A NEW DECOMPOSITION THEOREM FOR 3-MANIFOLDS

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1 Illinois Journal of Mathmatics Volum 46, Numbr 3, Fall 2002, Pags S A NEW DECOMPOSITION THEOREM FOR 3-MANIFOLDS BRUNO MARTELLI AND CARLO PETRONIO Abstract. Lt M b a (possibly non-orintabl) compact 3-manifold with (possibly mpty) boundary consisting of tori and Klin bottls. Lt X M b a trivalnt graph such that M \ X is a union of on disc for ach componnt of M. Building on prvious work of Matvv, w dfin for th pair (M, X) a complxity c(m, X) N and show that, whn M is closd, irrducibl and P 2 -irrducibl, c(m, ) is th minimal numbr of ttrahdra in a triangulation of M. Morovr c is additiv undr connctd sum, and, givn any n 0, thr ar only finitly many irrducibl and P 2 -irrducibl closd manifolds having complxity up to n. W prov that vry irrducibl and P 2 -irrducibl pair (M, X) has a finit splitting along tori and Klin bottls into pairs having th sam proprtis, and complxity is additiv on this splitting. As opposd to th JSJ dcomposition, our splitting is not canonical, but it involvs much asir blocks than all Sifrt and simpl manifolds. In particular, most Sifrt and hyprbolic manifolds appar to hav nontrivial splitting. In addition, a givn st of blocks can b combind to giv only a finit numbr of pairs (M, X). Our splitting thorm provids th thortical background for an algorithm which classifis 3-manifolds of any givn complxity. This algorithm has bn alrady implmntd and provd ffctiv in th orintabl cas for complxity up to 9. W dvlop in this papr a thory of complxity for pairs (M, X), whr M is a compact 3-manifold such that χ(m) = 0, and X is a collction of trivalnt graphs, ach graph τ bing mbddd in on componnt C of M so that C \ τ is on disc. In th spcial cas whr M is closd, so that X =, our complxity coincids with that introducd by Matvv [6]. Extnding his rsults w show that complxity of pairs is additiv undr connctd sum and that, whn M is closd, irrducibl, P 2 -irrducibl and diffrnt from S 3, L 3,1, P 3, its complxity is prcisly th minimal numbr of ttrahdra in a triangulation. Ths two facts show that complxity is indd a vry natural masur of how complicatd a manifold or pair is. Th formr fact was known to Matvv in th closd cas, th lattr in th orintabl cas. Rcivd Octobr 2, 2001; rcivd in final form Novmbr 11, Mathmatics Subjct Classification. Primary 57M50. Scondary 57M c 2002 Univrsity of Illinois

2 756 BRUNO MARTELLI AND CARLO PETRONIO Th most rlvant fatur of our thory is that it lads to a splitting thorm along tori and Klin bottls for irrducibl and P 2 -irrducibl pairs (so, in particular, for irrducibl and P 2 -irrducibl closd manifolds). Th blocks of th splitting ar thmslvs pairs, and th complxity of th original pair is th sum of th complxitis of th blocks. Rcalling that in [6] a complxity c(m) was dfind also for th cas whn M, w mphasiz hr that our complxity c(m, X) is typically diffrnt from c(m). So th splitting thorm crucially dpnds on th xtnsion of c from manifolds to pairs. Our splitting diffrs from th JSJ dcomposition [2][3] sinc it is not uniqu (s blow for a furthr discussion of this point), but it has th grat advantag that th blocks it involvs, which w call bricks, ar much asir than all Sifrt and simpl manifolds. As a mattr of fact, our splitting is non-trivial on almost all Sifrt and hyprbolic manifolds it has bn tstd on. Anothr advantag is that th graphs in th boundary rduc th flxibility of possibl gluings of bricks. As a consqunc, a givn st of bricks can only b combind in a finit numbr of ways. This proprty is of cours crucial for computations, and our thory actually lads to vry ffctiv algorithms for th numration of closd manifolds having small complxity. Rturning to th rlation btwn our splitting and th JSJ dcomposition, w mntion that all th bricks found so far [4] ar gomtrically atoroidal, which suggsts that our splitting is actually always a rfinmnt of th JSJ dcomposition (and w know it is in th orintabl cas for complxity up to 9; s [4]). Morovr, non-uniqunss for a Sifrt manifold typically corrsponds to non-uniqunss of its ralization as a graph-manifold. W hav also found som non-uniqunss instancs in th hyprbolic cas [5]. Th orintabl vrsion of th thory dvlopd in this papr, culminating in th splitting thorm, was stablishd in [4]. In th sam papr w hav provd svral strong rstrictions on th topology of bricks and, using a computr program, w hav bn abl to classify all orintabl bricks of complxity up to 9. Using th bricks w hav thn listd all closd irrducibl orintabl 3-manifolds up to complxity 9, showing in particular that th only four hyprbolic ons ar prcisly thos of last known volum. Th splitting thorm provd blow is th main thortical tool ndd to xtnd our program of numrating 3-manifolds of small complxity from th orintabl to th gnral cas. W ar planning to raliz this program in th clos futur. This will allow us to provid information on th smallst non-orintabl hyprbolic manifolds and on th dnsity, in ach givn complxity, of orintabl manifolds among all 3-manifolds. W hav dcidd to dvot th prsnt papr to th gnral thory and th splitting thorm, laving computr implmntation for a subsqunt papr, bcaus th non-orintabl cas displays crtain rmarkabl phnomna which

3 A NEW DECOMPOSITION THEOREM FOR 3-MANIFOLDS 757 do not appar in th orintabl cas. To bgin with, toric boundary componnts rstrict th shap of th trivalnt graph thy contain to only on possibility, whil Klin bottls allow two. Nxt, th assumption of P 2 -irrducibility has to b addd to irrducibility to gt th finitnss of closd manifolds of a givn complxity. Mor surprisingly, ths assumptions do not suffic whn a non-mpty boundary is allowd, bcaus th drilling of a boundary-paralll orintation-rvrsing loop nvr changs complxity. Bcaus of ths facts, th intrinsic dfinition of brick givn blow is somwhat subtlr than that givn in [4], and th proof of som of th ky rsults (including additivity undr connctd sum) is considrably hardr. 1. Manifolds with markd boundary If C is a connctd surfac, w call spin of C a trivalnt graph τ mbddd in C in such a way that C \ τ is an opn disc. If C is disconnctd thn a spin of C is a collction of spins for all its componnts. W dnot by X th st of all pairs (M, X), whr M is a connctd and compact 3-manifold with (possibly mpty) boundary mad of tori and Klin bottls, and X is a spin of M. Elmnts of X will b viwd up to homomorphism of pairs. Rmark 1.1. Sinc S 2 and P 2 do not admit spins with vrtics, a pair (M, X) with X a spin of M blongs to X if and only if χ(m) = 0 and all th lmnts of X hav vrtics. Spins of th torus T and th Klin bottl K. A spin of T or K must b a trivalnt graph with two vrtics, and thr ar prcisly two such graphs, namly th θ-curv and th fram σ of a pair of spctacls. Both θ and σ can srv as spins of K, as shown in Fig. 1, lft and cntr. Th following rsult will b shown in th appndix: Proposition 1.2. (1) Evry spin of K is isotopic to on of th two graphs shown in Fig. 1. (2) With notation as in Fig. 1, for both τ = θ and τ = σ thr xists f Aut(K) such that f(τ) = τ, f( ) =, and f( ) =, but for all f Aut(K) such that f(τ) = τ w hav f( ) =. Th situation for T is compltly diffrnt. First of all, σ is not a spin of T. In addition, θ can b usd as a spin of T in infinitly many non-isotopic ways, bcaus th position of θ on T is dtrmind by th tripl of loops on T which ar containd in θ. Not that any two of ths loops gnrat H 1 (T ; Z), and any such tripl dtrmins on spin θ. Howvr w hav th following rsult, which w lav to th radr to prov using th facts just statd.

