We extend the combinatorial Morse complex construction to the arbitrary free
chain complexes, and give a short, self-contained, and elementary proof of the
quasi-isomorphism between the original chain complex and its Morse complex.
  Even stronger, the main result states that, if $C_*$ is a free chain complex,
and $\cm$ an acyclic matching, then $C_*=C_*^\cm\oplus T_*$, where $C_*^\cm$ is
the Morse complex generated by the critical elements, and $T_*$ is an acyclic
complex

Kozlov, Dmitry N.

English

arXiv

Dmitry N. Kozlov

Comptes Rendus Mathématique

1/ntained,nger, theoireme entre leréen com-discrèteC. R. Acad. Sci. Paris, Ser. I 340 (2005) 867–872http://france.elsevier.com/direct/CRASSHomological AlgebraDiscrete Morse theory for free chain complexesDmitry N. Kozlov1Eidgenössische Technische Hochschule, Zürich, SwitzerlandReceived 24 January 2005; accepted after revision 13 April 2005Available online 17 June 2005Presented by Christophe SouléAbstractWe extend the combinatorial Morse complex construction to arbitrary free chain complexes, and give a short, self-coand elementary proof of the quasi-isomorphism between the original chain complex and its Morse complex. Even stromain result states that, ifC∗ is a free chain complex, andM an acyclic matching, thenC∗ = CM∗ ⊕ T∗, whereCM∗ is theMorse complex generated by the critical elements, andT∗ is an acyclic complex.To cite this article: D.N. Kozlov, C. R. Acad.Sci. Paris, Ser. I 340 (2005). 2005 Académie des sciences. Published by Elsevier SAS. All rights reserved.RésuméThéorie de Morse pour des complexes de chaînes libres.On étend la construction du complexe de Morse combinataux complexes de chaînes libres généraux, et on donne une démonstration brève et élémentaire du quasi-isomorphiscomplexe de chaînes original et son complexe de Morse. Plus précisément, le résultat principal dit que, siC∗ est un complexede chaînes libres etM est une correspondance acyclique, alorsC∗ = CM∗ ⊕ T∗, oùCM∗ est le complexe de Morse engendpar les éléments critiques etT∗ est un complexe acyclique.Pour citer cet article : D.N. Kozlov, C. R. Acad. Sci. Paris, Ser. I340 (2005). 2005 Académie des sciences. Published by Elsevier SAS. All rights reserved.Version française abrégéeLa théorie de Morse discrète a été introduite par Forman [1], elle s’est révélée utile pour des calculsbinatoire topologique. Il a été demontré dans [1, Theorem 8.2] que, étant donné une fonction de MorseE-mail address: dkozlov@inf.ethz.ch (D.N. Kozlov).URL: http://www.ti.inf.ethz.ch/people/kozlov.html (D.N. Kozlov).1 Research supported by Swiss National Science Foundation Grant PP002-102738/1.1631-073X/$ – see front matter 2005 Académie des sciences. Published by Elsevier SAS. All rights reserved.doi:10.1016/j.crma.2005.04.036868 D.N. Kozlov / C. R. Acad. Sci. Paris, Ser. I 340 (2005) 867–872eésente unenouvelleomme un-nase.e dee.seutationsefini-edendent,rivationgiven in[1, Definition 2.1] sur un complexe CW finiK , le complexe de chaînes cellulaireC∗(K;Z) est quasi-isomorphau complexe de Morse combinatoire associé.Dans cette Note, on étend cette construction au cas des complexes de chaînes libres généraux. On prdémonstration indépendante et simple dans cette généralité, en particulier on arrive à une démonstrationet élémentaire des résultats de Forman. À un niveau plus élévé, on peut regarder notre démonstration canalogue algébrique des arguments présentés dans [2, Theorem 3.2].Soit R un anneau commutatif général avec un élément neutre. On dit qu’un complexe de chaînesC∗ dR-modules· · · ∂n+2−→ Cn+1 ∂n+1−→ Cn ∂n−→ Cn−1 ∂n−1−→ · · · est libre si chaqueCn est unR-module libre finiment engendré. Si les indices sont clairs, on écrit∂ au lieux de∂n. On demande queC∗ soit borné à droite.Supposons qu’on ait choisit une base (un ensemble de générateurs libres)Ωn pour chaqueCn. Dans ce cas, odit qu’on a choisi une baseΩ = ⋃n Ωn pourC∗, et on écrit(C∗,Ω) pour un complexe de chaînes avec une bUn complexe de chaînes libre avec une base est l’objet principal de cette Note.Définition 0.1. Soit (C∗,Ω) un complexe de chaînes libre avec une base.