math.GT 2008-12-18 0

Combinatorial realization of the Thom-Smale complex via discrete Morse theory

In the case of smooth manifolds, we use Forman's discrete Morse theory to realize combinatorially any Thom-Smale complex coming from a smooth Morse function by a couple triangulation-discrete Morse function. As an application, we prove that any Euler…

Authors: Etienne Gallais (LMAM, LMJL)

COMBINA TORIAL REALIZA TION OF THE THOM-SMALE COMPLEX VIA DISCRETE MOR SE THEOR Y ´ ETIENNE GALLAIS Abstract. In the case of smo oth m anifolds, we use F orman’s discrete M orse theory to realize com binatorially any Thom-Smale complex coming from a smo oth M orse function by a couple triangulation-discrete Morse function. As an application, we prov e that an y Euler structure on a smo oth oriente d closed 3-manifold has a particular realization b y a complete matching on the Hasse diagram of a triangulation of the manif old. 1. Introduction R. F o rman defines a com binator ia l analog o f smo o th Morse theory in [4], [5], [6] for simplicial c o mplexes and more generally for CW-complexes. Discrete Morse theory has many applications (computer gra phics [10], graph theory [3]). An im- po rtant pro blem is the resea rch of optimal discr e te Morse functions in the sense that they ha ve the minimal num b er of critical cells ([8], [9],[1 8] for the minimality of hyperplane a rrangements). Thanks to a combinatorial Morse vector fie ld V , F orman cons tructs a combina- torial Thom-Smale complex ( C V , ∂ V ) who se homo lo gy is the simplicial homo lo gy of the simplicial complex. The differential is defined by counting algebr aically V - paths b etw een critical cells. Nev er theless, the pro of o f ∂ V ◦ ∂ V = 0 is an indirect pro of (see [5]). W e give t wo pr o ofs of ∂ V ◦ ∂ V = 0, o ne whic h focusse s o n th e geometry and another one which fo cusses on the a lgebraic p oint of view (compare with [2] and [19]). In fact, the algebr aic pro of giv es also the property tha t the combinatorial Thom-Smale complex is a chain c o mplex homoto p y equiv alent to the simplicial chain complex. After that, we investigate one step for ward the relation betw e e n the smo oth Mo rse theory a nd the discrete Mor se theory . W e prov e that any Thom-Smale complex has a combinatorial realiza tion. W e use this to pr ov e that any Spin c -structures on a closed oriented 3 -manifold can b e realized b y a complete matching on a triangula tio n of this manifold. This article is or ganised as follows. In section 2, we recall the discre te Mo rse theory from the v iewpo int of c ombinatorial Morse vector field. Section 2.2.3 is de- voted to the pr o ofs of ∂ V ◦ ∂ V = 0 a nd tha t the Thom-Smale co mplex is a chain complex homotop y equiv alent to the simplicial chain complex. In sectio n 3, we prov e that any combinatorial Thom- Smale complex is r ealizable as a combinatorial Thom-Smale complex. In section 4 , we obtain as a coro llary the existence of tri- angulations with complete match ing s on their Has se diagra m and pr ov e tha t any Spin c -structures on a clo sed oriented 3-manifold ca n b e r ealized by such co mplete matchings. 1 2 ´ ETIENNE GALLAIS 2. Discrete Morse theor y 2.1. Combinatorial Morse vector field. First of all, instead of co nsidering dis- crete Mors e functions on a simplicial complex, we will only cons ider co mb ina torial Morse vector fie lds . In fact, working with discrete Morse functions o r co mbinatorial vector fields is exactly the same [5 , Theorem 9.3 ]. In the fo llowing, X is a finite simplicial complex and K is the set of ce lls of X . A cell σ ∈ K of dimension k is deno ted σ ( k ) . Let < b e the partial o rder o n K given by σ < τ iff σ ⊂ τ . Given a simplicial c o mplex, one a sso ciates its Has se diagra m: the set of vertices is the set of cells K , an edge joins tw o cells σ and τ if σ < τ and dim ( σ ) + 1 = dim ( τ ). Definition 2.1. A combinatorial vector field V on X is an o riented matc hing on the ass o ciated Hasse diagr a m o f X that is a set of edges M such that (1) any tw o distincts edg e s of M do not s hare any co mmon vertex, (2) every edge b elonging to M is oriented toward the to p dimensional cell. A cell which do es not b elong to a n y edge of the matching is said to b e cr itica l. R emark. The origina l definition o f a combinatorial vector field is the fo llowing one: given a matching on the Hasse dia gram define V : K → K ∪ { 0 } σ 7→ V ( σ ) = ( τ iff ( σ, τ ) is an edge of the matching and σ < τ , 0 otherwise. W e will use b oth these p oints o f view in the following. A V -path of dimension k is a sequence of cells γ : σ 0 , σ 1 , . . . , σ r of dimension k such that (1) σ i 6 = σ i +1 for all i ∈ { 0 , . . . , r − 1 } , (2) for every i ∈ { 0 , . . . , r − 1 } , σ i +1 < V ( σ i ). A V -path γ is said to be clos e d if σ 0 = σ r , and no n- stationary if r > 0. Definition 2.2 . A co mbinatorial v ecto r field V which ha s no non-stationa ry closed path is called a combinatorial Morse vector. In this case, the co rresp onding ma tc h- ing is called a Mor se match ing . The terminolo gy Morse matching first a ppea red in [2]. R emark. Let V be a combinatorial (r esp. combinatorial Mo rse) vector field. It we remov e an edge from the underlying matching, it remains a combinatorial (resp. combinatorial Mors e ) vector field (there ar e tw o extra critica l cells). 2.2. The combinatorial Thom -Smale comple x. 2.2.1. Definition of the c ombinatorial Thom-Smale c omplex. The follo wing data are necessary to define the combinatorial Thom-Sma le complex (se e [5]). First, let X be a finite s implicia l complex, K its set of cells and V a combinatorial Mors e vector field. Suppose that every cell σ ∈ K is o riented. Let γ : σ 0 , σ 1 , . . . , σ r be a V -path. Then the multiplicit y of γ is given b y the formula m ( γ ) = r − 1 Y i =0 − < ∂ V ( σ i ) , σ i >< ∂ V ( σ i ) , σ i +1 > ∈ { ± 1 } COMBINA TORIAL REALIZA TION OF THE THOM-SMALE COMPLEX 3 where for every ce ll σ , τ , < σ, τ > ∈ {− 1 , 0 , 1 } is the incidence num b er b etw een the cells σ and τ (see [11]) and ∂ is the b oundar y map when we consider X a s a CW- complex. In fact, o ne ca n think of the multiplicit y as chec king if the or ientation of the first cell σ 0 mov ed along γ coincides or not with the orientation of the last cell σ r . Let Γ( σ , σ ′ ) b e the set of V -paths starting at σ a nd ending at σ ′ and C rit k ( V ) be the set of cr itical cells of dimension k . Definition 2. 3 . The combinatorial T ho m-Smale co mplex ass o ciated with ( X , V ) is ( C V ∗ , ∂ V ) where: (1) C V k = L σ ∈ C r it k ( V ) Z .σ , (2) if τ ∈ C rit k +1 ( V ) then ∂ V τ = X σ ∈ C r it k ( V ) n ( τ , σ ) .