4 758 BRUNO MARTELLI AND CARLO PETRONIO (K, θ) (K, σ) (T,θ) Figur 1. Spins of th Klin bottl and th torus. Proposition 1.3. If θ is a spin of T thn all automorphisms of θ ar inducd by automorphisms of T. If θ, θ ar spins of T thr xists f Aut(T ) such that f(θ) = θ. Exampls of pairs. If M is a closd 3-manifold thn (M, ) is an lmnt of X. For simplicity w will oftn writ only M instad of (M, ). W list hr svral mor lmnts of X ndd blow. Our notation will b consistnt with that of [4]. Th radr is invitd to us Propositions 1.2 and 1.3 to mak sur that all th pairs w introduc ar wll-dfind up to homomorphism. W start with th product pairs: B 0 = (T [0, 1], {θ {0}, θ {1}}), B 0 = (K [0, 1], {θ {0}, θ {1}}), B 0 = (K [0, 1], {σ {0}, σ {1}}). W nxt hav two pairs B 1 and B 2 basd on th solid torus T and shown in Fig. 2-lft, and two on th solid Klin bottl K, namly B 1 = (K, θ) and B 2 = (K, σ). For k 1 w tak now th 2-orbifold D 2 with k mirror sgmnts on D 2 and w dfin Z k X as th Sifrt manifold without singular fibrs ovr this orbifold [10], with a spin σ in ach of th k Klin bottls on th boundary. Not that Z k can also b viwd as th complmnt of k disjoint orintationrvrsing loops in S 2 S 1. Anothr dscription of Z k is givn in Fig. 2-right.

5 A NEW DECOMPOSITION THEOREM FOR 3-MANIFOLDS 759 B 1 k 1 B 2 Z k Figur 2. Th pairs B 1, B 2, and Z k for k 1. W also not that Z 1 = B 2 and Z 2 = B 0. W dfin now B 2 to b Z 3, for a spcific rason xplaind blow. W will now introduc thr oprations on pairs which allow us to construct nw pairs from givn ons. Th ultimat goal is to show that all manifolds can b constructd via ths oprations using only crtain building blocks. Connctd sum of pairs. Th opration of connctd sum obviously xtnds from manifolds to pairs. Namly, givn (M, X) and (M, X ) in X, w dfin (M, X)#(M, X ) as (M#M, X X ), whr M#M is on of th two possibl connctd sums of M and M. Of cours S 3 = (S 3, ) X is th idntity lmnt for opration #. It is now natural to dfin (M, X) to b prim or irrducibl if M is. Of cours th only prim non-irrducibl pairs ar S 2 S 1 and S 2 S 1. Assmbling of pairs. Givn (M, X) and (M, X ) in X, w pick spins τ X and τ X with τ C M and τ C M. If thr is a homomorphism ψ : C C such that ψ(τ) = τ w can construct th pair (N, Y ) = (M ψ M, (X X ) \ {τ, τ }). W call this an assmbling of (M, X) and (M, X ) and w writ (N, Y ) = (M, X) (M, X ). Of cours two givn lmnts of X can only b assmbld in a finit numbr of inquivalnt ways. Considring th pairs Bi and Z k introducd abov, th radr may asily chck as an xrcis that Z k Z h = Z h+k 2 and that th following holds: Rmark (M, X) B 0 = (M, X) for any (M, X) X. 2. For (i, j) qual to (1, 1), (1, 2), or (2, 1), it is possibl to assmbl B i and B j along a crtain map ψ in ordr to gt S 3. So, for any (M, X), if w assmbl (M, X)#B i to B j along ψ, w gt th original (M, X) as a rsult.

6 760 BRUNO MARTELLI AND CARLO PETRONIO 3. Th assmbling of B 2 with B 2 givs B 0, so ((M, X) B 2 ) B 2 = (M, X) providd B 2 is assmbld to on of th fr boundary componnts of B 2. This shows that w can discard various assmblings without impairing our capacity of constructing nw manifolds. So w will call trivial an assmbling (M, X) (M, X ) if, up to intrchanging (M, X) and (M, X ), on of th following holds: (1) (M, X ) is of typ B0. (2) (M, X ) = B j for j {1, 2} and (M, X) can b xprssd as (N, Y )#B i for i {1, 2} with (N, Y ) S 3 in such a way that th assmbling is prformd along th boundary of B i and B i B j = S 3. (3) (M, X ) = B 2 and (M, X) = (N, Y ) B 2 with B 2 bing assmbld to B 2. Slf-assmbling. Givn (M, X) X, w pick two distinct τ, τ X with τ C and τ C. If thr is a homomorphism ψ : C C w can choos on such that ψ(τ) and τ intrsct transvrsly in two points, and w dfin (N, Y ) as (M ψ, X \ {τ, τ }). W call this a slf-assmbling of (M, X) and w writ (N, Y ) = (M, X). As abov, only a finit numbr of slf-assmblings of a givn lmnt of X ar possibl. In th squl it will b convnint to rfr to a combination of assmblings and slf-assmblings of pairs just as an assmbling. Not that of cours w can first do th assmblings and thn th slf-assmblings. 2. Complxity, bricks, and th dcomposition thorm In th following sctions w will introduc and discuss a crtain function c : X N which w call complxity. In th prsnt sction w anticipat th dfinition of c vry brifly and stat svral rsults about this function, which could also b takn as axiomatic proprtis. Thn w show how to dduc th splitting thorm from ths proprtis only. Proofs of th proprtis ar givn in Sctions 3 5. Givn (M, X) X w dnot by c(m, X) th minimal numbr of vrtics of a polyhdron P mbddd in M such that P M is simpl, P M = X, and th complmnt of P M is an opn 3-ball. Hr simpl mans that th link of vry point mbds in th 1-sklton of th ttrahdron, and a point of P is a vrtx if its link is prcisly th 1-sklton of th ttrahdron. W obviously hav c(m, ) = c(m) if M is a closd 3-manifold and c(m) is Matvv s complxity [6]. Not that c(m) is also dfind in [6] for M, but typically c(m, X) c(m). Axiomatic proprtis. W start with thr thorms which suggst to rstrict th study of c(m, X) to pairs (M, X) which ar irrducibl and P 2 - irrducibl. Rcall that M is calld P 2 -irrducibl if it dos not contain any

