. Two simpliil omplxs r isomorphi iff thir vrtx shms r isomorphi s strt simpliil omplxs. Dfinition 3.9 (Gomtri Rliztion) If th simplis S of n strt simpliil omplx K 1 isomorphi with th vrtx shm S of th simpliil omplx K 2, w ll K 2 gomtri rliztion of K 1. It is uniquly trmin up to n isomorphism, linr on th simplis. A gomtri rliztion of n strt simpliil omplx is th nlog of n immrsion of mnifol, s th simplis my intrst on w pl th omplx insi sp. Simplis r onvx hulls whih r ompt, so w o not hv to worry out othr nsty immrsions. Hving onstrut finit simpliil omplx, w will ivi it into topologil n gomtri omponnts. Th formr will strt simpliil omplx, purly omintoril ojt, sily stor n mnipult in omputr systm. Th lttr is mp of th vrtis of th omplx into th sp in whih th omplx is rliz. Agin, this mp is finit, n n pproximtly rprsnt in omputr using floting point rprsnttion. Exmpl 3.1 (Wvfront OBJ formt) This rprsnttion of simpliil omplx trnslts wor for wor into most ommon fil formts for storing surfs. On stnr formt is th OBJ formt from Wvfront. Th formt first sris th mp whih pls th vrtis in R 3. A vrtx with lotion (x, y, z) R 3 gts th lin v x y z in th fil. Aftr spifying th mp, th formt sris n simpliil omplx y only listing its tringls, whih r th prinipl simplis (s Dfinition 3.5.) Th vrtis r numr oring to thir orr in th fil n numr from 1. A tringl with vrtis v 1, v 2, v 3 is spifi with lin f v 1 v 2 v 3. Th sription in n OBJ fil is oftn ll tringl soup, s th topology is spifi impliitly n must xtrt. v v v v v f f f f Figur 5. Portions of n OBJ fil spifying th surf of th Stnfor Bunny. 3.4 Suomplxs Rll tht simplx is th powr st of its simplis. Similrly, nturl viw of simpliil omplx is tht it is spil sust of th powr st of ll its vrtis. Th sust is spil us of th rquirmnts in Dfinition 3.6. Consir th smll omplx in Figur 6 (). Th igrm () shows how th simplis onnt within th omplx: it hs no for h simplx, n n g initing f-of rltionship. Th mrk prinipl simplis r th pks of th igrm. This igrm is, in ft, post. 4
5 CS 468 Wintr 2004 Simpliil Complxs Afr Zomoroin () A smll omplx () Post of th smll omplx, with prinipl simplis mrk. () An strt post: th wtr lvl of th post is fin y prinipl simplis. Figur 6. Post viw of simpliil omplx Dfinition 3.10 (post) Lt S finit st. A prtil orr is inry rltion on S tht is rflxiv, ntisymmtri, n trnsitiv. Tht is for ll x, y, z S, 1. x x, 2. x y n y x implis x = y, n 3. x y n y z implis x z. A st with prtil orr is prtilly orr st or post for short. It is lr from th finition tht th f rltion on simplis is prtil orr. Thrfor, th st of simplis with th f rltion forms post. W oftn strtly imgin post s in Figur 6 (). Th st is ft roun its wist us th numr of possil simplis ( n k) is mximiz for k n/2. Th prinipl simplis form lvl nth whih ll simplis must inlu. Thrfor, w my rovr simpliil omplx y simply storing its prinipl simplis, s in th s with tringultions in Exmpl 3.1. This viw lso givs us intuition for xtnsions of onpts in point st thory to simpliil omplxs. A simpliil omplx my viw s los st (it is los point st, if it is gomtrilly rliz.) Dfinition 3.11 (suomplx, link, str) A suomplx is simpliil omplx L K. Th smllst suomplx ontining sust L K is its losur, Cl L = {τ K τ σ L}. Th str of L ontins ll of th ofs of L, St L = {σ K σ τ L}. Th link of L is th ounry of its str, Lk L = Cl St L St (Cl L { }). Figur 7 monstrts ths onpts within th post for our omplx in Figur 6. A suomplx is th nlog of sust for simpliil omplx. Givn st of simplis, w tk ll th simplis low th st within th post to gt its losur (), n ll th simplis ov th st to gt its str (). Th f rltion is th prtil orr () Cl {, } () St {, } (light) n its losur Cl St {, } (rk) () Lk {, } Figur 7. Closur, str, n link of simplis 5