σ where n ( τ , σ ) = X e σ<τ < ∂ τ , e σ > X γ ∈ Γ( e σ ,σ ) m ( γ ) Thu s , this co mplex is exactly in the same spirit as the Thom-Smale co mplex for smo oth Morse functions (see s ection 3): it is g enerated by critical cells and the differential is g iven by counting a lgebraica lly V -paths. Theorem 2.4 (F or ma n [5]) . ∂ V ◦ ∂ V = 0 . Theorem 2. 5 (F or man [5]) . ( C V ∗ , ∂ V ) is homotopy e quivalent to t he simplicial chain c omplex. In p articular, its homolo gy is e qual to t he simplicial homolo gy. W e will give a dir ect pr o of of both of these theorems. The pro of of Theore m 2.4 is done by lo o king at V -paths and under s tanding their co nt r ibution to ∂ V ◦ ∂ V . Then, we prov e Theorem 2.5 (whic h gives a no ther proof of Theorem 2.4) using Gaussian elimination (this idea fir st app ear s in [2], se e also [1 9]). 2.2.2. Pr o of of The or em 2.4 . Let X b e a simplicial complex, K b e the set o f its ce lls and K n the s e t of cells of dimension n . The pro of is by induction on the num b er on edges b elong ing to the Morse matching. Initialization: matching with no edge. In this case, every cell is cr itical and the combinatorial Thom-Smale complex coinc ide s with the well-kno wn simplicia l chain complex. Therefore, ∂ V ◦ ∂ V = 0. Heredity : supp ose the prop erty is true for every matching with a t most k edg es defining a combinatorial Morse vector field. Let V b e a combinatorial Morse vector field with cor resp onding matching consisting of k + 1 edges. In par ticular, there is no non-s tationary closed V -path. Le t ( σ, τ ) b e an edge of this matching with σ < τ and let V be the comb ina torial Mor se v ector field corr esp onding to the original matching with the edge ( σ , τ ) r emov ed. By induction hypothesis ∂ V ◦ ∂ V = 0. In particular, for every n ∈ N , every τ ∈ K n +1 and every ν ∈ K n − 1 when ther e is a cell σ 1 ∈ K n such that there is a V -path from an hyperface of τ to σ 1 and another V -path from a n h yp er face of σ 1 to ν there exists another cell σ 2 ∈ K n with the same prop er ty s o that their contribution to ∂ V ◦ ∂ V are opp osite. First, we will prov e that ∂ V ◦ ∂ V = 0 when the chain complex is with co efficients in Z / 2 Z a nd after we will take ca re of signs. 4 ´ ETIENNE GALLAIS Suppo se that the disting uished edge o f the matching ( σ, τ ) is such that dim ( σ ) + 1 = di m ( τ ) = n + 1. Therefore, C V i = C V i for i 6 = n, n + 1 and ∂ V | C V i = ∂ V | C V i for i / ∈ { n, n + 1 , n + 2 } . So we hav e ∂ V ◦ ∂ V ( µ ) = 0 for all µ ∈ K − ( K n ∪ K n +1 ∪ K n +2 ). Remark that it is also true for every σ ′ ∈ C rit n ( V ) that ∂ V ◦ ∂ V ( σ ′ ) = 0 (since with resp ect to V it is true and σ ′ 6 = σ ). There are tw o cases left. Case 1. Let τ ′ ∈ C rit n +1 ( V ). T o see that ∂ V ◦ ∂ V ( τ ′ ) = 0 we mu s t consider t wo cas es. First case is when the t wo V -paths which annihilates don’t go through σ . Then, nothing is c ha ng ed and cont r ibutions to ∂ V ◦ ∂ V ( τ ′ ) cancel b y pair. The s econd case is when at least o ne the V -path whic h cancel b y pair for ∂ V go thr o ugh σ . The V -paths which go from τ ′ to ν are of tw o types: those who go via σ and the others. Let τ ′ → σ 2 → ν be a juxtap ositio n of tw o V -paths which ca ncel with the juxtap osition o f V -path τ ′ → σ → ν . Since ∂ V ◦ ∂ V ( τ ) = 0, there must b e a critica l cell σ 1 such that the juxtap os ition of V -paths τ → σ → ν and τ → σ 1 → ν cancels. Therefore, when consider ing ∂ V , three juxtapos itio ns of V -paths disapp ear and one is created: τ ′ → ( σ → τ ) → σ 1 → ν . It cancels with τ ′ → σ 2 → ν . It may happ ens that tw o juxtap os itions of V -paths g o through σ but this cas e works exa ctly in the same wa y . Case 2. This case is simila r to the previous ca se. Let ς b e a cell in K n +2 . There are tw o cases to see that ∂ V ◦ ∂ V ( ς ) = 0. The fir s t ca se is when the tw o V - paths whose contributions are opp osite don’t go throug h τ . Then, nothing is changed and contributions to ∂ V ◦ ∂ V ( τ ′ ) cancel b y pair. The second case is when the V -path which disapp ears is replaced by exactly a new one which g o es through the e dg e ( σ , τ ). The res ult follows simila rly . Note that to deal with this tw o cases w e used the fact that there is no non- stationary clos e d V -path (and so V -path). Now, let’s deal with the sig ns. W e will only consider the ca se 1. ab ov e, other cases work similarly . Denote n ( α → β ) (resp. n ( α → β )) the sign o f the co n tr ibution in the differential ∂ V (resp. ∂ V ) of a path going from α to β where b oth cells a re critical o f consecutive dimensio n. While considering V , we hav e by induction hypothesis (2.1) n ( τ ′ → σ 2 ) .n ( σ 2 → ν ) = − n ( τ ′ → σ ) .n ( σ → ν ) and (2.2) n ( τ → σ 1 ) .n ( σ 1 → ν ) = − n ( τ → σ ) .n ( σ → ν ) Since the juxtap osition of the V -paths τ ′ → σ 2 → ν don’t g o throug h σ we have that (2.3) n ( τ ′ → σ 2 ) .n ( σ 2 → ν ) = n ( τ ′ → σ 2 ) .n ( σ 2 → ν ) By definition of the m ultiplicity of paths we have (2.4) n ( τ ′ → σ 1 ) = n ( τ ′ → σ ) . ( − < ∂ τ , σ > ) .n ( τ → σ 1 ) COMBINA TORIAL REALIZA TION OF THE THOM-SMALE COMPLEX 5 Combining equa tions (2.1)-(2.4) we obtain the following equalities n ( τ ′ → σ 1 ) .n ( σ 1 → ν ) = n ( τ ′ → σ ) . ( − < ∂ τ , σ > ) .n ( τ → σ 1 ) .n ( σ 1 → ν )(2.4) = n ( τ ′ → σ ) . ( − < ∂ τ , σ > ) .n ( τ → σ 1 ) .n ( σ 1 → ν ) = n ( τ ′ → σ ) . < ∂ τ , σ > .n ( τ → σ ) .n ( σ → ν ) (2.2) = ( < ∂ τ , σ > . n ( τ → σ )) .n ( τ ′ → σ ) .n ( σ → ν ) = n ( τ ′ → σ ) .n ( σ → ν ) by definition = − n ( τ ′ → σ 2 ) .n ( σ 2 → ν ) (2.1) = − n ( τ ′ → σ 2 ) .n ( σ 2 → ν ) (2.3) which concludes the pro of o f the theo r em. 2.2.3. Pr o of of Th e or em 2.5. The main ingredient of the pro of is thinking ab out combinatorial Morse vector field as an instruction to remove acyclic co mplexes from the origina l s implicial chain complex, a s done b y Char i [2 , P rop osition 3.3 ] o r Sk¨ oldb erg [19]. Giv en a matching b etw een tw o c e lls σ < τ , we would lik e to r emov e the following s hort complex (which is acy c lic ) 0 → Z .τ <∂ τ ,σ > − − − − − → Z .