Non-homogeneous combinatorial manifolds
NNON-HOMOGENEOUS COMBINATORIAL MANIFOLDS
NICOLAS ARIEL CAPITELLI AND ELIAS GABRIEL MINIAN
Abstract.
In this paper we extend the classical theory of combinatorial manifolds to thenon-homogeneous setting. NH -manifolds are polyhedra which are locally like Euclideanspaces of varying dimensions. We show that many of the properties of classical mani-folds remain valid in this wider context. NH -manifolds appear naturally when studyingPachner moves on (classical) manifolds. We introduce the notion of NH -factorizationand prove that P L -homeomorphic manifolds are related by a finite sequence of NH -factorizations involving NH -manifolds. Introduction
The notion of manifold (piecewise linear, topological, differentiable) is central in math-ematics. An n -manifold is an object which is locally like the Euclidean space R n . Con-cretely, in the piecewise linear setting a PL-manifold of dimension n is a polyhedron inwhich every point has a (closed) neighborhood which is a PL-ball of dimension n .The theory of combinatorial manifolds (which are the triangulations of PL-manifolds)has been widely developed during the last ninety years. J.W. Alexander’s Theorem onregular expansions, Newman’s result on the complement of an n -ball in an n -sphere, White-head’s Regular Neighborhood theory and the s-cobordism theorem are some of its mostimportant advances [4, 5, 7, 12]. More recently Pachner [11] studied a set of elementarycombinatorial operations or moves , and showed that any combinatorial manifold can betransformed into any other PL-homeomorphic one by using these moves (see also [7]).It is well known that any combinatorial n -manifold is a homogeneous (or pure) simplicialcomplex, which means that all the maximal simplices have the same dimension. It isnatural to ask whether it is possible to extend the theory of combinatorial manifoldsto the non-homogeneous context. More concretely, the main goal of this article is toinvestigate the properties of those polyhedra which are locally like Euclidean spaces ofvarying dimensions (see Figures 1 and 2 below). In this paper we introduce the theory of non-homogeneous manifolds or N H -manifolds, for short. We will show that many of thebasic properties of (classical) manifolds are also satisfied in this much wider setting.We investigate shellability in the non-homogeneous context. It is well-known that anyshellable complex is homotopy equivalent to a wedge of spheres and that the only shellablemanifolds are balls and spheres (see [2] and [6]). We prove that every shellable
N H -manifold is in particular an
N H -bouquet, which extends the classical result for manifolds.We also study the notion of regular expansion for
N H -manifolds and prove a generalizationof Alexander’s Theorem.
Mathematics Subject Classification.
Key words and phrases.
Simplicial complexes, combinatorial manifolds, collapses, shellability, Pachnermoves.The authors’ research is partially supported by Conicet. a r X i v : . [ m a t h . G T ] A ug N.A. CAPITELLI AND E.G. MINIAN
Non-homogeneous manifolds appear naturally when studying Pachner moves betweenmanifolds. We introduce the notion of
N H -factorization and prove that any two PL-homeomorphic manifolds (with or without boundary) are related by a finite sequence offactorizations involving
N H -manifolds. When the manifolds are closed, the converse alsoholds. 2.
Preliminaries
We start by fixing some notation and terminology. In this paper, all the simplicialcomplexes that we deal with are assumed to be finite. If a simplex σ is a face of a simplex τ , we will write σ < τ and when σ is an immediate face we write σ ≺ τ . A principal or maximal simplex in K is a simplex which is not a proper face of any other simplexof K and a ridge in K is an immediate face of a maximal simplex. A complex is saidto be homogeneous of dimension n if all of its principal simplices have dimension n . Theboundary ∂K of an n -homogeneous complex K is the subcomplex generated by the mod2 union of the ( n − K will be denoted by V K .The join of two simplices σ, τ with σ ∩ τ = ∅ will be denoted by σ ∗ τ . Also K ∗ L will denote the join of the complexes K and L . Given a simplex σ ∈ K , lk ( σ, K ) willdenote its link , which is the subcomplex lk ( σ, K ) = { τ ∈ K : τ ∩ σ = ∅ , τ ∗ σ ∈ K } ,and st ( σ, K ) = σ ∗ lk ( σ, K ) will denote the (closed) star of σ in K . The union of twocomplexes K, L will be denoted by K + L .Following [4], arbitrary subdivisions of K will be denoted by αK, βK, . . . Derivedsubdivisions will be denoted by δK and the barycentric subdivision by K (cid:48) , as usual. If σ ∈ K and a ∈ ◦ σ , the interior of σ , then ( σ, a ) K will denote the elementary subdivisionof K by starring σ in a ; i.e. the replacing of st ( σ, K ) by a ∗ ∂σ ∗ lk ( σ, K ). A stellarsubdivision sK of K is a finite sequence of elementary starrings. The operation inverseto an elementary starring is called an elementary weld and denoted by ( σ, a ) − K . Twocomplexes K and L are stellar equivalent if they are related by a sequence of starrings,welds and (simplicial) isomorphisms. In this case we write K ∼ L . It is well knownthat the combinatorial and the stellar theories are equivalent (see for example [4, 7]), andtherefore K ∼ L if and only if they are PL-homeomorphic. A class of complexes will becalled PL-closed if it is closed under PL-homeomorphisms.We recall now the basic definitions and properties of combinatorial manifolds. For acomprehensive exposition of the theory of combinatorial manifolds we refer the reader to[4, 7, 12].