aa r X i v : . [ m a t h . G T ] A ug MATCHING CELLS
Ga¨el Meigniez
Abstract.
A (complete) matching of the cells of a triangulatedmanifold can be thought as a combinatorial or discrete version of anonsingular vector field. We give several methods for constructingsuch matchings.
M.S.C. 2010: 05C70, 05E45, 37C10, 37F20, 57Q15On a triangulated manifold (all triangulations being understood smo-oth), a “complete matching” (for short we just say “matching”) is apartition of the set of the cells into pairs such that in each pair, one ofthe two cells is a hyperface of the other. Such objects are regarded asa combinatorial equivalent to nonsingular vector fields — a viewpointinspired by Forman’s works [4], see also [5]. The present note intendsto provide some methods for the construction of matchings, either al-lowing oneself to subdivide the triangulation, or not. We feel that themethods are more important than the existence results themselves. Afirst approach is algebraic, playing with Hall’s “marriage theorem” andcellular homology; a second one is geometric: a matching is deducedfrom an ambiant nonsingular vector field transverse to the cells, or froma round handle decomposition of the manifold.In a first time, manifolds are not mandatory, nor simplices. Considergenerally a polyhedral cellular complex X (the cells are convex poly-hedra, finiteness is understood everywhere) and a subcomplex Y ⊂ X .Write Σ( X, Y ) the set of the cells of X not lying in Y . Call two cells incident to each other if one is a hyperface of the other. definition . A matching on X relative to Y , or a matching on thepair ( X, Y ) , is a partition of Σ( X, Y ) into incident pairs.As usual, for Y = ∅ we write Σ( X ) instead of Σ( X, ∅ ) and we speakof “a matching on X ”.The cases of the complexes of dimension 1 and of the triangulationsof surfaces will easily follow from a few general remarks. remark . If ( X, Y ) is matchable, then the rel-ative Euler characteristic χ ( X, Y ) = χ ( X ) − χ ( Y ) vanishes. remark . Every collapse of a polyhedral complex X ontoa subcomplex Y gives a matching on X rel. Y . Indeed, a collapse is nothing but filtration of X by subcomplexes X n , 0 ≤ n ≤ N , such that X = Y and X N = X and Σ( X n , X n − )consists, for each 1 ≤ n ≤ N , of exactly two incident cells. (A collapseof X onto Y amounts more precisely to a matching of X rel. Y withoutcyclic orbit, where an orbit is defined as a finite sequence σ , σ , · · · ∈ Σ( X, Y ), alternately of dimensions d and d + 1 for some d ≥
0, suchthat σ k − and σ k are incident for every k ≥
1, and mates for k odd,and not mates for k even.) remark . Every cellulation of the circle ad-mits two matchings.More generally, let X be a polyhedral cellulation of a manifold; let ℓ be a simple loop in the -skeleton of the dual cellulation; let Y ⊂ X be the union of the cells of X disjoint from ℓ . Then, X admits twomatchings rel. Y . example . Every connected graph whose Euler character-istic vanishes is matchable.