σ → 0 where ∂ is the b ounda r y op erator of the simplicia l chain co mplex. T o do this, we use Gaussian e limina tion (see e.g . [1]): Lemma 2.6 (Gaussian elimination) . L et C = ( C ∗ , ∂ ) b e a chain c omplex over Z fr e ely gener ate d. L et b 1 ∈ C i (r esp. b 2 ∈ C i − 1 ) b e su ch that C i = Z .b 1 ⊕ D (r esp. C i − 1 = Z .b 2 ⊕ E ). If φ : Z .b 1 → Z .b 2 is an isomorphism of Z -mo dules, then t he four term c omplex se gment of C (2.5) . . . →  C i +1  0 @ α β 1 A − − − − →  b 1 D  0 @ φ δ γ ε 1 A − − − − − − →  b 2 E  “ µ ν ” − − − − − →  C i − 2  → . . . is isomorphic to the fol lowing chain c omplex se gment (2.6) . . . →  C i +1  0 @ 0 β 1 A − − − →  b 1 D  0 @ φ 0 0 ε − γ φ − 1 δ 1 A − − − − − − − − − − − − − →  b 2 E  “ 0 ν ” − − − − − →  C i − 2  → . . . Both these c omplexes ar e homotopy e quivalent to the c omplex se gment (2.7) . . . →  C i +1  “ β ” − − − →  D  “ ε − γ φ − 1 δ ” − − − − − − − − − − →  E  “ ν ” − − − →  C i − 2  → . . . Her e we use d matrix notation for the differ ential ∂ . Pr o of. Since ∂ 2 = 0 in C , we obtain φα + δ β = 0 and µφ + ν γ = 0. By do ing the following change o f basis A =  1 φ − 1 δ 0 1  on  b 1 D  and B =  1 0 − γ φ − 1 1  on  b 2 E  we see that the complex segments 2.5 and 2.6 ar e isomo r phic. Then, we remov e the short complex 0 →  b 1  φ →  b 2  → 0 which is acy c lic.  Now, we are ready to prov e Theor em 2.5. Pr o of of The or em 2.5. Like for the pro of of Theo rem 2 .4, we make an induction o n the num b er of edg es b e lo nging to the matching defining the combinatorial Mo r se vector field. Let X b e a simplicial co mplex, K b e the set of its cells. 6 ´ ETIENNE GALLAIS Initialization: matching with no edge. In this ca se, there is nothing to prove since the combinatorial Tho m-Smale complex is exactly the simplicial chain complex. Heredity : supp ose the prop erty is true for every matching with a t most k edg es defining a combinatorial Morse vector field. Let V b e a combinatorial Morse vector field who se underly ing matching consis ts of k + 1 edges. Let σ ( n ) < τ ( n +1) be a n element of this matching and V be the co mbin a torial Morse vector field equa l to V with the match ing σ < τ removed (it is ac tually a combinatorial Morse vector field). So, C V i = C V i for all i 6 = n, n + 1 a nd ∂ V = ∂ V when r estricted to C V i for all i / ∈ { n, n + 1 , n + 2 } . Moreov er , we have the following equalities: ( ∂ V ) | C V n +1 = ( ∂ V ) | C V n +1 and ∂ V | C V n = ∂ V | C V n . By induction hypo thesis, the combinatorial Thom- Smale complex ( C V ∗ , ∂ V ) is a chain complex homotopy equiv alent to the simplicial chain complex of X . Thus, the combinatorial Thom-Smale co mplex as s o ciated with V is eq ual to the one o f V except on the following chain segment (where ε = ( ∂ V ) | C V n | C V n +1 ): (2.8) . . . →  C V n +2  0 @ α ∂ V 1 A − − − − − →  τ C V n +1  0 @ < ∂ τ , σ > δ γ ε 1 A − − − − − − − − − − − − − →  σ C V n  “ µ ∂ V ” − − − − − − − →  C V n − 1  → . . . Since X is a simplicial complex we have < ∂ τ , σ > ∈ {± 1 } . Applying lemma 2.6, we obtain the follo wing new co m bina to rial chain complex which is homotopic to the combinatorial Tho m-Smale complex o f V (2.9) . . . →  C V n +2  “ ∂ V ” − − − − →  C V n +1  “ α ” − − − →  C V n  “ ∂ V ” − − − − →  C V n − 1  → . . . where α = ε − γ < ∂ τ , σ > δ = ∂ V . Thus, the only thing to prove is tha t α = ∂ V ov er C V n +1 . T o do this, w e in vestigate ∂ V . Ther e are t wo types of co n tr ibutions. First type cor resp ond to V - paths which do not go thr ough σ , and they a r e counted in ε . Second t y pe are V -paths which go thr ough σ . They b egin at an hyperface of a cr itical cell τ ′ and go through σ : this is the contribution of δ . Then, they jump to τ : this is the contribution of < ∂ τ , σ > . Finally they be gin at an hyper face o f τ and go to a critical cell in C V n : this is the co ntribution of γ . It rema ins to check that the sign is cor rect, but this is e xactly the same a s in the fir st pro o f of Theo rem 2.4.  Corollary 2.7. Let X b e a finite simplicial c omplex, C = ( C ∗ , ∂ ) b e the corre- sp onding simplicial chain co mplex and M b e a matching ( σ i < τ i ) i ∈ I on its Has se diagram defining a combinatorial vector field V . Then the following pro per ties ar e equiv alent: (1) M is a Mors e matching, (2) for any se q uence ( σ i 1 < τ i 1 ) , ( σ i 2 < τ i 2 ) , . . . , ( σ i | I | < τ i | I | ) such that i j 6 = i k if j 6 = k , Gaussian eliminations can b e p erformed in this o rder. In particula r , any se quence of Gauss ian eliminations co rresp onding to M lead to the same chain complex which is the combinatorial Tho m-Smale complex o f V . Pr o of. 1 ⇒ 2 This is an immediate co nsequence of the pro o f of Theorem 2.5 and the fact that it leads to the Thom-Smale co mplex ass o ciated to ( X , V ). COMBINA TORIAL REALIZA TION OF THE THOM-SMALE COMPLEX 7 2 ⇒ 1 It is enoug h to show tha t there is no non-stationa ry clo sed path under the hypothesis. Supp ose there is a closed V -path γ : σ 1 , . . . , σ r , σ 1 and cons ide r any sequence of Gaussian eliminatio n which co incides with ( σ j < V ( σ j )) un til step r . In particular , r ≥ 3 since X is a simplicial co mplex. Let V be the co rresp onding combinatorial vector field. Let γ ′ : σ 1 , . . . , σ r be the V -path with length decrea se by one. Then < ∂ r − 1 V ( σ r ) , σ r > = < ∂ V ( σ r ) , σ r > + m ( γ ′ ) Since m ( γ ′ ) = ± 1, < ∂ r − 1 V ( σ r ) , σ r > is not in vertible ov er Z and the Gaussian eliminatio n cannot b e p erformed (see lemma 2.6). This is a co n- tradiction.  3. Rela tion between smooth and discrete Mo rse theories In this section, we inv estig ate the link b etw een smo o th and discrete Morse the- ories. W e first recall briefly the main ing redients of s mo oth Mo rse theory . In particular, we des crib e the Thom- Smale complex and prove the following: Theorem 3.1 (Combinatorial rea liz a tion) . L et M b e a smo oth close d oriente d Rie m ann ian manifold and f : M → R b e a generic Morse funct ion. Supp ose that every stable m anifold has b e en given an orientation so that the smo oth Thom- Smale c omplex is define d. Then, ther e exists a C 1 -triangulation T of M and a c ombinatorial Morse ve ctor field V on it which r e alize the s m o oth Thom-Smale c omplex (after a choic e of orientation of e ach c el ls of T ) in the fol lowing sense: (1) ther e is a bije ction b etwe en the set of critic al c el ls and the set of critic al p oints, (2) for e ach p air of critic al c el ls σ p and σ q such that dim ( σ p ) = dim ( σ q ) + 1 , V -p aths fr om hyp erfac es of σ p to σ q ar e in bije ction with int e gr al curves of v u p to r enormalization c onne cting q t o p , (3) this bije ct ion induc e an isomorphism b etwe en the smo oth and the c ombina- torial Thom-Smale c omplexes. Throughout this section, we follow co nven tions of Milnor ([12 ], [13]). 3.1. Smo oth Morse theory. Let M be a smo oth closed oriented Riemannian manifold of dimension n . Given a smo o th function f : M → R , a p o int p ∈ M is said critical if D f ( p ) = 0. Let C r it ( f ) be the set of critical p oints. A t a critical po int p , we consider the bilinear form D 2 f ( p ). The num be r of neg ative eighenv alues of D 2 f ( p ) is called the index of p (denoted ind ( p )). W e denote C rit k ( f ) the set o f critical p oints of index k . Definition 3. 