7 A NEW DECOMPOSITION THEOREM FOR 3-MANIFOLDS 761 two-sidd mbddd projctiv plan P 2 (s [1] for gnralitis about this notion, and in particular for th proof that a connctd sum is P 2 -irrducibl if and only if th individual summands ar). Whn M is closd, w call singular a triangulation of M with multipl and slf-adjacncis btwn ttrahdra. Th first and scond thorms, rspctivly, xtnd rsults of Matvv [6] from th closd to th markd-boundary cas, and from th orintabl to th possibly-non-orintabl cas. Th xtnsion is asy for th scond thorm, but lss so for th first thorm. Th third thorm shows that th nonorintabl thory is far richr than th orintabl on. Thorm 2.1 (Additivity undr #). hav For any (M, X) and (M, X ) w c((m, X)#(M, X )) = c(m, X) + c(m, X ). Morovr c(s 2 S 1 ) = c(s 2 S 1 ) = 0. Thorm 2.2 (Naturality). If M is closd, irrducibl, P 2 -irrducibl, and diffrnt from S 3, P 3, L 3,1, thn c(m) = c(m, ) is th minimal numbr of ttrahdra in a singular triangulation of M. Thorm 2.3 (Finitnss). For all n 0 th following holds: (1) Thr xist finitly many irrducibl and P 2 -irrducibl pairs (M, X) such that c(m, X) = n and (M, X) cannot b xprssd as an assmbling (N, Y ) B 2. (2) If (N, Y ) X is irrducibl and P 2 -irrducibl and c(n, Y ) = n thn (N, Y ) can b obtaind from on of th pairs (M, X) dscribd abov by rpatd assmbling of copis of B 2. Any such assmbling has complxity n. Th lattr rsult is of cours crucial for computational purposs. To bttr apprciat its finitnss contnt, not that whnvr w assmbl on copy of B 2 th numbr of boundary componnts incrass by on. Thrfor th thorm implis that for all n, k 0 th st M k n = {(M, X) X irrd. and P2 irrd., c(m, X) n, #X k} is finit. It should b mphasizd that not only w can prov that M k n is finit, but th proof itslf provids an xplicit algorithm to produc a finit list of pairs from which M k n is obtaind by rmoving duplicats. Th thorm also implis that dropping th rstriction #X k w gt infinitly many pairs, but only finitly many orintabl ons. This fact, which is ultimatly du to th xistnc of th Z k sris gnratd by B 2 undr assmbling, is on of th ky diffrncs btwn th orintabl and th gnral cas (anothr important diffrnc will aris in th proof of Thorm 2.1 s Proposition 5.2). Not

8 762 BRUNO MARTELLI AND CARLO PETRONIO also that an assmbling with B 2 gomtrically corrsponds to th drilling of a boundary-paralll orintation-rvrsing loop. Th following mor spcific vrsion of th prvious thorm for n = 0 is ndd blow. Proposition 2.4. Th only irrducibl and P 2 -irrducibl pairs having complxity 0 ar S 3, L 3,1, P 3 and all th pairs Bi and Z k dfind abov. W turn now to th bhaviour of complxity undr assmbling. Proposition 2.5 (Subadditivity). For any (M, X), (M, X ) X w hav c((m, X) (M, X )) c(m, X) + c(m, X ), c( (M, X)) c(m, X) + 6. W dfin now an assmbling (M, X) (M, X ) to b sharp if it is nontrivial and c((m, X) (M, X )) = c(m, X) + c(m, X ). Similarly, a slfassmbling (M, X) is sharp if c( (M, X)) = c(m, X) + 6. Proposition 2.5 radily implis th following facts. Rmark If a combination of sharp (slf-)assmblings is rarrangd in a diffrnt ordr thn it still consists of sharp (slf-)assmblings. 2. Evry assmbling with B 2 is sharp (unlss it is trivial, which only happns whn B 2 is assmbld to B 0 or to B 2). To s this, not again that (M, X) B 2 B 2 = (M, X) and c(b 2 ) = c(b 2) = 0. Thorm 2.7 (Sharp splitting). Lt (N, Y ) b irrducibl and P 2 -irrducibl. If (N, Y ) can b xprssd as a sharp assmbling (M, X) (M, X ) or as a slf-assmbling (M, X ) thn (M, X), (M, X ), and (M, X ) ar irrducibl and P 2 -irrducibl. Proof. In both cass w ar cutting N along a two-sidd torus or Klin bottl, so P 2 -irrducibility is obvious. If (N, Y ) = (M, X ), this torus or Klin bottl is incomprssibl in N, and irrducibility of M is a gnral fact [1]. W ar lft to show that if (N, Y ) = (M, X) (M, X ) sharply thn M and M ar irrducibl. Sinc ths manifolds hav boundary, it is nough to show that thy ar prim. Suppos thy ar not, and considr prim dcompositions of (M, X) and (M, X ) involving summands (M i, X i ) and (M j, X j ). So on summand (M i, X i ) is assmbld to on (M j, X j ), and th othr (M i, X i ) s and (M j, X j ) s surviv in (N, Y ). It follows that, up to prmutation, (M, X) is prim, (M, X ) = (M 1, X 1)#(M 2, X 2) with (M 1, X 1) and (M 2, X 2) prim, (M, X) (M 1, X 1) = S 3 and (M 2, X 2) = (N, Y ). Sharpnss of th original assmbling and additivity undr # now imply that c(m, X) = c(m 1, X 1) = 0. So Proposition 2.4 applis to (M, X) and

9 A NEW DECOMPOSITION THEOREM FOR 3-MANIFOLDS 763 (M 1, X 1). Knowing that (M, X) (M 1, X 1) = S 3 it is asy to dduc that (M, X) and (M 1, X 1) ar ithr B 1 or B 2, and that th original assmbling was a trivial on. This is a contradiction. Bricks and dcomposition. Taking th rsults statd abov for grantd, w dfin hr th lmntary building blocks and prov th dcomposition thorm. Latr w will commnt on th actual rlvanc of this thorm. A pair (M, X) X is calld a brick if it is irrducibl and P 2 -irrducibl and cannot b xprssd as a sharp assmbling or slf-assmbling. Thorm 2.3 and Rmark 2.6 asily imply that thr ar finitly many bricks of complxity n. From Proposition 2.4 it is asy to dduc that in complxity zro th only bricks ar prcisly th pairs Bi introducd abov, which xplains why w hav givn a spcial status to Z 3 = B 2, and that th othr irrducibl and P 2 -irrducibl pairs ar assmblings of bricks. Now, w show mor gnrally: Thorm 2.8 (Existnc of splitting). Evry irrducibl and P 2 -irrducibl pair (M, X) X can b xprssd as a sharp assmbling of bricks. Proof. Th rsult is tru for c(m, X) = 0, so w procd by induction on c(m, X) and suppos c(m, X) > 0. By Thorm 2.3 w can assum that (M, X) cannot b split as (N, Y ) B 2, bcaus vry assmbling with B 2 is sharp, and w hav sn that B 2 is a brick. Now if (M, X) is a brick w ar don. Othrwis (M, X) is ithr a sharp slf-assmbling (N, Y ), in which cas c(n, Y ) = c(m, X) 6 and w can conclud by induction using Thorm 2.7, or (M, X) is a sharp assmbling (N, Y ) (N, Y ). Thorm 2.7 stats that (N, Y ) and (N, Y ) ar irrducibl and P 2 -irrducibl. If both (N, Y ) and (N, Y ) hav positiv complxity w conclud by induction. Othrwis w can assum that c(n, Y ) = 0 and apply Proposition 2.4. Sinc th assmbling is non-trivial, (N, Y ) is not of typ B0. It is also not B 2 or Z k for k 3, by th proprty of (M, X) w ar assuming. So (N, Y ) is on of B 1, B 1, B 2, B 2. In particular, it is a brick. Now w claim that (N, Y ) cannot b split as (N, Y ) B 2. Assuming it can, w hav two cass. In th first cas th assmbling of (N, Y ) is prformd along a fr boundary componnt of B 2, but thn w must hav (N, Y ) = B 2, and th assmbling is trivial, which is absurd. In th scond cas (N, Y ) is assmbld to a fr boundary componnt of (N, Y ), and w hav (M, X) = ( (N, Y ) (N, Y ) ) B 2, which is again absurd. Our claim is provd. Now w know that (N, Y ) again blongs to th finit list of irrducibl and P 2 -irrducibl pairs which hav complxity n and cannot b split as an assmbling with B 2. Howvr (N, Y ) has on mor boundary componnt than (M, X), which implis that by rpatdly applying this argumnt w must vntually nd up with a brick.