(1) Unecorrespondance partielle M ⊆ Ω × Ω sur(C∗,Ω) est une correspondance partielle sur le diagrammHasse deP(C∗,Ω), tel que sib  a et b et a sont en correspondance, i.e., si(a, b) ∈ M , alorsw(b  a) estinversible.(2) Une correspondance partielle sur(C∗,Ω) estacyclique, s’il n’y a pas de cycled(b1) ≺ b1  d(b2) ≺ b2  d(b3) ≺ · · ·  d(bn) ≺ bn  d(b1), (1)avecn  2 et tous lesbi ∈ U(Ω) différents.Définition 0.2.Soit (C∗,Ω) un complexe de chaînes libre avec une base et soitM une correspondance acycliquLe complexe de Morse · · · ∂Mn+2−→ CMn+1∂Mn+1−→ CMn∂Mn−→ CMn−1∂Mn−1−→ · · · est défini comme suit. LeR-moduleCMn estlibrement engendré par les elements deCn(Ω). L’opérateur de borne est défini par∂Mn (s) =∑p w(p) · p• pours ∈ Cn(Ω), où la somme est prise sur tous les chemins alternésp qui satisfontp• = s.Le complexe de chaînes· · · −→ 0 −→ R id−→ R −→ 0 −→ · · · dans lequels les seuls modules non-triviauxtrouvent en dimensiond etd − 1, est appelé uncomplexe de chaînes atomique, qu’on écritAtom(d).Le résultat principal de cette Note est comme suit :Théorème 0.3.Supposons qu’on ait un complexe de chaînes libre avec une base (C∗,Ω) et une correspon-dance acyclique M. Alors, C∗ se décompose en une somme directe de complexes de chaînes CM∗ ⊕ T∗ etT∗  ⊕(a,b)∈M Atom(dimb).1. Acyclic matchings on chain complexes and the Morse complexDiscrete Morse theory was introduced by Forman, see [1], and it proved to be useful in various compin topological combinatorics. It was shown, [1, Theorem 8.2], that, given a discrete Morse function, [1, Dtion 2.1], on a finite CW complexK , the cellular chain complexC∗(K;Z) is quasi-isomorphic to the associatcombinatorial Morse complex.In this Note, we extend this construction to the case of arbitrary free chain complexes. We give an indepsimple, and self-contained proof in this generality, in particular furnishing a new elementary and short deof Forman’s result. On a higher level, our proof can be viewed as an algebraic analog of the argument[2, Theorem 3.2].D.N. Kozlov / C. R. Acad. Sci. Paris, Ser. I 340 (2005) 867–872 869eaint,ttion,t sufficesfLet R be an arbitrary commutative ring with a unit. We say that a chain complexC∗ consisting ofR-modules· · · ∂n+2−→ Cn+1 ∂n+1−→ Cn ∂n−→ Cn−1 ∂n−1−→ · · · , is free if each Cn is a finitely generated freeR-module. When theindexing is clear, we simply write∂ instead of∂n. We requireC∗ to be bounded on the right.Assume that we have chosen a basis (i.e., a set of free generators)Ωn for eachCn. In this case we say that whave chosen a basisΩ = ⋃n Ωn for C∗, and we write(C∗,Ω) to denote a chain complex with a basis. A free chcomplex with a basis is the main character of this Note.Given a free chain complex with a basis(C∗,Ω), and two elementsα ∈ Cn andb ∈ Ωn, we denote the coefficienof b in the representation ofα as a linear combination of the elements ofΩn by KΩ(α,b), or, if the basis is clearsimply byK(α, b). Forx ∈ Cn we write dimx = n. By convention, we setKΩ(α,b) = 0 if the dimensions do nomatch, i.e., if dimα 	= dimb.Note that a free chain complex with a basis(C∗,Ω) can be represented as a ranked posetP(C∗,Ω), withR-weights on the order relations. The elements of rankn