2 . A smo o th map f : M → R is called a Morse function if at each critical p oint p of f , D 2 f ( p ) is no n-degenera te. More generally , a Morse function on a (smo oth) cob ordism ( M ; M 0 , M 1 ) is a smo oth map f : M → [ a, b ] such that (1) f − 1 ( a ) = M 0 , f − 1 ( b ) = M 1 , (2) all cr itical p oints of f are in ter ior (lie in M − ( M 0 ∪ M 1 )) and are non- degenerate. F or technical reasons, we m ust consider the following o b ject: 8 ´ ETIENNE GALLAIS Definition 3. 3 . Let f b e a Mo rse function on a cobo rdism ( M n ; M 0 , M 1 ). A vector field v o n M n is a gradient-lik e vector field for f if (1) v ( f ) > 0 througho ut the complement of the set of cr itical p oints of f , (2) given any critical po in t p of f there is a Mor se chart in a neighbourho o d U of p so that f ( x ) = f ( p ) − k X i =1 x 2 i + n X i = k +1 x 2 i and v has c o ordinates v ( x ) = ( − x 1 , . . . , − x k , x k +1 , . . . , x n ). Given any Morse function, ther e alwa ys exists a gradie nt-like vector field (see [13]). In the following, we shall abrevia te “g r adient like vector field” by “ gr adient ” . Thu s , when needed, we will assume that we hav e chosen one. Given a ny x 0 ∈ M , w e consider the following Cauchy problem  γ ′ ( t ) = v ( γ ( t )) γ (0) = x 0 and call in teg ral cur ve (denoted γ x 0 ) the solution of this Cauch y pro blem. The stable manifold of a critica l p oint p is by definition the set W s ( p, v ) := { x ∈ M | lim t → + ∞ γ x ( t ) = p } . The unstable manifold o f a c r itical p oint p is by definition the set { x ∈ M | lim t →−∞ γ x ( t ) = p } . When stable and unstable ma nifolds are tra nsverse (this is c a lled Morse-Sma le condition), w e ca lled v a Mo rse–Smale g radient: such gradient always exists in a neighbour ho o d o f a gradient (see e.g . [16]). W e shall call a Mor se function f generic if we have chosen for f a Mo rse–Smale gr a dient. T o define the smo oth Thom-Smale co mplex we need the following data: • a generic Mors e function f , • an orientation o f ea ch stable manifold. Under these conditions, the num b e r of integral curves of v up to r enormalizatio n (that is γ x ∼ γ y iff there exists t ∈ R such that γ x ( t ) = y ) co nnecting tw o critical po int s of co nsecutive index is finite. Moreov er , when we consider a n int eg ral cur ve from q to p where ind ( p ) = ind ( q ) + 1, it carries a co orientation induced by the orientation of the stable manifold and the orientation of the in tegr al curve. O ne can mo ve this co or ient a tion from p to q along the integral curve and compare it with the orientation of the sta ble manifold of q . This gives the sign which is car ried by the integral curve connecting q to p . The Thom-Sma le complex ( C f ∗ , ∂ f ) is defined as : (1) C f k = L p ∈ C r it k ( f ) Z .p , (2) if p ∈ C r it k ( f ) then ∂ p = P q ∈ C r it k − 1 ( f ) n ( p, q ) .q where n ( p, q ) is the al- gebraic n umber of integral curves up to renormalization connec ting q to p . Theorem 3.4. The homolo gy of the Thom-Smale c omplex is e qual t o the singular homolo gy of M . The pro o f of this theor em can b e extracted fr om [13]. 3.2. Eleme ntary cob ordisms. In this subsection, we will pr ov e that we can real- ize combinatorially the smooth Thom-Smale complex of a ny elementary cob or disms. Thu s , by cutting the manifold M into elemen tar y cob ordism we will obtain the first COMBINA TORIAL REALIZA TION OF THE THOM-SMALE COMPLEX 9 part of Theo rem 3.1: there exists a bijection b etw een the set of critical cells and the set of critical p oints. W e will only co nsider C 1 -triangulatio n o f manifolds for technical reaso ns (see [21]). So, whenever we use the word tr iangulation it means C 1 -triangulatio n. A triangulation o f a n + 1-c o b o rdism ( M n +1 ; M 0 , M 1 ) is a triplet ( T ; T 0 , T 1 ) such that T is a C 1 -triangulatio n o f M , T 0 (resp. T 1 ) is a sub complex of T which is a C 1 -triangulatio n o f M 0 (resp. M 1 ). A combinatorial Morse vector field V on a tr ia ngulated n + 1-cob or dis m ( T ; T 0 , T 1 ) is a co mbinatorial Morse vector fie ld on T such tha t no ce lls of T 1 is critical and every cell of T 0 is critical. Definition 3.5. Let V b e a c o mbin a torial Mor se vector field on a triangulated n + 1- cob ordism ( T ; T 0 , T 1 ). V satisfie s the ancesto r’s prop er t y if given any n -ce ll σ 0 ∈ T 0 , there exis ts an n -cell σ 1 ∈ T 1 and a V -path starting at σ 1 and ending a t σ 0 . R emark. Ther e is a key difference b etw een integral curves up to renor malization of a gr adient v and V - paths. Given a point x ∈ M , there is only o ne solution to the Cauch y pro blem. Moreov er, the pa s t a nd the future of a p o int pushed along the flo w is uniquely determined. A c ontr ario g iven a cell σ , there ar e (in general) many V -paths sta rting a t σ . The a ncestor’s prop erty ca racterise s n + 1- cob ordism equipp ed with a combinatorial Morse v ector field which k nows its history in maximal dimension ( n, n + 1). T o pr ove that elementary cob or dism can b e realized , we need a combinatorial description of b eing a defor mation retract. Let X be a simplicial complex and σ be an hyper fa ce of τ which is free (that is σ is a face of no other cell). In this case, we say that X c ollapses to X − ( σ ∪ τ ) by a n elementary co llapsing and write X ց X − ( σ ∪ τ ). A co lla psing is a finite s equence of such elemen ta r y collapsings. In particular, a colla psing defines a matc hing o n the Hasse diagram of the simplicial complex. Moreov er, one can prove that X − ( σ ∪ τ ) is a defor mation retract of X . Prop ositio n 3. 6 . Let X b e a simplicial complex and X 0 be a sub complex. Suppo se X ց X 0 . Then the match ing given by this collaps ing defines a combinatorial Mo rse vector field whose set of critical c e lls is the set of cells of X 0 . Pr o of. The only thing to chec k is tha t there is no non-stationa ry closed path. Since elementary colla psings ar e p erformed by choos ing a free hyperface of a ce ll, there is no non-sta tionary closed pa th.  Let ∆ m = ( a 0 , . . . , a m ) b e the standard simplex of dimensio n m . The car tesian pro duct X = ∆ m × ∆ n is the cellular complex whose set of cells is { µ × ν } where µ (resp. ν ) is a cell o f ∆ m (resp. ∆ n ) (see [22]). Prop ositio n 3.7 ([17, Pro p. 2 .9 ]) . The cartesian pro duct ∆ m × ∆ n has a simplicial sub div ision without any new vertex. More gener ally , the ca rtesian pro duct of tw o simplicial complexes has a simplicia l sub division witho ut any new vertex. Lemma 3. 