10 764 BRUNO MARTELLI AND CARLO PETRONIO Exprimntal facts. Thorm 2.8 shows that listing irrducibl and P 2 - irrducibl manifolds up to complxity n is asy onc th bricks up to complxity n ar classifid. Th finitnss faturs of our thory imply that thr xists an algorithm which rducs such a classification to a rcognition problm. Morovr it turns out xprimntally that rcognition nds to b carrid out only on a comparativly short list of pairs. Th complt list of orintabl bricks up to complxity 9 was found in [4] (s also [11]), and it consists of 30 pairs, whras thr ar 1901 closd, irrducibl, and orintabl 3-manifolds of complxity up to 9. As a mattr of fact, only 7 bricks ar alrady sufficint to obtain 1882 closd manifolds (th othr 19 bing thmslvs bricks). In addition, all bricks found ar gomtrically atoroidal, which maks it asy to rcogniz thir assmblings; s also [5]. 3. Sklta W introduc hr th notion of sklton of a pair (M, X), w dfin th complxity of (M, X) as th minimal numbr of vrtics of a sklton, and w discuss th first proprtis of minimal sklta, dducing som of th rsults statd abov. Th othr rsults rquir a dpr analysis and nw tchniqus and will b provd latr. Simpl sklta and complxity. W rcall that a compact polyhdron P is calld simpl if th link of vry point of P mbds in th spac givn by a circl with thr radii. Th points having th whol of this spac as a link ar calld vrtics. Thy ar isolatd and thrfor finit in numbr. Givn a pair (M, X) X, a polyhdron P mbddd in M is calld a sklton of (M, X) if th following holds: P M is simpl; M \ (P M) is an opn ball; P M = X. Rmark 3.1. If P is a sklton of (M, X) thn P is simpl, and th vrtics of P cannot li on M. Whn #X = 1 thn P is a spin of M (i.., M collapss onto P ), and whn #X = 0 (i.., whn M is closd) thn P is a spin of M \ {point}. Whn #X 2 no such intrprtation is possibl. Th proof that vry (M, X) X has a sklton, alrady givn in [4, Rmark 2.1], xtnds vrbatim to th non-orintabl contxt. For a simpl polyhdron P w dnot by v(p ) th numbr of vrtics of P, and w dfin th complxity c(m, X) of a givn (M, X) X as th minimum of v(p ) ovr all sklta P of (M, X). So w hav a function c : X N. Som sklta without vrtics. If w rmov on point from th closd manifolds S 3, L 3,1, P 3, S 2 S 1, and S 2 S 1 w can collaps th rsult rspctivly to a point, to th tripl hat, to P 2, and to th join of S 2 and

11 A NEW DECOMPOSITION THEOREM FOR 3-MANIFOLDS 765 S 1 (for both th last two cass). Hr th tripl hat is th spac obtaind by attaching D 2 to S 1 so that D 2 runs thr tims around S 1. This shows that S 3, L 3,1, P 3, S 2 S 1, and S 2 S 1 all hav complxity zro. It is a wllknown fact, which w will prov again blow, that ths ar th only prim and P 2 -irrducibl manifolds having complxity zro. Turning to th pairs Bi and Z k dfind in th prvious sction, w now show that thy also hav complxity 0. This is rathr obvious for th product pairs B 0, B 0, and B 0, bcaus thy hav th product sklta P 0 = θ [0, 1] T [0, 1], P 0 = θ [0, 1] K [0, 1], and P 0 = σ [0, 1] K [0, 1]. For B 1 = (T, {θ}) w not that θ contains a mridian of th torus, so w can attach to X a mridional disc and gt th sklton P 1 shown in Fig. 3. Th sam construction applis to B 1 = (K, {θ}) and lads to th sklton P 1 also shown in th figur. Of cours P 1 and P 1 ar isomorphic as abstract polyhdra (just as P 0 and P 0 ar), but w us diffrnt nams to kp track also of thir mbddings. P P 1 1 Figur 3. Th sklta P 1 and P 1 of B 1 and B 1. Figur 4. Th sklta P 2 and P 2 of B 2 and B 2. Th sklta P 2 and P 2 of B 2 and B 2, rspctivly, ar shown in Fig. 4, both as abstract polyhdra, and as polyhdra mbddd in T and K. W conclud

12 766 BRUNO MARTELLI AND CARLO PETRONIO with th sris Z k for k 3, for which a sklton is shown in Fig. 5. Rcalling that B 2 was dfind as Z 3, w dnot this sklton by P 2 whn k = 3. Figur 5. Th sklton of Z k for k = 4. Nuclar and standard sklta. A sklton of (M, X) is calld nuclar if it dos not collaps to a propr subpolyhdron which is also a sklton of (M, X). A nuclar sklton P of (M, X) X having c(m, X) vrtics is calld minimal. Of cours vry (M, X) has minimal sklta. W will introduc now two mor rstrictd classs of simpl polyhdra. Latr w will show that, undr suitabl assumptions, minimal polyhdra must blong to ths classs. A simpl polyhdron Q is calld quasi-standard with boundary if vry point has a nighborhood of on of th typs (1)-(5) shown in Fig. 6. A point of typ (3) was alrady dfind abov to b a vrtx of Q. W dnot now by V (Q) th st of all vrtics, and w dfin th singular st S(Q) as th st of points of typ (2), (3), or (5), and th boundary Q as th st of points of typ (4) or (5). Morovr w call 1-componnts of Q th connctd componnts of S(Q) \ V (Q) and 2-componnts of Q th connctd componnts of Q \ (S(Q) Q). (1) (2) (3) (4) (5) Figur 6. Typical nighborhoods of points in a quasistandard polyhdron with boundary. If th 2-componnts of Q ar opn discs (and hnc ar calld just facs), and th 1-componnts ar opn sgmnts (and hnc calld just dgs), thn

13 A NEW DECOMPOSITION THEOREM FOR 3-MANIFOLDS 767 w call Q a standard polyhdron with boundary. For short w will oftn just call Q a standard polyhdron, and possibly spcify that Q should b mpty or non-mpty. W prov now th first proprtis of nuclar sklta. Lmma 3.2. If P is a nuclar sklton of a pair (M, X) X, thn P = Q s 1... s m G, whr: (1) Q is a quasi-standard polyhdron with boundary Q X. (2) For all componnts (C, τ) of ( M, X), ithr Q τ or Q appars nar C as in Fig. 7, so that Q τ is on or two circls, dpnding on th typ of (C, τ). (3) s 1,..., s m ar th dgs of th τ s in X which do not alrady blong to Q. (4) G is a graph with G (Q s 1... s m ) finit and G V (Q M) mpty. Figur 7. Local aspct of Q nar C if Q τ. Proof. Nuclarity is a proprty of local natur, and th rsult is trivial if M =. For M, dfining Q as th 2-dimnsional portion of P and G as P \ (Q X), th only non-obvious point to show is (2). Of cours Q C Q is ithr τ or a union of circls. To chck that th only possibilitis ar thos of Fig. 7 on rcalls that M \ (P M) is a ball, so C \ (Q G) is planar, and thn C \ Q is also planar. Rmark 3.3. Evry (M, X) X has a minimal sklton P = Q s 1... s m G as abov, whr in addition G M =. This is bcaus, without changing v(p ), w can tak th nds of G lying on M and mak thm slid ovr Q s 1... s m until thy rach int(m). Not that th rgular nighborhood of τ X in P is now ithr a product τ [0, 1] or as shown in Fig. 7.