Indeed, such a graph collapses onto a circle. example . Let M be a compact, connected -manifold suchthat χ ( M ) = 0 . Then, every polyhedral cellulation X of M is matchableabsolutely, and relatively to ∂M .Proof. First case: M is the annulus or the M¨obius strip. Then, the 1-skeleton of the cellulation dual to X contains an essential simple loop ℓ such that M \ ℓ collapses onto ∂M . Consider the union Y ⊂ X ofthe cells of X disjoint from ℓ . The pair ( X, Y ) is matchable (Remark4), the pair (
Y, ∂M ) is matchable (Remark 3), and X | ∂M is matchable(Remark 4).Second case: M is the 2-torus or the Klein bottle. Then, the 1-skeleton of the cellulation dual to X contains an essential simple loop ℓ such that M \ ℓ is an annulus. Consider the union Y ⊂ X of the cellsof X disjoint from ℓ . The pair ( X, Y ) is matchable (Remark 4) andthe annulus Y is matchable. (cid:3) Next, recall Hall’s so-called “marriage theorem”. Let Σ := Σ ⊔ Σ be a finite, Z / Z -graded set and let I be a symmetric relation in Σ, ofdegree 1. For every subset A ⊂ Σ, denote by | A | its cardinality, anddenote by I ( A ) ⊂ Σ the subset of the elements I -related to at leastone element of A . A matching on Σ w.r.t. I is a partition of Σ into I -related pairs. ATCHING CELLS 3 theorem . The following properties are equivalent:(1) The relation I is matchable;(2) One has | A | ≤ | I ( A ) | for every A ⊂ Σ ;(3) | Σ | = | Σ | and one has | A | ≤ | I ( A ) | for every A ⊂ Σ . Also recall that the Ford-Fulkerson algorithm [3][2] computes a match-ing, if any, in time O ( | Σ | | I | ), thus giving some (moderate) effectivenessto the existence results below.Coming back to a pair of polyhedral complexes ( X, Y ), we writeΣ ( X, Y ) (resp. Σ ( X, Y )) for the set of the cells of X of even (resp.odd) dimension not lying in Y . Some counterexamples of unmatchablecomplexes will follow from the trivial sense of Hall’s criterium. example . A connected simplicial -complex whose Euler character-istic vanishes, unmatchable as well as its subdivisions. Let S (resp. T ) be a triangulated 2-sphere (resp. circle); let X := S ∗ T ∗ T be the bouquet, at some common vertex v , of S with two T ’s. Then, χ ( X ) = 0, but X does not admit any matching. Indeed,for A := Σ ( S, v ), one has I ( A ) = Σ ( S ), thus | I ( A ) | = | A | −
1. Thesame holds for any subdivision of X . example . An unmatchable triangulated closed -manifold whose Eu-ler characteristic vanishes. Let n be even and at least 4. Let M be a closed ( n − n -manifold. Let X be a triangu-lation of M . It is easy to make two compact n -manifolds M i , i = 1 , M , and whose Euler characteristics verify : χ ( M ) = − χ ( M ) > | Σ ( X ) | Extend X to some triangulation X of M and to some triangulation X of M . Let M (resp. X ) be the union of M (resp. X ) with M (resp. X ) over M (resp. X ). Then χ ( M ) = 0, but its triangulation X does not admit any matching. Indeed, the set A := Σ ( X , X ) has I ( A ) = Σ ( X ); hence: | I ( A ) | = | Σ ( X ) | − χ ( X ) = | A | + | Σ ( X ) | − χ ( X ) < | A | On the other hand, in dimension 3, E. Gallais has proved that everyclosed 3-manifold admits a matchable triangulation [5]. question . Is every triangulation of every closed -manifold match-able? Is every triangulation of every closed odd-dimensional manifoldmatchable? lemma
11 (Acyclic pair) . If H ∗ ( X, Y ) = 0 , then the pair ( X, Y ) ismatchable. MATCHING CELLS
Rational coefficients are understood everywhere; one could as welluse Z / Z . Proof.