8. L et X 1 = ∆ k b e t he standar d simplicial c omplex of dimension k and X 0 b e a s implicial sub divisio n o f X 1 . Consider the CW-c omplex which is e qu al to the c artesian pr o duct ∆ k × ∆ 1 and wher e we sub divide ∆ k × { 0 } so that it is e qual to X 0 . Then, ther e exists a simplicial sub division X of this CW- c omplex such that X | ∆ ×{ i } = X i for i ∈ { 0 , 1 } . 10 ´ ETIENNE GALLAIS Mor e over for i ∈ { 0 , 1 } ther e exists a c ol lapsing X ց X i and the c ombinatorial Morse ve ctor field asso ciate d V i satifies the anc estor’s pr op erty on ( X ; X i , X i +1 ) ( j is the class in Z / 2 Z ). Pr o of. The simplicial s ubdivis ion and the collapsing is constructed by induction on k . If k = 0 , cho ose a new vertex in the in ter ior of the simplex ∆ 0 × ∆ 1 and the elementary collapsing ∆ 1 ց { i } g ives the tw o collapsing ∆ 0 × ∆ 1 ց ∆ 0 × { i } for i ∈ { 0 , 1 } . In par ticular, the co rresp onding combinatorial Morse vector field satifies the ancesto r’s prop erty . Suppo se the le mma is true until rank k − 1 . At ra nk k , let Y b e the corre- sp onding CW-complex and x be a p oint in the in terio r of the cell of dimension k + 1. By induction hypo thesis Y | ∂ ∆ k × ∆ 1 admits a simplicial s ubdivis ion. There- fore, Y | ∂ (∆ k × ∆ 1 ) admits a simplicia l s ubdivis ion de no ted Z (just add the simplexes ∆ k × { i } whic h ar e equal to X i for i ∈ { 0 , 1 } ). The simplicial subdivisio n X is given by ma king the join of the s implicial subdivisio n o f the b oundar y ov er { x } : X = Z ∗ { x } . Now, the c ollapsing X ց X 0 is p erformed in three steps. Step 1. The cell σ ∈ X 1 of dimension k is the free hyperfac e o f the cell σ ∗ { x } . W e do the following elementary collaps ing : (3.1) X ց X − ( σ ∪ σ ∗ { x } ) Step 2. By induction h yp othesis, X | ∂ ∆ k × ∆ 1 ց X | ∂ ∆ k ×{ 0 } . Performing the join ov er x induces the following collapsing : (3.2) X | ∂ ∆ k × ∆ 1 ∗ { x } ց X | ∂ ∆ k ×{ 0 } ∗ { x } Step 3. It remains to collapse X 0 ∗ { x } on X 0 . L e t y b e a vertex in X 0 which is a vertex of the original simplex X 1 . Since X 0 is a simplicial subdivisio n o f ∆ k , ther e exists a co llapsing X 0 ց { y } . This co llapsing gives the following collapsing: (3.3) X 0 ∗ { x } ց X 0 ∪ ( { y } × { 0 } ) ∗ { x } ց X 0 Combining collaps ing s (3.1), (3.2) a nd (3 .3) gives X ց X 0 . The corr esp onding combinatorial Mors e vector field satisfies the ancestor ’s prop erty by construction. The collapsing X ց X 1 is constructed in the same w ay and conclusions of lemma follows.  R emark. The pr o of of lemma 3 .8 is b y induction. Let δ ( j ) be the j -th sk ele ton of ∆ k . Denote by X ( j ) (resp. X ( j ) i ) the s implicial co mplex X | δ ( j ) × ∆ 1 (resp. ( X i ) | δ ( j ) × ∆ 1 ). F or i ∈ { 0 , 1 } , the collapsing X ց X i can be re s tricted to X ( j ) ց X ( j ) i for any 0 ≤ j ≤ k a nd the induced combinatorial Morse vector field sa tisfies the ancestor ’s prop erty . The nex t tw o lemmas are technical lemmas. The first one is the basic to ol to glue tog ether tr iangulated cob or disms. The sec o nd one will b e useful to c onstruct a co m bina torial rea lization of a cob ordis m with exactly o ne critica l po int and is a generaliza tion of lemma 3.8. Lemma 3.9. L et ( T M i , T N i ) b e two C 1 -triangulations of the p air ( M , N ) wher e N k is a submanifo ld (p ossibly with b oundary) of M n ( k ≤ n ). Then, ther e exists a C 1 -triangulation T of ( M × [0 , 1 ] , N × [0 , 1]) such that ( T | M ×{ i } , T | N ×{ i } ) = ( T M i , T N i ) COMBINA TORIAL REALIZA TION OF THE THOM-SMALE COMPLEX 11 for i ∈ { 0 , 1 } and 2 c ol lapsings (3.4) T ց T M 0 ∪ T | N × [0 , 1] (3.5) T | N × [0 , 1] ց T N 0 Mor e over, t he induc e d c ombinatorial Morse ve ctor fi elds V s atifies the anc estor’s pr op ert y on the c ob or disms ( T | N × [0 , 1] ; T N 0 , T N 1 ) and ( T ; T M 0 , T M 1 ) . Pr o of. First, supp ose N = ∅ . Both triang ulations T 0 and T 1 are C 1 -triangulatio n of the s ame manifold therefore they hav e a common simplicial sub divisio n T 1 / 2 [21] (this is wher e we use the fact that tria ng ulations a re C 1 -triangulatio ns). Sub divide ∆ 1 = [0 , 1 ] in tw o standard simplexes [0 , 1 / 2] and [1 / 2 , 1]. Lemma 3.8 gives a C 1 - triangulation o f M × [0 , 1 / 2] (resp. M × [1 / 2 , 1]) denoted T [0 , 1 / 2] (resp. T [1 / 2 , 1] ). The union T [0 , 1 / 2] ∪ T [1 / 2 , 1] is a tria ngulation o f M × [0 , 1] denoted T . By constr uc - tion, T | M ×{ i } = T M i for i ∈ { 0 , 1 } and we hav e the tw o following collapsings: T [1 / 2 , 1] ց T 1 / 2 T [0 , 1 / 2] ց T 0 Comp osing these tw o collapsing s giv e the desired collapsing and lemma 3.8 give the ancestor’s pro per ty . In the cas e w her e the s ubma nifold N is non-empty , the constr uction ab ov e gives a tr iangulation T of the pair ( M × [0 , 1] , N × [0 , 1 ]) and we have ( T | M ×{ i } , T | N ×{ i } ) = ( T M i , T N i ) for i ∈ { 0 , 1 } . The collapsing T ց T 0 can b e restricted to T | N × [0 , 1] . W e remov e from the matching edges corr esp onding to T N × [ 0 , 1] ց T N × { 0 } to obtain the desired colla psing. Aga in lemma 3.8 give the ancestor’s pro p er t y .  Lemma 3.10. L et ( m, n ) b e a p air of p ositive inte gers. L et ∆ n = ( a 0 , . . . , a n ) b e the st andar d simplex of dimension n and δ n − 1 = ( b a 0 , . . . , a n ) b e the hyp erfac e which do es not c ontain a 0 . In p articular ∆ n = { a 0 } ∗ δ n − 1 . Then ther e exists a simplicial sub division X of the c artesian pr o duct ∆ m × ∆ n such that • X | ∆ m × δ n − 1 is a simplicial sub division without any n ew vertex given by lemma 3.7, • X | ∆ m ×{ a 0 } = ∆ m , • X ց X | ( ∂ ∆ m × ∆ n ) ∪ (∆ m ×{ a 0 } ) . Mor e over, for e ach simplex ∆ 1 i = ( a 0 , a i ) ( i 6 = 0 ), • X | ∆ m × ∆ 1 i c oincides with t he simplicial c omplex given by lemma 3.8, • the c ol lapsing X ց X | ( ∂ ∆ m × ∆ n ) ∪ (∆ m ×{ a 0 } ) r estricte d to X | ∆ m × ∆ 1 i c oincides with the c ol lapsing of lemma 3.8 , • the induc e d c ombinatorial Morse ve ctor field satisfies t he anc estor’s pr op erty on ( X | ∆ m × ∆ 1 i ; X | ∆ m ×{ a 0 } , X | ∆ m ×{ a i } ) . Pr o of. The pro of is by induction o n k = m + n > 0. At rank k = 1 there are tw o cases. The case m = 0 and n = 1 is trivia l: there is nothing to pr ov e. The case m = 1 and n = 0 is given b y lemma 3.8. Suppo se the lemma is true until rank k − 1. Let ( m, n ) ∈ N 2 be such that m + n = k . W e will first sub divide the b oundar y of ∆ m × ∆ n . Since ∂ (∆ m × ∆ n ) = ( ∂ ∆ m × ∆ n ) ∪ (∆ m × ∂ ∆ n ) = ( ∂ ∆ m × ∆ n ) ∪ (∆ m × ( { a 0 } ∗ ∂ δ n − 1 )) ∪ (∆ m × δ n − 1 ) we define for each cellular complex ab ov e a simplicial sub divisio n. 12 ´ ETIENNE GALLAIS • The simplicial sub divisio n o f ∆ m × δ n − 1 is giv en by Propo sition 3.7: in particular, we do not create any new vertex. • The induction hypothesis gives a simplicial sub divisio