14 768 BRUNO MARTELLI AND CARLO PETRONIO Subadditivity. Som proprtis of complxity radily follow from th dfinition and from th first facts shown about minimal sklta. To bgin with, if P and P ar sklta of (M, X) and (M, X ) and w add to P P a sgmnt which joins P \V (P ) to P \V (P ), w gt a sklton of (M, X)#(M, X ) with v(p )+v(p ) vrtics. Thrfor c((m, X)#(M, X )) c(m, X)+c(M, X ). Turning to assmbling, lt P and P b as in Rmark 3.3, and lt an assmbling (M, X) (M, X ) b prformd along a map ψ : C C with ψ(τ) = τ. Thn P ψ P is simpl, and it is a sklton of (M, X) (M, X ). W dduc that c((m, X) (M, X )) c(m, X) + c(m, X ). Now w considr a slf-assmbling (M, X). If P is a sklton of (M, X) as in Rmark 3.3 and th slf-assmbling is prformd along a crtain map ψ : C C such that τ ψ(τ) consists of two points, thn (P C C )/ ψ is a sklton of (M, X). It has th sam vrtics as (M, X) plus at most two from th vrtics of τ, two from th vrtics of τ, and two from τ ψ(τ). This shows that c( (M, X)) c(m, X) + 6. Surfacs dtrmind by graphs. W will nd vry soon th ida of splitting a sklton along a graph, so w spll out how th construction gos. M γ γ P M W P M γ M W P P γ γ γ P W W P P Figur 8. Surfac dtrmind by a trivalnt graph. P W Lmma 3.4. Lt P b a quasi-standard sklton of (M, X) and lt γ b a trivalnt graph containd in P M, locally mbddd as in Fig. 8-lft. Thn thr xists a surfac S proprly mbddd in M such that S (P M) = γ

15 A NEW DECOMPOSITION THEOREM FOR 3-MANIFOLDS 769 and S \ γ is a union of discs. Morovr S is sparating in M if and only if γ is sparating in P M. Proof. To construct S w first tak a surfac W with boundary as shown in Fig. 8-right, so that W (P M) = γ. Thn w attach disjoint discs to th componnts of W lying in th intrior of M. S [4, Rmark 4.1] for dtails. Rmark 3.5. With th sam notation as in th prvious lmma, assum furthr that γ {θ, σ} is containd in P, that S is sparating in M, and that S \ γ is only on disc. Thn cutting M along S and choosing γ as a spin for th two nw boundary componnts w gt a dcomposition (M, X) = (M 1, X 1 ) (M 2, X 2 ) which, at th lvl of sklta, corrsponds prcisly to th splitting of P along γ. Minimal sklta ar standard. W now prov a thorm on which most of our rsults ar basd. In particular, Proposition 2.4 radily follows from it, and Thorm 2.2 will b asily provd using it. W start with an asy rmark. Rmark 3.6. If P is a nuclar and standard sklton of (M, X) thn it is proprly mbddd, namly P = M P = X, and P M is standard without boundary. Morovr P M is a spin of a manifold boundd by on sphr and som tori and Klin bottls, so χ(p M) = 1. Knowing that S(P M) is 4-valnt, w thn s that P has #X + v(p ) + 1 facs. Thorm 3.7. Lt (M, X) X b an irrducibl and P 2 -irrducibl pair, and lt P b a minimal sklton of (M, X). Thn: (1) If c(m, X) > 0 thn P is standard. (2) If c(m, X) = 0 and X = thn M {S 3, L 3,1, P 3 } and P is not standard. (3) If c(m, X) = 0 and X thn (M, X) is on of th B i or Z k, and P is prcisly th sklton dscribd in Sction 3, so P is standard unlss (M, X) is B 1 or B 1. Proof. W first show that if P is not standard thn ithr X = and M {S 3, L 3,1, P 3 }, or (M, X) {B 1, B 1} and P {P 1, P 1}. Latr w will dscrib standard sklta without vrtics. If P rducs to on point, thn of cours M = S 3. Lt us first assum that P is not purly 2-dimnsional, so thr is sgmnt containd in th 1-dimnsional part of P. W distinguish two cass dpnding on whthr lis in int(m) or on M. If int(m), w tak a small disc which intrscts transvrsly in on point. As in th proof of Lmma 3.4 w attach to a disc containd in th ball M \ (P M), gtting a sphr S M intrscting P in on point of.

16 770 BRUNO MARTELLI AND CARLO PETRONIO By irrducibility S bounds a ball B, and P B is asily sn to b a spin of B. Nuclarity now implis that P B contains vrtics, so P \ B is a sklton of (M, X) with fwr vrtics than P. This is a contradiction. If M, lt C b th componnt of M on which lis. Sinc on C thr is a circl which mts τ transvrsly in on point of, looking at th ball M \ (P M) again w s that in M thr is a proprly mbddd disc D intrscting P in a point of. W hav now thr cass dpnding on th typ of th pair (C, τ). If (C, τ) = (T, θ) thn D is a comprssing disc for T, so by irrducibility M is th solid torus. Knowing that D mts P only in on point it is now asy to show also that (M, X) = B 1 and P = P 1. If (C, τ) = (K, θ) thn must b containd in th dg of θ by Lmma 3.2, and th sam rasoning shows that (M, X) = B 1 and P = P 1. If (C, τ) = (K, σ) thn must b containd in th dg of σ by Lmma 3.2. Th complmnt in K of D is now th union of two Möbius strips. If w choos any on of ths strips and tak its union with D, w gt an mbddd P 2 in M. Sinc M is irrducibl and P 2 -irrducibl, it would thn hav to b P 3, but M, so w hav obtaind a contradiction. W ar lft to dal with th cas whr P is purly two-dimnsional, so that P is quasi-standard, but not standard. Lt us first suppos that som 2-componnt F of P is not a disc. Thn ithr F is a sphr, in which cas P also rducs to only a sphr, which is clarly impossibl bcaus M would thn b S 2 [0, 1], or thr xists a loop γ in F such that on of th following holds: (1) γ is orintation-rvrsing on F. (2) γ sparats F in two componnts non of which is a disc. W considr now th closd surfac S dtrmind by γ as in Lmma 3.4, and not that S is ithr S 2 or P 2. If S = P 2 w dduc that (M, X) = P 3. If S = S 2 irrducibility implis that S bounds a ball B in M. This is clarly impossibl in cas (1), so w ar in cas (2). Now w not that P B must b a nuclar spin of B. Knowing that F B is not a disc it is asy to dduc that P B must contain vrtics. This contradicts th minimality bcaus w could rplac th whol of P B by only on disc, gtting anothr sklton of (M, X) with fwr vrtics. If P is quasi-standard and its 2-componnts ar discs thn ithr P is standard or S(P ) rducs to a singl circl. Thn it is asy to show that P must b th tripl hat and (M, X) = L 3,1. W ar lft to analyz th cas whr P is standard and c(m, X) = 0, so that X. Dnoting #X by n, Rmark 3.6 shows that P has n + 1 facs.