correspond to the elements ofΩn, and the weight ofthe covering relationb  a, for b ∈ Ωn,a ∈ Ωn−1, is simply defined bywΩ(b  a) := KΩ(∂b, a). In other words,∂b = ∑ba wΩ(b  a)a, for eachb ∈ Ωn. Again, if the basis is clear, we simply writew(b  a).Definition 1.1.Let (C∗,Ω) be a free chain complex with a basis. Apartial matching M ⊆ Ω ×Ω on (C∗,Ω) is apartial matching on the covering graph ofP(C∗,Ω), such that ifb  a, andb anda are matched, i.e., if(a, b) ∈ M ,thenw(b  a) is invertible.Remark 1. Note that the Definition 1.1 is different from [1, Definition 2.1]. The latter is a topological definiand has the condition that the matched cells form a regular pair (in the CW sense). In our algebraic setting ito require the invertibility of the covering weight.Given such a partial matchingM, we writeb = u(a), anda = d(b), if (a, b) ∈ M. We denote byUn(Ω) theset of allb ∈ Ωn, such thatb is matched with somea ∈ Ωn−1, and, analogously, we denote byDn(Ω) the set ofall a ∈ Ωn, which are matched with someb ∈ Ωn+1. We setCn(Ω) := Ωn \ {Un(Ω) ∪ Dn(Ω)} to be the set oall unmatched basis elements; these elements are calledcritical. Finally, we setU(Ω) := ⋃nUn(Ω), D(Ω) :=⋃n Dn(Ω), andC(Ω) :=⋃n Cn(Ω).Given two basis elementss ∈ Ωn andt ∈ Ωn−1, analternating path is a sequencep = (s  d(b1) ≺ b1  d(b2) ≺ b2  · · ·  d(bn) ≺ bn  t), (2)wheren  0, and allbi ∈ U(Ω) are distinct. We use the notationsp• = s andp• = t . Theweight of such an alter-nating path is defined to be the quotientw(p) := (−1)n w(s  d(b1)) · w(b1  d(b2)) · · · · · w(bn  t)w(b1  d(b1)) · w(b2  d(b2)) · · · · · w(bn  d(bn)) .Definition 1.2.A partial matching on(C∗,Ω) is calledacyclic, if there does not exist a cycled(b1) ≺ b1  d(b2) ≺ b2  d(b3) ≺ · · ·  d(bn) ≺ bn  d(b1), (3)with n  2, and allbi ∈ U(Ω) being distinct.There is a nice alternative way to reformulate the notion of acyclic matching.Proposition 1.3.A partial matching on (C∗,Ω) is acyclic if and only if there exists a linear extension of P(C∗,Ω),such that in this extension u(a) follows directly after a, for all a ∈D(Ω).This extension can always be chosen so that, restricted to D(Ω) ∪ C(Ω), it does not decrease the rank.870 D.N. Kozlov / C. R. Acad. Sci. Paris, Ser. I 340 (2005) 867–872rdiction.reedlnt fromn-entrticulareralProof. If such an extensionL exists, then following a cycle (3) from left to right we always go down in the ordeL(more precisely, moving one position up is followed by moving at least two positions down), hence a contraAssume that the matching is acyclic, and defineL inductively. LetQ denote the set of elements which aalready ordered inL. We start withQ = ∅. Let W denote the set of the lowest rank elements inP(C∗,Ω) \ Q. Ateach step we have one of the following cases:Case 1. One of the elementsc in W is critical. Then simply addc to the orderL as the largest element, and procewith Q ∪ {c}.Case 2. All elements inW are matched. The covering graph induced byW ∪ u(W) is acyclic, hence the totanumber of edges is at most 2|W | − 1. It follows that there existsa ∈ W , such thatP(C∗,Ω)<u(a) \ Q = {a}.Hence, we can add elementsa andu(a) on top ofL and proceed withQ ∪ {a,u(a)}. Definition 1.4. Let (C∗,Ω) be a free chain complex with a basis, and letM be an acyclic matching. TheMorsecomplex · · · ∂Mn+2−→ CMn+1∂Mn+1−→ CMn∂Mn−→ CMn−1∂Mn−1−→ · · · is defined as follows. TheR-moduleCMn is freely generatedby the elements ofCn(Ω). The boundary operator is defined by∂Mn (s) =∑p w(p) · p•, for s ∈ Cn(Ω), where thesum is taken over all alternating pathsp satisfyingp• = s.Again, if the indexing is clear, we simply write∂M instead of∂Mn .Given a free chain complex with a basis(C∗,Ω), we