n of ( ∂ ∆ m × ( { a 0 } ∗ δ n − 1 )) ∪ (∆ m × ( { a 0 } ∗ ∂ δ n − 1 )) . Let x be a p oint which is in the interior of the ( m + n )-cell o f ∆ m × ∆ n . The simplicial sub division X of ∆ m × ∆ n is given by mak ing the cone ov er { x } of the simplicial sub division of the b ounda ry of ∆ m × ∆ n . By co nstruction we have the following collapsing (3.6) X |{ x }∗ (∆ m × δ n − 1 ) ց X |{ x }∗ ∂ (∆ m × δ n − 1 ) which is realized b y a downw ard induction on the dimension of c e lls of ∆ m × ( δ n − 1 − ∂ δ n − 1 ): ev ery cell σ ∈ ∆ m × ( δ n − 1 − ∂ δ n − 1 ) is a free hyper face of { x } ∗ σ . The induction hypothesis says that there exits a simplicial sub divisio n Y of ∆ u × ∆ v such tha t Y ց Y | ( ∂ ∆ u × ∆ v ) ∪ (∆ u ×{ a 0 } ) whenever u + v < k ( a 0 is the first vertex of ∆ v ). In fact, we hav e also the fo llowing co llapsing since the co nstruction is made by induction: Y ց Y | ∆ u ×{ a 0 } Therefore, we have the following colla psings (3.7) X | ∂ ∆ m × ( { a 0 }∗ δ n − 1 ) ց X | ∂ ∆ m ×{ a 0 } (3.8) X | ∆ m ×{ a 0 }∗ ∂ δ n − 1 ց X | ∆ m ×{ a 0 } Collapsing (3.6) follow ed b y the co ne over x of the co llapsing (3 .7) a nd the cone ov er x of the collapsing (3.8) g ive the fo llowing collapsing : X ց X | ( ∂ ∆ m × ∆ n ) ∪ ( { x }∗ (∆ m ×{ a 0 } )) Finally there exists a collapsing { x } ∗ (∆ m × { a 0 } ) ց ∆ m × { a 0 } (by choo sing a vertex y ∈ ∆ m and co ns idering the colla psing ∆ m ց { y } ) whic h gives the result. In case n = 1 , this cons tr uction is the same a s the one o f lemma 3.8.  Theorem 3. 11. L et f b e a generic Morse function on a c ob or dism ( M ; M 0 , M 1 ) with exactly one critic al p oint p of index k . Then, ther e exists a C 1 -triangulation of the c ob or dism ( T ; T 0 , T 1 ) such t hat (1) the stable manifold of p is a sub c omplex of T denote d T s p and T ց T s p ∪ T 0 , (2) ther e is a c el l σ p of dimension ind ( p ) such that p ∈ σ p ⊂ T s p and T s p − σ p ց ( T s p ∩ T 0 ) In p articular, the c ombinatorial Mo rse ve ctor field give n by t hese two c ol lapsings has exactly one critic al c el l σ p outside c el ls of T 0 . Pr o of. Supp ose a Mor se–Smale gradient v for f is fixed. Let W s ( p, v ) b e the cor- resp onding stable ma nifold of p . W e follow the pro of o f Milnor which pr ov es that M 0 ∪ W s ( p, v ) is a deformation r etract of M (see the pro of o f Theor em 3.14 [1 2]). Let C b e a (small eno ugh) tubular neighbour ho o d of W s ( p, v ). The origina l pro of consists of tw o steps. First, M 0 ∪ C is a deformation r e tr act of M : this is done by pushing along the gra dient lines of v . The n, M 0 ∪ W s ( p, v ) is a defor mation retr act of M 0 ∪ C . W e prove the theor em in tw o s teps. COMBINA TORIAL REALIZA TION OF THE THOM-SMALE COMPLEX 13 First step: construction of a go o d triangulation of C . The tubular neighbourho o d C is diffeomorphic to D k × D n − k (for i ∈ N ∗ , D i is the unit disk in R i ). Thanks to this diffeomo rphism, the stable manifold is ident ified with D k × { 0 } a nd the a dher ence of the unstable manifold is iden tified with { 0 } × D n − k . T riangulate the stable manifold by the standar d simplex ∆ k and denote σ p its interior (so T s p = σ p ). W e triangulate D n − k by cho osing a n arbitrary triangula tion of ∂ D n − k = S n − k − 1 and considering D n − k as the c o ne over its center { 0 } : this gives a triangulatio n of D n − k . The triangulation of σ p × D n − k is the following o ne: c ho o se a simplicial sub div ision of σ p × ∂ D n − k without any new vertex given by prop osition 3.7. Then, triangulate the cartesia n pro duct σ p × D n − k with the triangula tion of σ p × ∂ D n − k already fixed thanks to lemma 3.10: • for ea ch simplex ν ∈ ∂ D n − k , the lemma constructs a triangula tion of σ p × ( { 0 } ∗ ν ), • for each pair of simplex e s ( ν 0 , ν 1 ) ∈ ( ∂ D n − k ) 2 , the s implicial subidivis io ns of σ p × ( { 0 } ∗ ν i ) coincides ov er σ p × ( { 0 } ∗ ( ν 0 ∩ ν 1 )). Let T C be the triang ula tion of σ p × D n − k constructed ab ov e. By co nstruction, we hav e the fo llowing co llapsing (3.9) T C ց T s p ∪ T C | ∂ σ p × D n − k Second step: com bi natorial realization of the first retraction. Let T 0 be a triangulation of M 0 which coincides ov er M 0 ∩ C with the triang ula- tion ab ove. Consider the following submanifolds with bo undary: ∂ C − = M 0 ∩ C , M C 0 = M 0 − I nt ( ∂ C − ) a nd ∂ C + = ∂ C − I nt ( ∂ C − ). Let V b e ∂ C − ∩ ∂ C + : it is diffeomorphic to ∂ D k × ∂ D n − k = S k − 1 × S n − k − 1 a manifold of dimension n − 2. The manifold ( ∂ C + , V ) is a manifold with bo undary which is triangula ted. The gradient lines of v starting a t any p oint of this manifold are transverse to it: we push along the gradient lines of v the triang ulation until it meets M 1 . It gives a triangulation o f ( M ∂ C + 1 , M V 1 ) which is a submanifold o f M 1 with b ounda r y . This triangulation is C 1 since pushing along the flow in this case is a diffeomor phism. Then, we get a pro duct cob ordism (with b ounda ry) with triang ulation of the top and the bo ttom a lready fixed: lemm a 3.9 gives a triangulation of this co b o r dism with the des ired collaps ing . The same co nstruction ho lds fo r ( M C 0 , V ) (w e suppose that the tria ngulation of V × [0 , 1] is the sa me as the one given above). Let T b e the cor resp onding triangulation of M . Then, we hav e the following co llapsing (3.10) T ց T 0 ∪ T C Conclusion. The co mpo sition of co llapsings (3.10) a nd (3.9) give T ց T 0 ∪ T s p Since T s p = σ p we get the following co llapsing: T s p − σ p ց ∂ T s p . Th us a combinatorial Morse vector field which satifies the conclus ion of the theorem has b een constructed. Nevertheless, note that the triangula tion ab ove in not C 1 : the tr iangulation of the stable manifold done b y ∆ k gives only a top ologica l triangula tion. T o corr ect this, push the level M 0 (denote this level M ′ 0 ) a long the g r adient line a little inside the cob ordism so that the stable manifold ca n b e C 1 -triangulated by the s tandard 14 ´ ETIENNE GALLAIS simplex. Then, we endow the cob ordis m whose b oundary is M 0 ∪ M ′ 0 with a C 1 - triangulation given b y Lemma 3.8.  