17 A NEW DECOMPOSITION THEOREM FOR 3-MANIFOLDS 771 W considr first th cas n = 1. Sinc P has on dg and two facs, it is asy to s that it must b homomorphic to ithr P 2 or P 2 (s Fig. 4) as an abstract polyhdron. This dos not quit imply that (M, X) is B 2 or B 2, bcaus in gnral a sklton P alon is not nough to dtrmin a pair (M, X). Howvr P M crtainly dos dtrmin (M, X), bcaus it is a standard spin of M minus a ball, and X = P M. W ar lft to analyz all th polyhdra of th form P 2 ψ T for ψ : P 2 θ T, of th form P 2 ψ K for ψ : P 2 θ K, and of th form P 2 ψ K for ψ : P 2 σ K. Among ths polyhdra w must slct thos which can b thicknd to manifolds with two boundary componnts (a sphr plus ithr a torus or a Klin bottl). Th symmtris of (T, θ), (K, θ), and (K, σ) dscribd in Propositions 1.3 and 1.2 imply that thr ar actually not many such polyhdra. Mor prcisly, thr is just on P 2 ψ T, which givs B 2. Thr ar two P 2 ψ T, on of which is not thicknabl (i.., not th spin of any manifold), and th othr can b thicknd to a manifold with thr boundary componnts (a sphr and two Klin bottls). Finally, thr ar two P 2 ψ K, on of which is not thicknabl, and th othr givs B 2. This concluds th proof for n = 1. Having workd out th cas n = 1, w turn to th cas n 2, so that P has n dgs and n + 1 facs. If a fac of P mts M in on arc only, thn it mts S(P ) in on dg only, and this dg joins a componnt of M to itslf, which asily implis that n = 1, contradicting our prsnt assumption n 2. If a fac of P is an mbddd rctangl, with two opposit dgs on M and two in S(P ), thn it radily follows that n = 2 and P is ithr θ [0, 1] or σ [0, 1]. As abov, to conclud that (M, X) {B 0, B 0, B 0 }, w must considr th various polyhdra obtaind by attaching (T, θ), (K, θ), and (K, σ) to th uppr and lowr bass of θ [0, 1] and σ [0, 1]. Using again Propositions 1.3 and 1.2 on ss that thr ar only six such polyhdra. Thr of thm ar not thicknabl, and th othr thr giv B 0, B 0, B 0. Rturning to th gnral cas with n 2, w not that thr ar a total of 3n dgs on M, so thr ar 3n grms of facs starting from M. Knowing that thr is a total of n + 1 facs and non of thm uss only on grm, w s that at last on fac uss only two grms, and so it is a rctangl R, possibly an immrsd on. If n = 2 w hav thr rctangls, on of which must b mbddd, and w ar ld back to a cas alrady discussd. If n 3 thn R must b immrsd, so in particular it joins a componnt (K 1, σ 1 ) of ( M, X) to anothr (K 2, σ 2 ) componnt. A rgular nighborhood in P of R σ 1 σ 2 is shown in Fig. 9. Th boundary of this nighborhood is again a graph σ which dtrmins a sparating Klin bottl according to Rmark 3.5. If w cut P along σ w gt a disjoint union P 2 P, which at th lvl of manifolds givs a splitting (M, X) = B 2 (M, X ). Morovr P is a nuclar sklton of (M, X ), so c(m, X ) = 0, P is minimal, and #X = n 1. Now ithr (M, X ) = B 0 and P = P 0 or w can procd, vntually gtting that (M, X) = B 2... B 2, so (M, X) = Z k for som

18 772 BRUNO MARTELLI AND CARLO PETRONIO k 3, and P is th corrsponding sklton constructd in Sction 3. Th proof is now complt. Figur 9. An immrsd rctangl joins two (K, σ) componnts. Proof of Thorm 2.2. By th prvious rsult, a minimal spin of M is standard with vrtics, and dual to it thr is a singular triangulation with c(m) ttrahdra (and on vrtx). A singular triangulation of M with n ttrahdra and k vrtics dually givs a standard polyhdron Q with n vrtics whos complmnt is a union of k balls. If w punctur k 1 suitably chosn facs of Q w gt a sklton of (M, ), whnc th conclusion at onc. 4. Finitnss Th proof of Thorm 2.3 will b basd on th following rsult. Proposition 4.1. Lt (M, X) b an irrducibl and P 2 -irrducibl pair which dos not split as (M, X) = (N, Y ) B 2. Assum that c(m, X) > 0 and lt P b a standard sklton of (M, X). Thn vry dg of P is incidnt to at last on vrtx of P. Proof. Assum by contradiction that an dg of P is not incidnt to any vrtx of P, i.., that both th nds of li on M. If th nds of li on th sam spin τ X thn τ is a connctd componnt of S(P ) M. Th standardnss of P implis that P has no vrtics, which contradicts th assumption that c(m, X) > 0. So th nds of li on distinct spins τ, τ X. Lt C and C b th componnts of M on which τ and τ li, and lt R b a rgular nighborhood in P of C C. By construction R is a quasi-standard polyhdron with boundary R = τ τ γ. Hr γ is a trivalnt graph with on componnt homomorphic to θ or to σ, and possibly anothr componnt homomorphic to th circl. Lt us first considr th cas whr γ has a circl componnt γ 0. This circl lis on P and is disjoint from S(P ). Th standardnss of P thn implis that γ 0 bounds a disc D containd in P and disjoint from S(P ). In this cas w st γ = γ \ γ 0 and R = R D. In cas γ is connctd w just st γ = γ and R = R. In both cass w hav found a graph γ homomorphic to θ or to σ

19 A NEW DECOMPOSITION THEOREM FOR 3-MANIFOLDS 773 which sparats P. Morovr on componnt R of P \ γ is standard without vrtics and is boundd by τ τ γ. According to Lmma 3.4, th graph γ dtrmins a sparating surfac S in M such that S P = γ. Sinc χ(γ ) = 1 and S \ γ consists of discs, w hav χ(s) 0. Of cours χ(s) 1, for othrwis S would b an mbddd P 2, but w ar assuming that M is irrducibl and P 2 -irrducibl and has non-mpty boundary. W will now show that if χ(s) = 2 thn c(m, X) = 0, and if χ(s) = 0 thn (M, X) splits as (M, X) = (N, Y ) B 2. This will imply th conclusion. Assum that χ(s) = 2, so S is a sphr. W dnot by B th opn 3- ball M \ (P M) and not that S B = S \ γ consists of thr disjoint opn 2-discs, which cut B into four opn 3-balls. By irrducibility, S bounds a closd 3-ball D, and B \ D is th union of som of th four opn 3-balls just dscribd. Viwing (D, γ ) abstractly w can now asily construct a nw simpl polyhdron Q D without vrtics such that Q S = γ and D \ Q consists of thr distinct 3-balls, ach incidnt to on of th thr opn 2-discs which constitut S \ γ. Lt us considr now th simpl polyhdron P = R γ Q viwd as a subst of M. By construction P M = τ τ = X. Morovr M \ (P M) is obtaind from B \ D (which consists of opn 3- balls) by attaching ach of th thr 3-balls of D \ Q along only on 2-disc (a componnt of S \ γ ). It follows that M \ (P M) still consists of opn 3-balls. By puncturing som of th 2-componnts of P w can thn construct a sklton of (M, X) without vrtics, so indd c(m, X) = 0. Assum now that χ(s) = 0, so S is a sparating torus or Klin bottl. Rmark 3.5 now shows that (M, X) is obtaind by assmbling som pair (N, Y ) with a pair (N, Y ) which has sklton R. By construction R is standard without vrtics and N has thr componnts, and it was shown in th proof of Thorm 3.7 that (N, Y ) must thn b B 2. This complts th proof. Corollary 4.2. Lt (M, X) b irrducibl and P 2 -irrducibl. Assum c(m, X) > 0 and thr is no splitting (M, X) = (N, Y ) B 2. Thn #X 2c(M, X). Proof. A minimal sklton P of (M, X) is standard by Thorm 3.7, and w hav just shown that ach dg of P joins ithr V (P ) to itslf or V (P ) to X. Sinc P has c(m, X) quadrivalnt vrtics, thr can b at most 4c(M, X) dgs raching X. Each componnt of X is rachd by prcisly two dgs, so thr ar at most 2c(M, X) componnts. Proof of Thorm 2.3. Th rsult is valid for n = 0 by th classification carrid out in Thorm 3.7, so w assum n > 0. Lt F n b th st of all irrducibl and P 2 -irrducibl pairs (M, X) which cannot b split as (M, X ) B 2. By Thorm 3.7, ach such (M, X) has a minimal standard spin P with