6 CS 468 Wintr 2004 Simpliil Complxs Afr Zomoroin tht fins ov n low. Most of th tim, th str of st is n opn st (viw s point st) n not simpliil omplx. Th str orrspons to th notion of nighorhoo for simplx, n lik nighorhoo, it is opn. Th losur oprtion omplts th ounry of st s for, mking th str simpliil omplx (). Th link oprtion givs us th ounry. In our xmpl, Cl {, } = {, }, so w rmov th simplis from th light rgions from thos in th rk rgion in () to gt th link (). Thrfor, th link of n is th g n th vrtx. Chk on Figur 6 () to s if this mths your intuition of wht ounry shoul. 3.5 Tringultions Th primry rson w stuy simpliil omplxs is to rprsnt mnifols. Dfinition 3.12 (unrlying sp) Th unrlying sp K of simpliil omplx K is K = σ K σ. Not tht K is topologil sp, s fin in th lst ltur. Dfinition 3.13 (tringultion) A tringultion of topologil sp X is simpliil omplx K suh tht K X. For xmpl, th ounry of 3-simplx (ttrhron) is homomorphi to sphr n is tringultion of th sphr, s shown in Figur 8. ~ Figur 8. Th ounry of ttrhron is tringultion of sphr, s its unrlying sp is homomorphi to th sphr. Th trm tringultion is us y iffrnt fils with iffrnt mnings. For xmpl, in omputr grphis, th trm most oftn rfrs to tringl soup sriptions of surfs. Th finit lmnt ommunity oftn rfrs to tringl soups s msh, n my llow othr lmnts, suh s qurngls, s si uiling loks. In rs, thr-imnsionl mshs ompos of ttrhr r ll ttrhrliztions. Within topology, tringultion rfrs to omplxs of ny imnsion, howvr. 3.6 Orintility W h finition of orintility in th nots for th first ltur tht pn on iffrntiility. W now xtn this finition to simpliil omplxs, whih r not smooth. This xtnsion furthr ffirms tht orintility is topologil proprty not pnnt on smoothnss. Dfinition 3.14 (orinttion) Lt K simpliil omplx. An orinttion of k-simplx σ K, σ = {v 0, v 1,..., v k }, v i K is n quivln lss of orrings of th vrtis of σ, whr (v 0, v 1,..., v k ) (v τ(0), v τ(1),..., v τ(k) ) (1) r quivlnt orrings if th prity of th prmuttion τ is vn. W not n orint simplx, simplx with n quivln lss of orrings, y [σ]. Not tht th onpt of orinttion rivs from tht ft tht prmuttions my prtition into two quivln lsss (if you hv forgottn ths onpts, you my rviw prmuttions n prtitions in th nots from ltur 1 n 2, rsptivly.) Orinttions my shown grphilly using rrows, s shown in Figur 9. W my us orint simplis to fin th onpt of orintility to tringult -mnifols. 6
7 CS 468 Wintr 2004 Simpliil Complxs Afr Zomoroin vrtx g [, ] tringl ttrhron [,, ] [,,, ] Figur 9. k-simplis, 0 k 3. Th orinttion on th ttrhron is shown on its fs. Dfinition 3.15 (orintility) Two k-simplis shring (k 1)-f σ r onsistntly orint if thy inu iffrnt orinttions on σ. A tringull -mnifol is orintl if ll -simplis n orint onsistntly. Othrwis, th -mnifol is non-orintl Lst ltur, w sw two si non-orintl 2-mnifols: th Klin ottl n th projtiv pln. Our xposition shows tht non-orintl mnifols n xist in ny imnsions, howvr. Exmpl 3.2 (Rnring) Th surf of thr-imnsionl ojt is 2-mnifol n my mol with tringultion in omputr. In omputr grphis, ths tringultions r rnr using light mols tht ssign olor to h tringl oring to how it is sitution with rspt to th lights in th sn, n th viwr. To o this, th mol ns th norml for h tringl. But h tringl hs two normls pointing in opposit irtions. To gt orrt rnring, w n th normls to onsistntly orint. 