Corollary 3.12. Let f b e a ge ne r ic Morse function o n a Riemannian closed man- ifold M . Then, there ex ists T a C 1 -triangulatio n o f M and a combinatorial Morse vector field V defined on T suc h that fo r every k ∈ N the set of critical p o ins o f index k is in bijection with the set o f critical cells of dimension k . Pr o of. Since the Morse function f is gene r ic, we have that for any critical points p 6 = q , f ( p ) 6 = f ( q ). Let a 1 < a 2 < . . . < a l be the order ed set of cr itical v alues of f . F or each k ∈ { 1 , . . . , l } , let ε k > 0 be small eno ugh so that the co b o r dism ( M a k ; M a k − , M a k + ) =  f − 1 ([ a k − ε k , a k + ε k ]); f − 1 ( a k − ε k ) , f − 1 ( a k + ε k )  is a co bo rdism with exactly one critical p oint. Define for k ∈ { 1 , . . . , l − 1 } the pro duct cob ordisms ( M b k ; M a k − 1 + , M a k − ) =  f − 1 ([ a k − 1 + ε k − 1 , a k − ε k ]); f − 1 ( a k − 1 + ε k − 1 ) , f − 1 ( a k − ε k )  The manifold M is eq ua l to: M a 1 ∪ M b 1 ∪ . . . ∪ M b l − 1 ∪ M a l Theorem 3.11 gives for k = 1 , . . . , l a co mbinatorial realization of the cob ordism ( M a k ; M a k − , M a k + ). Lemma 3.9 gives a combinatorial r ealization o f each cob ordism ( M b k ; M a k − 1 + , M a k − ) for k = 1 , . . . , l − 1 (with the conv ention that M a 0 = ∅ ). Then, we constr uct a C 1 -triangulatio n o f M a nd define on it a combinatorial vector field. It is in fact a combinatorial Mors e vector field since along V -paths we only can go down and the conclusion o f the cor ollary follows.  3.3. Pro of of Theorem 3.1. Since f is gener ic, we use the Rearrang ement The- orem [13, Theorem 4.8] to consider g a generic self-indexed Morse function suc h that • the set o f critical p oints of index k of g co incide s with the one o f f fo r every k ∈ N , • for each pair of critical p oints p and q of suc c essive index, the set of int eg ral curves up to renor malization connecting q to p for g is in bijection with the corres p o nding set for f (w e supp ose here that Morse– Smale gradients have bee n chosen for f and for g ), • this bijection induces an isomorphism b etw een the Thom-Smale complexes of f and g (we suppos e that orientations o f stable manifolds hav e been chosen). Thu s , we supp ose that f : M n → R is a generic self-index ed Mors e function i.e. for every k ∈ N , for every p ∈ C ri t k ( f ), f ( p ) = k . In particular f ( M ) = [0 , n ]. W e suppo se whenever we need it that a Mo rse–Smale gr a dient v for f is given. One mor e time, w e will cut M in cob or dis ms (a lmost) elementary and control combinatorially the b ehavior o f V -paths. F or i ∈ { 0 , . . . , n } choo se 0 < ε i < 1 / 2. F or i ∈ { 0 , . . . , n } , let ( M i ; M i − , , M i + ) b e the cob ordis m ( f − 1 ([ i − ε i , i + ε i ]); f − 1 ( i − ε i ) , f − 1 ( i + ε i )) Similarly , define ( M i,i +1 ; M i + , M i +1 − ) the pr o duct cob or dism e q ual to ( f − 1 ([ i + ε i , i + 1 − ε i +1 ]); M i + , M i +1 − ) COMBINA TORIAL REALIZA TION OF THE THOM-SMALE COMPLEX 15 Then M = M 0 ∪ M 0 , 1 ∪ M 1 ∪ . . . ∪ M n − 1 ,n ∪ M n F or all i ∈ { 0 , . . . , n } , ( M i ; M i − , M i + ) is a cob ordis m with | C r it i ( f ) | critical p oints of index i (ma yb e there is no cr itica l p o int). The tria ngulation o f M is constr ucted in the following wa y: (1) triangulatio n of co bo rdisms ( M i ; M i − , M i + ) for all i ∈ { 0 , . . . , n } given by Theorem 3.1 1, (2) triangulatio n o f co bo rdisms ( M i,i +1 ; M i + , M i +1 − ) for all i ∈ { 0 , . . . , n } given by le mma 3.9. R emark. Theor em 3 .11 is prov ed in the case wher e there is exactly one critical po int . This pr o of extends directly to the case of k c r itical p oints of the same index under the condition that tubular neig hbo urho o ds o f stable manifolds are chosen to be disjoints one from each other. Let p b e a critical p oint of index k a nd C ( p ) b e a tubular neig hbourho o d (sma ll enough) o f the stable manifold of p in the corresp onding cobor dism. Denote ∂ C − ( p ) (resp. ∂ C + ( p )) the submanifold diffeomorphic to ∂ D k × D n − k (resp. D k × ∂ D n − k ). Denote σ p the critica l cell of dimensio n k c orresp onding to p (see Theo rem 3.11). Hyp othesis on the triangulation of ∂ C + ( p ) . (1) stable ma nifolds of critical p oints o f index k + 1 intersect ∂ C + ( p ) a long a sub c omplex of dimensio n k and in ter sect σ p × ∂ D n − k along cells of dimen- sion k of the type σ × { a i } where a i is a vertex of ∂ D n − k , (2) each integral curve up to renorma lization γ from p to q ∈ C rit k +1 ( f ) in- tersects ∂ C + ( p ) in the interior of a k - cell σ γ ∈ σ p × ∂ D n − k , (3) given t wo distinct integral curves up to re no rmalization γ and γ ′ from p to critical p oints of index k + 1 then σ γ 6 = σ γ ′ . R emark. The firs t h yp othesis is satisfied by choosing small enough ε k and since stable and unstable manifolds are transverse. F or such a s ma ll enough ε k the last hypothesis will b e sa tis fied. The second hypothesis is automatica lly satisfied if the first hypothesis is satisfie d. In each tr ia ngulated cob ordism ( M k ; M k − , M k + ), stable manifolds o f critical p oints of index k are subcomplexe s. F ollowing notations of Theor e m 3 .11, we have the following collaps ing T s p − σ p ց ∂ T s p Using lemma 3.9, we o btain the following collapsing M k − 1 ,k ց M k − 1 + which can be restricted to the stable manifold o f p since it is a submanifold of M k − . With resp ect to the stable ma nifold, the combinatorial Mors e vector field s a tisfies the ancesto r’s prop erty . Let γ b e an in tegra l curve of v up to re no rmalization from q ∈ C rit k − 1 ( f ) to p . It int er sects ∂ C + ( q ) in a point which by hypothesis b elongs to a cell σ q × { a γ } . There is a 1 − 1 cor resp ondance b etw een the s e t of int eg ral curves up to reno r malization from q to p (with ind ( p ) = ind ( q ) + 1) and V -paths from hyperfa c es of σ p to σ q given by γ ← → σ γ . F rom σ γ , ther e is a unique V -pa th ending at σ q . Since V satisfies the ancestor’s prop erty in the stable manifold and σ γ is a cell of dimensio n k − 1 ther e is an a ncestor 16 ´ ETIENNE GALLAIS of σ γ which is an h yp er face of σ p . This gives a V -path b etw een an hyperface of σ p to σ q which corr esp onds to γ . W e endow each critical cell with the orientation of the cor resp onding stable manifold and e very other cell is endow ed w ith an ar bitrary orientation. By constructio n, the multiplicit y of V -pa th coincides with the sign o f the co rre- sp onding gra dient pa th and the theorem follows. 4. Complete ma tchings and Euler structures In this section, we use Theorem 3.1 to prov e the follo wing: given a closed orient ed 3-manifold a nd an Euler structure on it, there is a triang ulation such that a complete matching on the Hasse dia gram of a triangulatio n r ealizes this Euler structure. 