20 774 BRUNO MARTELLI AND CARLO PETRONIO n vrtics. By Corollary 4.2, w hav that S(P M) is a quadrivalnt graph with at most 5n vrtics. Sinc P M is a standard polyhdron, thr ar only finitly many possibilitis for P M and hnc for (M, X). Givn an irrducibl and P 2 -irrducibl pair (M, X) with c(m, X) = n, ithr (M, X) F n or (M, X) splits along a Klin bottl K as (M, X ) B 2. Th only cas whr K is comprssibl in M is whn (M, X ) = B 2, but B 2 B 2 = B 0 and c(b 0 ) = 0. So K is incomprssibl, whnc M is irrducibl and P 2 -irrducibl. Morovr c(m, X ) = n by Rmark 2.6 (which dpnds on th now provd Propositions 2.4 and 2.5). Sinc (M, X ) has on boundary componnt lss than (M, X), w can itrat th procss of splitting copis of B 2 only a finit numbr of tims, and thn w gt to an lmnt of F n. 5. Additivity In this sction w prov additivity undr connctd sum. This will rquir th thory of normal surfacs and mor tchnical rsults on sklta. W start with an asy gnral fact (s [4, Proposition 2.9] for a proof). Proposition 5.1. Givn a pair (M, X) X, lt Q M b a quasistandard polyhdron with Q M = Q X. Assum that M \ Q has two componnts N and N. Thn th 2-componnts of Q that sparat N from N form a closd surfac Σ(Q) Q int(m) which cuts M into two componnts. Normal surfacs. Givn a pair (M, X) X, lt P b a nuclar sklton of (M, X). Th simpl polyhdron P M is now a spin of M with a ball B M rmovd. Choos a triangulation of P M, and lt ξ P b th handl dcomposition of M \ B obtaind by thickning th triangulation of P M, as in [7]. In this subsction w will study normal sphrs in ξ P. Not that thr is an obvious xampl, namly th sphr paralll to B and slightly pushd insid ξ P. Th following rsult dals with th othr normal sphrs. Its proof displays anothr rmarkabl diffrnc btwn th orintabl and th gnral cas. Namly, it was shown in [4] that, whn M is orintabl, any normal surfac raching M actually contains a componnt of M. By contrast, whn ( M, X) contains som (K, σ) componnt, an arbitrary normal surfac can rach K without containing it. As our proof shows, howvr, this cannot happn whn th surfac is a sphr. Proposition 5.2. Lt P b a nuclar sklton of (M, X) X, and lt S b a normal sphr in ξ P. Thn thr xists a simpl polyhdron Q such that v(q) v(p ), Q M = X and M \ (Q M) is a rgular nighborhood of S. If in addition P is standard, c(m, X) > 0, and S is not th obvious sphr N(P M), thn thr xists Q as abov with v(q) < v(p ).

21 A NEW DECOMPOSITION THEOREM FOR 3-MANIFOLDS 775 Proof. Evry rgion R of P carris a color n N givn by th numbr of shts of th local projction of S to R. Now w cut P M opn along S as xplaind in [7], i.., w rplac ach R by its (n + 1)-shtd covr containd in th normal bundl of R in M. As a rsult w gt a polyhdron P M which contains M, such that M \ P is th disjoint union of an opn ball B and an opn rgular nighborhood N of S in M. By rmoving from ach boundary componnt C M th opn disc C \ τ w gt a polyhdron P intrscting M in X. Now w punctur a 2-componnt which sparats B from N and claim that th rsulting polyhdron Q is as dsird. Only th inqualitis btwn v(p ) and v(q) ar non-obvious. W first prov that all th vrtics of P M which li on M disappar ithr whn w cut P along S gtting P or latr whn w rmov M \ X from P to gt P. This of cours implis th first assrtion of th statmnt. W concntrat on on componnt (C, τ) of ( M, X). By Lmma 3.2 ithr both vrtics of τ ar vrtics of P M or non of thm is. In th lattr cas thr is nothing to show, so w assum that thr ar thr (possibly non-distinct) 2-componnts of P incidnt to τ. Lt v and v b th vrtics of τ. Looking first at v, w dnot by (n, n, n, p, q, r) th colors of th six grms at v of 2-componnt of P M. Hr n corrsponds to C \ τ, which is triply incidnt to v. Th compatibility quations of normal surfacs now radily imply that (up to prmutation) r is vn, p = q r, and that n p/2 whn p = q = r. Morovr: v disappars in P if p = q > r; v survivs in P and rmains on M, so it disappars in P, if p = q = r and n = p/2; v survivs in P and movs to int(m) if p = q = r and n > p/2. Now if τ = θ thn th sam cofficints appar at v. Th only cas whr v and v do not both disappar in P is whn p = q = r and n > p/2. But in this cas S would thn contain n p/2 paralll copis of C, which is impossibl. Th cas τ = σ is asir, bcaus if v survivs in P th situation is as in Fig. 10. This is absurd bcaus S would contain Möbius strips. p S p p n n-p/2 Figur 10. Möbius strips in a normal surfac.

22 776 BRUNO MARTELLI AND CARLO PETRONIO Now w turn to th scond assrtion. If v(p ) < v(p ) th conclusion is obvious, so w procd assuming v(p ) = v(p ). It is now sufficint to show that som fac of P which sparats B from N contains vrtics of P, bcaus w can thn punctur such a fac and collaps th rsulting polyhdron until it bcoms nuclar, gtting fwr vrtics. Assum by contradiction that thr is no such fac. W not that P is th union of a quasi-standard polyhdron P and som arcs in X. Th 2-componnts of P which sparat B from N ar th sam as thos of P, so thy giv a closd surfac Σ P by Proposition 5.1. From th fact that v(p ) = v(p ) w dduc that nar a vrtx of P th transformation of P into P can b dscribd as in Fig. 11, namly P can Figur 11. Transformation of P into P nar a vrtx of P. b idntifid nar th vrtx with P S. Of cours this dos not imply that globally P = P S, bcaus th componnts of P playing th rol of P nar vrtics may not match across facs. Th closd surfac Σ cannot b disjoint from S(P ), bcaus othrwis S would b th obvious sphr B. On th othr hand w ar supposing Σ V (P ) =, so Σ S(P ) must b a non-mpty union of loops. In particular, S(P ) contains a loop γ disjoint from V (P ). Figur 11 now shows that S(P ) coincids with S(P ) away from M. Using th analysis of th transition from P to P nar M alrady carrid out abov, w also s that nar a componnt (C, τ) of ( M, X) ithr S(P ) coincids with S(P ) or it is obtaind from S(P ) by adding on dg of τ, and thn slightly pushing th rsult insid M. Whn (C, τ) = (K, σ) th dg addd is ncssarily. This implis that th loop γ dscribd abov can b viwd as a loop in S(P M) such that γ V (P ) =. In addition, if γ contains a vrtx of P M on a crtain componnt of M thn it contains also th othr vrtx in that componnt. This radily implis that th union of γ with all th τ s in X touchd by γ is a connctd componnt of S(P M). But P M is standard, so S(P M) is connctd, and w dduc that P has no vrtics. This is a contradiction.