3.7 Eulr Chrtristi Hving sn orintility for simpliil surfs, w finish this ltur y looking t our first topologil invrint. Dfinition 3.16 (invrint) A (topologil) invrint is mp tht ssigns th sm ojt to sps of th sm topologil typ. Not tht n invrint my ssign th sm ojt to sps of iffrnt topologil typ. In othr wors, n invrint n not omplt. All tht is rquir y th finition is tht if th sps hv th sm typ, thy r mpp to th sm ojt. Gnrlly, this hrtristi of invrints implis thir utility in ontrpositivs: if two sps r ssign iffrnt ojts, thy hv iffrnt topologil typs. On th othr hn, if two sps r ssign th sm ojt, w usully nnot sy nything out thm. Lt us formlly stt ths sttmnts for n invrint f: X Y = f(x) = f(y) f(x) f(y) = X Y (ontrpositiv) f(x) = f(y) = nothing A goo invrint, howvr, will hv nough iffrntiting powr to usful through ontrpositivs. Hr, w fmous invrint th Eulr hrtristi. Dfinition 3.17 (Eulr hrtristi) Lt K simpliil omplx n s i = {σ K im σ = i}. Th Eulr hrtristi χ(k) is im K χ(k) = ( 1) i s i = ( 1) im σ. (2) i=0 σ K { } Whil it is fin for simpliil omplx, th Eulr hrtristi is n intgr invrint for K, th unrlying sp of K. Givn ny tringultion of sp M, w lwys will gt th sm intgr, whih w will ll th Eulr hrtristi of tht sp χ(m). 7
8 CS 468 Wintr 2004 Simpliil Complxs Afr Zomoroin Mnifols Arm with tringultions, orintility, n th Eulr hrtristi, w rturn to 2-mnifols to onvrt our xistntil proof from lst ltur to omputtionl on. W gin with lulting th Eulr hrtristi for th si surfs from th lst ltur. W hv tringultion of sphr S 2 in Figur 8, so χ(s 2 ) = = 2. To omput th Eulr hrtristi of th othr mnifols, w must uil tringultions for thm. This is simpl, howvr, y tringulting th igrms for onstruting flt 2-mnifols from th lst ltur, s in Figur 10. This tringultion givs us χ(t 2 ) = = 0. W my omplt th tl in Figur 10 () in similr fshion. As χ(t 2 ) = χ(k 2 ) = 0, th Eulr hrtristi y itslf is not powrful nough to iffrntit twn surfs Mnifol χ Sphr S 2 2 Torus T 2 0 Klin ottl K 2 0 Projtiv pln RP 2 1 () A tringultion for th igrm of th torus T 2 () Th Eulr hrtristis of our si 2-mnifols Figur 10. A tringultion of th igrm of th torus T 2 Lst ltur, w lso isuss onstruting mor omplit surfs using th onnt sum. Suppos w form th onnt sum of two surfs M 1, M 2 y rmoving singl tringl from h, n intifying th two ounris. Clrly, th Eulr hrtristi shoul th sum of th Eulr hrtristis of th two surfs, minus 2 for th two missing tringls. In ft, this is tru for ritrry shp isks. Thorm 3.2 For ompt surfs M 1, M 2, χ(m 1 # M 2 ) = χ(m 1 ) + χ(m 2 ) 2. For ompt surf M, lt gm th onnt sum of g opis of M. Comining this thorm with th tl in Figur 10 (), w gt th following. Corollry 3.1 χ(gt 2 ) = 2 2g n χ(grp 2 ) = 2 g. Ths surfs, long with th sphr, form th quivln lsss of 2-mnifols isuss in th lst ltur. Dfinition 3.18 (gnus) Th onnt sum of g tori is ll surf with gnus g. Th gnus rfrs to how mny hols th multi-onut surf hs. W r now ry to giv omplt nswr to th homomorphism prolm for los ompt 2-mnifols. Thorm 3.3 (Homomorphism prolm of 2-mnifols) Clos ompt surfs M 1 n M 2 r homomorphi, M 1 M 2 iff 1. χ(m 1 ) = χ(m 2 ) n 2. ithr oth surfs r orintl or oth r non-orintl. Osrv tht th thorm is if n only if. W n sily omput th Eulr hrtristi of ny 2-mnifol. Computing orintility is lso sy y orinting on tringl n spring th orinttion throughout th mnifol if it is orintl. Thrfor, w hv full omputtionl mtho for pturing topology of 2-mnifols. As w shll s in th futur lturs, th prolm is muh hrr in highr imnsions, foring us to rsort to mor lort mhinry. 8
9 CS 468 Wintr 2004 Simpliil Complxs Afr Zomoroin Aknowlgmnts Th mtril for this ltur is mostly from Munkrs [4] n Firy n Grinr [2], with inspirtions from Hnl [3] n prsonl nots. Th ttriut quot is from Cmron [1]. Thnks to Dnil Russl n Niloy Mitr for proofring. Rfrns [1] CAMERON, P. J. Comintoris: topis, thniqus, lgorithms. Cmrig Univrsity Prss, Nw York, NY, [2] FIRBY, P. A., AND GARDINER, C. F. Surf Topology. Ellis Horwoo Limit, Chihstr, Engln, [3] HENLE, M. A Comintoril Introution to Topology. Dovr Pulitions, In., Nw York, [4] MUNKRES, J. R. Elmnts of Algri Topology. Aison-Wsly, Ring, MA,
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