This is an application of Hall’s criterium. For n ≥
0, consideras usual the set Σ n of the n -dimensional cells of X not lying in Y ; thechain vector space C n of basis Σ n ; the differential ∂ n : C n → C n − ; andits kernel Z n . Consider the union X n of Y with the ( n − X and with some n -cells which span a linear subspace complementaryto Z n in C n . The sequence ( X n ) is a filtration of the pair ( X, Y ) bysubcomplexes, and H ∗ ( X n , X n − ) = 0.One is thus reduced to the case where moreover, Σ( X, Y ) = Σ n ∪ Σ n − for some n ≥
1. Note that necessarily, | Σ n | = | Σ n − | . For every A ⊂ Σ( X, Y ), let h A i ⊂ C ∗ denote the spanned linear subspace; recallthat I ( A ) ⊂ Σ( X, Y ) is the set of the cells incident to at least one cellbelonging to A ; hence ∂ h A i ⊂ h I ( A ) i . If A ⊂ Σ n , since ∂ n is linear andone-to-one: | A | = dim( h A i ) = dim( ∂ h A i ) ≤ | I ( A ) | By the equivalence of (1) with (3) in the marriage theorem, the pair(
X, Y ) is matchable. (cid:3) corollary
12 (Subdivision) . Let ( X, Y ) be a pair of polyhedral com-plexes. Assume that ( X, Y ) is matchable.Then, every polyhedral subdivision ( X ′ , Y ′ ) of ( X, Y ) is also match-able.Proof. Consider a matching on (
X, Y ). For each matched pair σ, τ ∈ Σ( X, Y ) with τ ⊂ σ , consider the union ˆ ∂σ := ∂σ \ Int ( τ ) of theother hyperfaces of σ . The restriction ( X ′ | σ, X ′ | ˆ ∂σ ) is a pair of poly-hedral complexes, matchable by Lemma 11. Clearly, the collection ofall these partial matchings constitutes a global matching for the pairof complexes ( X ′ , Y ′ ). (cid:3) corollary
13 (Rational homology sphere) . Let M be a rational ho-mology sphere of odd dimension n .Then, every polyhedral cellulation X of M is matchable.Proof. One can assume that n ≥
3. Fix a ( n − σ of X and ahyperface τ ⊂ σ . Consider in X the union Y of τ with the cells of X not containing τ . First, the pair ( X, Y ) is matchable (Remark 4).Second, H ∗ ( Y, ∂σ ) = 0, hence the pair (
Y, ∂σ ) is matchable (Lemma11). Third, the polyhedral complex ∂σ , being homeomorphic to the( n − n . (cid:3) corollary
14 (Betti number 1) . Let M be a closed -manifold whosefirst Betti number is . ATCHING CELLS 5
Then, every polyhedral cellulation X of M is matchable.Proof. The 1-skeleton of X and the 1-skeleton of the dual cellulationcontain respectively two homologous essential simple loops ℓ , ℓ ∗ . Con-sider the union Y ⊂ X of the cells of X disjoint from ℓ ∗ . First, thepair ( X, Y ) is matchable (Remark 4). Second, H ∗ ( Y, ℓ ) = 0, hence thepair (
Y, ℓ ) is matchable (Lemma 11). Third, the circle ℓ is matchable(Remark 4). (cid:3) Now, consider a triangulation X of a compact manifold M of dimen-sion n ≥ ∂M (maybe empty). If a nonsingularvector field ∇ is transverse to every ( n − X , we say forshort that ∇ is transverse to X . Note that in particular, ∇ is thentransverse to ∂M ; thus ∂M splits as the disjoint union of ∂ s ( M, ∇ ),where ∇ enters M , with ∂ u ( M, ∇ ), where ∇ exits M . theorem
15 (Transverse nonsingular vector field) . If the nonsingularvector field ∇ is transverse to the triangulation X , then X is matchablerel. ∂ u ( M, ∇ ) .Proof. Because of the transversality, for every simplex σ ∈ Σ( X ) ofdimension less than n and not contained in ∂ u ( M, ∇ ) (resp. ∂ s ( M, ∇ )),there is a unique downstream (resp. upstream ) n -simplex d ( σ ) (resp. u ( σ )) ∈ Σ n ( X ) containing σ and such that the vector field ∇ enters d ( σ ) (resp. exits u ( σ )) at every point of Int ( σ ). For σ ∈ Σ n ( X ), weput d ( σ ) := u ( σ ) := σ .Consider any n -simplex δ ∈ Σ n ( X ) and any face σ ⊂ δ (the case σ = δ is included.) We call σ stable (resp. unstable ) with respect to δ if σ ∂ u ( M, ∇ ) (resp. σ ∂ s ( M, ∇ )) and if d ( σ ) = δ (resp. u ( σ ) = δ ).Note that • Every hyperface of δ is either stable or unstable; • δ has at least one stable hyperface (for