23 A NEW DECOMPOSITION THEOREM FOR 3-MANIFOLDS 777 Proof of Thorm 2.1. W hav alrady shown that c(s 2 S 1 ) = c(s 2 S 1 ) = 0 and that c is subadditiv. Lt us considr now a non-prim pair (M, X) and a minimal sklton P of (M, X). Sinc (M, X) is not prim, thr xists a normal sphr S in ξ P which is ssntial in M, namly it ithr is nonsparating or it sparats M into two manifolds both diffrnt from B 3. Thn w apply th first point of Proposition 5.2 to P and S, gtting a polyhdron Q. If S is sparating and splits (M, X) as (M 1, X 1 )#(M 2, X 2 ), w must hav that Q is th disjoint union of polyhdra Q 1 and Q 2, whr Q i is a sklton of (M i, X i ). Sinc v(q 1 ) + v(q 2 ) = v(q) v(p ) w dduc that c(m, X) c(m 1, X 1 ) + c(m 2, X 2 ), so quality actually holds. If S is not sparating w idntify a rgular nighborhood of S in M with S ( 1, 1) and not that thr must xist a fac of Q having S ( 1, 1 + ε) on on sid and S (1 ε, 1) on th othr sid. W punctur this fac gtting a polyhdron Q. Now Q is a sklton of a pair (M, X) such that (M, X) = (M, X)#E, whr E is S 2 S 1 or S 2 S 1. Morovr v(q ) = v(q) v(p ), and hnc c(m, X) c(m, X), so quality actually holds. W hav shown so far that an ssntial normal sphr in (M, X) lads to a non-trivial dcomposition (M, X) = (M 1, X 1 )#(M 2, X 2 ) on which complxity is additiv. If (M 1, X 1 ) and (M 2, X 2 ) ar prim w stop; othrwis w itrat th procdur until w find a dcomposition of (M, X) into prims on which complxity is additiv. Sinc any othr dcomposition into prims actually consists of th sam summands, w dduc that complxity is always additiv on dcompositions into prims. If w tak th connctd sum of two non-prim manifolds thn a prim dcomposition of th rsult is obtaind from prim dcompositions of th summands, so additivity holds also in gnral. Appndix: Som facts about th Klin bottl In this appndix, following Matvv [7], w classify all simpl closd loops on th Klin bottl K and w dduc Proposition 1.2 from this classification. W also mntion two mor rsults on K which asily follow from th classification. Ths rsults ar strictly spaking not ncssary for th prsnt papr, and thy ar probably wll-known to xprts, but w hav dcidd to includ thm bcaus thy show a striking diffrnc which xists btwn th orintabl and non-orintabl cass. Proposition A.1. Thr xist on th Klin bottl only four non-trivial loops up to isotopy, as shown in Fig. 12. Ths loops ar dtrmind by thir imag in H 1 (K; Z) = a, b a+b = b+a, 2a = 0, as also shown in th pictur. Morovr a and ±2b ar orintation-prsrving on K, whil ±b and a ± b ar orintation-rvrsing.

24 778 BRUNO MARTELLI AND CARLO PETRONIO b a a b a a+b +2b +b Figur 12. Non-trivial loops on th Klin bottl. Proof. A non-trivial loop is isotopic to on which is normal with rspct to a triangulation of K, i.., it appars as in Fig. 13. W must hav n+m = n +m, p m n m p n n p m n m p Figur 13. Normal loops in a triangulation of K. n + p = n + p, m + p = m + p, so n = n, m = m, p = p. If p > m, w furthr distinguish btwn th following cass: If n < p, sinc w look for a connctd curv, w gt n = m = 0 and p = 1, whnc th loop a; if n > p w do not gt any solution; if n = p w gt m = 0 and n = p {1, 2}, whnc th loops ±b and ±2b. If m > p w must hav p = n = 0 and m {1, 2}, whnc th loops ±b and ±2b again. If m = p, sinc th connctd curv w look for is also non-trivial, w must hav m = p = 0 and n {1, 2}, whnc th loops a ± b and ±2b. Proof of Proposition 1.2. W start by showing that σ mbds uniquly as a spin of K. Th closd dgs and of σ ar disjoint simpl loops in K, and thy must b orintation-rvrsing. It asily follows that {, } must b {±b, a ± b}. Now th nds of can b isotopically slid ovr and to rach th position of Fig. 1-cntr, and uniqunss is provd. Turning to th uniqunss of th mbdding of θ, not that two of th thr simpl closd loops containd in θ must b orintation-rvrsing on K. Lt b th dg containd in both of ths loops. If w prform th mov shown

25 A NEW DECOMPOSITION THEOREM FOR 3-MANIFOLDS 779 in Fig. 14 along w gt a spin σ of K, and th nwborn dg is th dg Figur 14. A mov changing a spin θ of K into a spin σ. of σ. So θ is obtaind from σ by th sam mov along σ. Sinc th mbdding of σ is uniqu, w obtain th sam conclusion for θ. Having provd uniqunss, w must undrstand symmtris. Our dscription obviously implis that, in both σ and θ, th dgs and play symmtric rols, whil th rol of is diffrnt, and th conclusion asily follows. Th sam conclusion could also b dducd from Proposition A.3 blow. Proposition A.2. If K is th solid Klin bottl and K = K thn vry automorphism of K xtnds to K. In particular, thr is only on possibl Dhn filling of a Klin bottl in th boundary of a givn manifold. Proof. Proposition A.1 shows that th mridian a of K can b charactrizd in K = K as th only orintation-prsrving loop having connctd complmnt. So vry automorphism of K maps th mridian to itslf and th conclusion follows. Proposition A.3. Th mapping class group of K is isomorphic to Z/ 2Z Z/ 2Z and vry automorphism of K is dtrmind up to isotopy by its action on H 1 (K; Z). Proof. It is quit asy to construct commuting ordr-2 automorphisms φ and ψ of K such that thir action on H 1 (K; Z) is givn by φ(a) = a, φ(b) = b, ψ(a) = a, ψ(b) = a + b. Givn any othr automorphism f, combining th gomtric charactrization of a with th obsrvation that a is isotopic (not only homologous) to itslf with opposit orintation, w dduc that (up to isotopy) f is th idntity on a. Up to composing f with φ w can assum that f is actually th idntity also nar a, so f rstricts to an automorphism of th annulus K \a which is th idntity on th boundary. Th mapping class group rlativ to th boundary of th annulus is now infinit cyclic gnratd by th rstriction of ψ (but ψ has ordr 2 whn viwd on K), and th conclusion follows.