can choose a different basis̃Ω by replacing eacha ∈Dn(Ω) by ã = w(u(a)  a) · a. SinceKΩ̃ (x, ã) = KΩ(x, a)/w(u(a)  a), (4)for anyx ∈ Ωn, we see that the weights of those alternating paths, which do not begin or end with an elemeDn(Ω), remain unaltered, as the quotientw(x  z)/w(y  z) stays constant as long asx, y 	= a. In particular,the Morse complex will not change. On the other hand, by (4),wΩ̃(u(a)  a) = 1, for all a ∈ D(Ω̃), so the totalweight of the alternating path in (2) will simply becomewΩ̃(p) = (−1)nwΩ̃(s  d(b1)) · wΩ̃(b1  d(b2)) · · · · · wΩ̃(bn  t).Because of these observations, we may always replace any given basis ofC∗ with the basisΩ̃ satisfyingwΩ̃(u(a)  a) = 1, for all a ∈ D(Ω̃).2. The main theoremThe chain complex· · · −→ 0−→ R id−→ R−→ 0−→ · · · where the only nontrivial modules are in the dimesionsd andd − 1, is called anatom chain complex, and is denoted byAtom(d).The main result brings to light a certain structure inC∗. Namely, by choosing a different basis, we will represC∗ as a direct sum of two chain complexes, one of which is a direct sum of atom chain complexes, in paacyclic, and the other one is isomorphic toCM∗ . For convenience, the choice of basis will be performed in sevsteps, one step for each matched pair of the basis elements.Theorem 2.1.Assume that we have a free chain complex with a basis (C∗,Ω), and an acyclic matching M. ThenC∗ decomposes as a direct sum of chain complexes CM∗ ⊕ T∗, where T∗ ⊕(a,b)∈M Atom(dimb).Proof. To start with, let us choose a linear extensionL of the partially ordered setP(C∗,Ω) satisfying the condi-tions of the Proposition 1.3, and let<L denote the corresponding total order.Assume first thatC∗ is bounded; without loss of generality, we can assume thatCi = 0 for i < 0, andi > N .Let m = |M| denote the size of the matching, and letl = |Ω| − 2m denote the number of critical cells.D.N. Kozlov / C. R. Acad. Sci. Paris, Ser. I 340 (2005) 867–872 871illfpact.totheftys togWe shall now inductively construct a sequence of basesΩ0,Ω1, . . . ,Ωm of C∗. Furthermore, each basis wbe divided into three parts:C(Ωk) = {ck1, . . . , ckl }, D(Ωk) = {ak1, . . . , akm}, andU(Ωk) = {bk1, . . . , bkm}, such thataki = d(bki ), for all i ∈ [m].We start withΩ0 = Ω and the initial conditionb0i <L b0i+1, for all i ∈ [m − 1]. Since the lower index oK−(−,−) andw−(−  −) will be clear from the arguments, we shall omit it to make the formulae more comWhen constructing the bases, we shall simultaneously prove by induction the following statements:(i) C∗ = C∗[k]⊕Ak1⊕· · ·⊕Akk , whereC∗[k] is the subcomplex ofC∗ generated byΩk \{ak1, . . . , akk , bk1, . . . , bkk},andAki is isomorphic toAtom(dimbki ), for i ∈ [k];(ii) for every xk ∈ U(Ωk) ∪ C(Ωk), y ∈ C(Ωk), we havew(xk  yk) = ∑p w(p), where the sum is restrictedthose alternating paths fromx0 to y0 which only use the pairs(a0i , b0i ), for i ∈ [k].Clearly, all of the statements are true fork = 0. Assumek  1.Transformation of the basis Ωk−1 into the basis Ωk : setakk := ∂bk−1k , bkk := bk−1k , andxk := xk−1 − w(xk−1 ak−1k ) · bk−1k , for all xk−1 ∈ Ωk−1, x 	= ak, bk .First, we see thatΩk is a basis. Indeed, assumebk−1k ∈ Cn. For i 	= n,n − 1, we haveΩki = Ωk−1i , hence, byinduction, it is a basis.