degree reasons); • σ is stable if and only if every hyperface of δ containing σ isstable.Next, for each δ ∈ Σ n ( X ) we pick arbitrarily a base vertex v ( δ ) inthe intersection ∂ − δ of the unstable hyperfaces of δ (here of course, it ismandatory that δ is a simplex rather than a general convex polytope.)To this choice there corresponds canonically a matching, as follows.For every simplex σ ∈ Σ( X, ∂ u ( M, ∇ )) we define its mate ¯ σ by:(1) If v ( d ( σ )) ∈ σ then ¯ σ is the hyperface of σ opposed to v ( d ( σ ));(2) If v ( d ( σ )) / ∈ σ then ¯ σ is the join of σ with v ( d ( σ )).These rules do define a matching on the pair ( X, ∂ u ( M, ∇ )): the pointhere is that ¯ σ is also a stable face of d ( σ ). Indeed, if not, then ¯ σ would MATCHING CELLS be contained in some unstable hyperface η of d ( σ ); but in both cases(1) and (2) above, this would imply that σ itself would be containedin η , a contradiction. In other words, d (¯ σ ) = d ( σ ): the map σ ¯ σ induces locally, for each n -simplex δ , an involution in the set of thestable faces of δ ; and thus globally a matching on Σ( X, ∂ u ( M, ∇ )).Note — It can be suggestive, for n = 2 and n = 3 and for each0 ≤ i ≤ n −
1, to figure out in R n , endowed with the parallel vectorfield ∇ := − ∂/∂x n , a linear n -simplex δ in general position with respectto ∇ and such that dim( ∂ − δ ) = i ; to list the stable faces and theunstable faces; to choose a base vertex v ∈ ∂ − δ ; and to compute thecorresponding matching between the stable faces. (cid:3) In particular, the Hall cardinality conditions also constitute some combinatorial necessary conditions for a triangulation to admit a trans-verse nonsingular vector field. For example, in Example 9, not only X does not admit any transverse nonsingular vector field (which is ob-vious since such a field would be transverse to M , in contradictionwith χ ( M , M ) = 0), but this holds also for every triangulation of M combinatorially isomorphic with X (e.g. every jiggling of X ). corollary
16 (Matchable subdivision) . Let M be a compact con-nected manifold with smooth boundary, and let ∂ M be a union of con-nected components of ∂M such that χ ( M, ∂ M ) = 0 .Then, every triangulation X of M admits a subdivision matchablerel. ∂ M . Either of the two sets ∂ M and ∂ M := ∂M \ ∂ M may be empty,or both. Proof.
Since χ ( M, ∂ M ) = 0, there is on M a nonsingular vector field ∇ transverse to ∂M , which exits M through ∂ M , and which enters M through ∂ M . Then, by W. Thurston’s famous Jiggling lemma [7],one has on M a triangulation X ′ which is combinatorially isomorphicto some iterated crystalline subdivision of X , and which is transverseto ∇ , hence matchable by Theorem 15. (cid:3) Finally, we give an alternative construction for Corollary 16 whichworks in every dimension, but 3. Note that, by Corollary 12 and theHauptvermutung for smooth triangulations, it is enough to construct one triangulation of M matchable relatively to ∂ M .Since n ≥ χ ( M, ∂ M ) = 0, the pair ( M, ∂ M ) admits a roundhandle decomposition [1]. One chooses a triangulation of M for whicheach handle is a subcomplex. Hence, one is reduced to the case of around handle M = S × D i × D n − i − ATCHING CELLS 7 ∂ M = S × S i − × D n − i − for some 0 ≤ i ≤ n − S − = ∅ ). Since D i × D n − i − retracts by deformation onto the union of D i × S i − × D n − i − ,by Lemma 11 the proof is reduced to the case where M = S × D i and ∂ M = S × S i − . In that case, let X be any triangulation of M . The1-skeleton of the dual subdivision contains a simple loop ℓ homologousto the core S ×
0. Consider the union Y ⊂ X of the cells of X disjointfrom ℓ . On the one hand, the pair ( X, Y ) is matchable (Remark 4). Onthe other hand, since H ∗ ( Y, ∂ M ) = 0, the pair ( Y, ∂ M ) is matchable(Lemma 11). References [1] D. Asimov,
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