4.1. Complete m atc hings. Definition 4.1. A co mplete matching on a gra ph is a matching such that ev er y vertex b elongs to an edg e of the matching. As a co rollar y of theorem 3.1 we obtain: Corollary 4.2. Let M b e a closed s mo o th ma nifold o f dimensio n 3 . Then ther e exits a C 1 -triangulatio n o f M such that a complete matching on its Ha s se diag r am exists. Pr o of. Since M is a clo sed smo o th manifold of dimension 3 we have χ ( M ) = 0 where χ denotes the Euler character is tic. T ake a p ointed Heega ard splitting of M (Σ g ; α = ( α 1 , . . . , α g ) , β = ( β 1 , . . . , β g ); z ) o f genus g so that ther e is an n -uplets o f int er section p oints x b etw een the α ’s and the β ’s which defines a bijection b etw een the se ts α and β . It is always p ossible to find such a p ointed Heega ard splitting after a finite num b er o f iso topies of the α ’s a nd β ’s curv es (see [7]). The Mo rse function f cor resp onding to this Heegaard splitting has o ne cr itica l p oint of index 0 a nd 3 and g critica l p oints of index 1 and 2 . Denote the set of index 1 (resp. 2) critical points by { q i } g i =1 (resp. { p i } g i =1 ) where q i (resp. p i ) corresp onds to α i (resp. β i ) for all i ∈ { 1 , . . . , g } . The n -uplet o f intersection p oints x = ( x 1 ,i 1 , . . . , x n,i n ) gives for each j ∈ { 1 , . . . , g } an integral c ur ve connecting q j to p i j . The p oint z gives a n integral cur ve connecting the index 0 cr itical p oint to the index 3 critical po int . T ake a co mbin a torial realiza tio n ( T , V ) as given b y Theo rem 3.1 of ( M , f ). Then to each p oint x i,j i corres p o nd now a V -path γ from an h y per face of the cr itical cell σ p i j to σ q i : we change the matching along this path so that b oth τ p i j and σ q i are no more cr itical cells. If γ : σ 0 , . . . , σ r = σ q i , then do the following: • match σ 0 with τ , • for every i ∈ { 1 , . . . , r } match σ i with V ( σ i − 1 ). Now suupp ose that z b elo ngs to the interior of a 2- c ell τ z (if not, subdivide T ). Denote by ς the critica l cell of dimension 3 and by υ the critica l c ell of dimension 0. There is by construction a V -path γ : τ 0 , . . . , τ r = τ z from an hyperface of ς to τ z since z is in the stable manifold of the index 3 critical po int . W e mo dify the matching alo ng γ this way: • match τ 0 with ς , • for every i ∈ { 1 , . . . , r } match τ i with V ( τ i − 1 ). COMBINA TORIAL REALIZA TION OF THE THOM-SMALE COMPLEX 17 In fact, it is no more a ma tc hing since τ z belo ngs to tw o edges of the matching. Nevertheless, τ z belo ngs to the unstable ma nifo ld of υ the critica l cell of dimen- sion 0. By the cons tr uction done in Theorem 3.1, the tub ula r neighbourho o d o f the critical point of index 0 is equal to D 3 = ∂ D 3 ∗ { 0 } . The triangulation of this tubular neighbo urho o d is given by making the c o ne ov er { 0 } = υ of a tr ian- gulation of ∂ D 3 . W e mo dify the matching a s follows. Le t ν denotes the critical 0-cell and supp ose τ z ∗ ν is the tetrahedr on AB C D where A corr esp onds to ν . The collapsing ∂ D 3 ∗ { 0 } ց { 0 } gives in pa rticular the following matching on AB C D : ( B C D , AB C D ), ( B C, AB C ) and ( B , AB ) ( A is critical). Mo dify the ma tch ing by ( A, AB ), ( B , B C ) a nd ( AB C, AB C D ). Then, B C D (which is τ z ) is critica l. This gives a co mplete matching ov er T .  4.2. Euler structures and homologo us v ector fields. Thr o ughout this sub- section we us e conv entions of T uraev [20 ]. 4.2.1. Combinatorial Euler st ructur es. Complete matchings hav e an interpretation as Euler chains. First, we r ecall Euler s tructures as defined b y T ur aev [2 0]. Let ( M , ∂ M ) be a smo oth ma nifold of dimension n and T b e a C 1 -triangulatio n of M . Suppo se ∂ M = ∂ 0 M ` ∂ 1 M b e such that χ ( M , ∂ 0 M ) = 0 a nd let T i be equal to T | ∂ i M for i ∈ { 0 , 1 } . Denote K the set of cells o f T and K i the set of c ells of T i for i ∈ { 0 , 1 } . F o r each cell σ ∈ K , let sg n ( σ ) be equal to ( − 1) dim ( σ ) and pick a σ a p oint in the interior of σ . An Euler chain in ( T , T 0 ) is a one-dimensional singula r chain ξ in T with the b oundar y of the form P σ ∈ K − K 0 sg n ( σ ) a σ . Since χ ( M , ∂ 0 M ) = 0 , the set o f Euler chains is no n- empt y . Given t wo Euler chains ξ and η , the difference ξ − η is a cycle. If ξ − η = 0 ∈ H 1 ( M ) then we say that ξ and η a re homolo g ous. A c la ss of homologo us Euler chains in ( T , T 0 ) is called a combinatorial Euler str ucture on ( T , T 0 ). Let E ul ( T , T 0 ) b e the set of Euler structur es on ( T , T 0 ). If ξ is an Euler chain, denote b y [ ξ ] its class as a com binator ial Euler structure. Euler chains b ehav e well with r esp ect to the sub div ision of a triangulation: this allows us to consider the se t E u l ( M , ∂ 0 M ) of Euler structures on ( M , ∂ 0 M ). T ak ing ξ an element of this set means choosing a tr ia ngulation ( T , T 0 ) of ( M , ∂ 0 M ) and co nsidering an Euler chain on ( T , T 0 ). R emark. Let C b e a complete matching o n a C 1 -triangulatio n ( T , T 0 ) of ( M , ∂ 0 M ). Then it defines an Euler chain [ ξ c ] ∈ E ul ( T , T 0 ): orien t every edge of the ma tc hing from o dd dimensio nal cells to ev en dimensional cells. Complete matchings are sp ecial Euler chains that do not pas s through a cell mor e than one time. 4.2.2. Homolo gous ve ctor fields. By a vector field on ( M , ∂ 0 M ) we mean (except in clearly mentioned ca se) a non-singula r co ntin uo us vector field of tangent v ec to rs on M directed into M on ∂ 0 M and dire c ted outw ards on ∂ 1 M . Since χ ( M , ∂ 0 M ) = 0, there exists such vector fields on ( M , ∂ 0 M ). V ector fields u and v on ( M , ∂ 0 M ) are called homolog ous if for so me clo sed ball B ⊂ I nt ( M ) the restriction o f the fields u and v ar e homotopic in the class of non-singular vector fields on M − I nt ( B ) directed into M o n ∂ 0 M , o utw a rds o n ∂ 1 M , a nd a rbitrarily on ∂ B . Denote by v ect ( M , ∂ 0 M ) the set of homologo us vector fields on ( M , ∂ 0 M ) a nd the class of a vector field u is denoted by [ u ]. 4.2.3. The c anonic al bije ction. T uraev proved the following: 18 ´ ETIENNE GALLAIS Theorem 4 .3 (T uraev [2 0]) . L et ( M , ∂ 0 M ) b e a smo oth p air such that dim ( M ) ≥ 2 . F or e ach C 1 -triangulation ( T , T 0 ) of t he p air ( M , ∂ 0 M ) ther e exists a bije ction ρ : E ul ( T , T 0 ) → v ect ( M , ∂ 0 M ) R emark. In fact, this bijection is a n H 1 ( T )-isomor phism, but we’ll make no use of it. Let us describ e the constructio n of T uraev in the case ∂ M = ∅ . Let T b e a C 1 -triangulatio n of M and T ′ be the fir st barycentric sub division o f T . W e recall the definition of the vector field F 1 with singularities on M . F or a simplex a o f the triangulation T , let a denotes its bar ycenter. If A = < a 0 , a 1 , . . . , a p > is a simplex of the tr ia ngulation T ′ , wher e a 0 < a 1 < . . . < a p are simplexes of T , then, at a po int x ∈ I nt ( A ), F 1 ( x ) = X 0 ≤ i

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