Ωkn−1 is obtained fromΩk−1n−1 by adding a linear combination of other basis elements tobasis elementak−1k , henceΩkn−1 is again a basis. Finally,Ωkn is obtained fromΩk−1n by subtracting multiples othe basis elementbk−1k from the other basis elements, hence it is also a basis.Next, we investigate how the posetP(C∗,Ωk) differs from P(C∗,Ωk−1). If x 	= bk , we havew(xk  akk ) =K(∂xk, akk ) = K(∂xk, ak−1k ) = K(∂xk−1, ak−1k )−w(xk−1  ak−1k ) ·K(∂bk−1k , ak−1k ) = 0, where the second equalifollows from the fact thatΩkn−1 is obtained fromΩk−1n−1 by adding a linear combination of other basis elementthe basis elementak−1k , and the last equality follows fromK(∂bk−1k , ak−1k ) = 1.Furthermore, sinceΩkn is obtained fromΩk−1n by subtracting multiples of the basis elementbk−1k from the otherbasis elements, we see that forx ∈ Ωkn+1, y ∈ Ωkn , y 	= bk , we havew(xk  yk) = w(xk−1  yk−1). Additionally,since the differential of the chain complex squares to 0, we have 0= ∑zk∈Ωkn w(xk  zk) · w(zk  akk ) = w(xk bkk) · w(bkk  akk ) = w(xk  bkk), where the second equality follows fromw(zk  akk ) = 0, for z 	= bk .We can summarize our findings as follows: all weights in the posetP(C∗,Ωk) are the same as inP(C∗,Ωk−1),with the following exceptions:(1) w(xk  bkk) = 0, andw(bkk  xk) = 0, for x 	= ak ;(2) w(akk  xk) = 0, andw(xk  akk ) = 0, for x 	= bk ;(3) w(xk  yk) = w(xk−1  yk−1) − w(xk−1  ak−1k ) · w(bk−1k  yk−1), for x ∈ Ωkn , y ∈ Ωkn−1, x 	= bk , y 	= ak .In particular, the statement (i) is proved. Furthermore, the following fact(∗) can be seen by induction, usin(1), (2), and (3): ifw(xk  yk) 	= w(xk−1  yk−1), thenb0k L y0. Indeed, eithery ∈ {ak, bk}, or y is critical, ory = ak̃, for k̃ > k, such thatw(bk−1k  yk−1) 	= 0. In the first two casesb0k L y0 by the construction ofL, and thelast case is impossible by induction, and again, by the construction ofL.We havew(bkj  akj ) = w(bk−1j  ak−1j ), for all j, k. Indeed, this is clear forj = k. The casej < k follows byinduction, and the casej > k is a consequence of the fact(∗).Next, we see that the partial matchingMk := {(aki , bki ) | i ∈ [m]} is acyclic. Forj  k, the poset elementsbkj , akjare incomparable with the rest, hence they cannot be a part of a cycle. Fori > k, we havew(bkj  aki ) = w(bk−1j ak−1), by the fact(∗). Hence, by induction, no cycle can be formed by these elements either.i872 D.N. Kozlov / C. R. Acad. Sci. Paris, Ser. I 340 (2005) 867–872r.eth),sions, soclicexcerptFinally, we trace the boundary operator. Letxk ∈ U(Ωk)∪C(Ωk), y ∈ C(Ωk). Forx = bk the statement is cleaIf x 	= bk , we havew(xk  yk) = w(xk−1  yk−1)− w(xk−1  ak−1k )w(bk−1k  yk−1). By induction, the first termis counting the contribution of all the alternating paths fromx0 to y0 which do not use the edgesb0l  a0l , for l  k.The second term contains the additional contribution of the alternating paths fromx0 to y0 which use the edgb0k  a0k . Observe, that if this edge occurs then, by the construction ofL, it must be the second edge of the pa(counting fromx0), and, by the fact(∗), we havew(xk−1  ak−1k ) = w(x0  a0k ). This proves the statement (iiand therefore concludes the proof of the finite case.It is now easy to deal with the infinite case, since the basis stabilizes as we proceed through the dimenwe may take the union of the stable parts as the new basis forC∗. Remark 2. Even if the chain complexC∗ is infinite in both directions, one can still define the notion of the acymatching and of the Morse complex. Since each particular homology group is determined by a finitefrom C∗, we may still conclude thatH∗(C∗) = H∗(CM∗ ).References[1] R. Forman, Morse theory for cell complexes, Adv. Math. 134 (1) (1998) 90–145.[2] D.N. Kozlov, Collapsibility of(Πn)/Dn and some related CW complexes, Proc. Amer. Math. Soc. 128 (8) (2000) 2253–2259.

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Discrete Morse theory for free chain complexes

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Discrete Morse Theory for free chain complexes

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