∆ n will denote the n -simplex. A combinatorial n -ball is a complex which is PL-homeomorphic to ∆ n . A combinatorial n -sphere is a complex PL-homeomorphic to ∂ ∆ n +1 .By convention, ∅ = ∂ ∆ is considered a sphere of dimension −
1. A combinatorial n -manifold is a complex M such that for every v ∈ V M , lk ( v, M ) is a combinatorial ( n − n − n -manifolds are homogeneous complexes ofdimension n . It is well known that the link of any simplex in a manifold is also a ball or asphere and that the class of n -manifolds is PL-closed. It follows that combinatorial ballsand spheres are combinatorial manifolds. ON-HOMOGENEOUS COMBINATORIAL MANIFOLDS 3
The boundary ∂M can be regarded as the set of simplices whose links are combinatorialballs. By a classical result of Newman [10] (see also [4, 5, 7]), if S is a combinatorial n -sphere containing a combinatorial n -ball B , then the closure S − B is a combinatorial n -ball.Some global properties of combinatorial manifolds can be stated in terms of pseudomanifolds. An n -pseudo manifold is an n -homogeneous complex K satisfying the followingtwo properties: (a) for every ( n − σ , lk ( σ, K ) is a combinatorial 0-ball or 0-sphere (or equivalently, every ( n − n -simplices), and (b)given two n -simplices σ, σ (cid:48) , there exists a sequence of n -simplices σ = σ , . . . , σ k = σ (cid:48) such that σ i ∩ σ i +1 is ( n − i = 0 , . . . , k −
1. It is well known thatany connected combinatorial n -manifold, or more generally, any triangulated homologicalmanifold, is an n -pseudo manifold.A simplex τ of a complex K is said to be collapsible in K if it has a free face σ , i.e. aproper face which is not a face of any other simplex of K . Note that, in particular, τ isa maximal simplex and σ is a ridge. In this situation, the operation which transforms K into K − { τ, σ } is called an elementary (simplicial) collapse , and it is usually denoted by K (cid:38) e K − { τ, σ } . The inverse operation is called an elementary (simplicial) expansion . Ifthere is a sequence K (cid:38) e K (cid:38) e · · · (cid:38) e L we say that K collapses to L (or equivalently, L expands to K ) and write K (cid:38) L or L (cid:37) K respectively. A complex K is said to be collapsible if it has a subdivision which collapses to a single vertex. A celebrated theoremof J.H.C. Whitehead states that collapsible combinatorial n -manifolds are combinatorial n -balls [4, Corollary III.17].A more general type of collapse is the geometrical collapse . If K = K + B n , where B n is a combinatorial n -ball and B n ∩ K = B n − is a combinatorial ( n − B n , then the move K → K it called an elementary geometrical collapse .A finite sequence of elementary geometrical collapses (resp. expansions) is a a geometricalcollapse (resp. expansion).If M is an n -manifold, an elementary geometrical expansion M → N = M + B n suchthat M ∩ B n ⊂ ∂M is called an elementary regular expansion . By a Theorem of Alexander,an elementary regular expansion is a PL-equivalence (see [4, 7]). A sequence of elementaryregular expansions (resp. collapses) is a regular expansion (resp. collapse). Note that thedimension of all the balls being expanded in such a sequence must be n .If M is a combinatorial n -manifold with boundary and there is an n -simplex η = σ ∗ τ ∈ M with dim σ, dim τ ≥ σ ∈ ◦ M , the interior of M , and ∂σ ∗ τ ⊂ ∂M , then themove M sh −→ M = M − σ ∗ τ is called an elementary shelling . This operation producesagain a combinatorial n -manifold. The inverse operation is called an inverse shelling .Pachner showed in [11] that two combinatorial n -manifolds with non-empty boundaryare PL-homeomorphic if and only if one can obtain one from the other by a sequence ofelementary shellings, inverse shellings and isomorphisms.3. NH-manifolds A non-homogeneous manifold , or N H -manifold for short, is a simplicial complex whichlocally looks as in Figure 1. We will define such complexes by induction on the dimension.We need first a definition.
N.A. CAPITELLI AND E.G. MINIAN
Definition 3.1.
Let K be a complex. A subcomplex L ⊆ K is said to be top generated in K if it is generated by principal simplices of K , i.e. every maximal simplex of L is alsomaximal in K . Figure 1.
Local structure of NH -manifolds. Definition 3.2. An N H -manifold (resp.
N H -ball , N H -sphere ) of dimension 0 is a mani-fold (resp. ball, sphere) of dimension 0. An
N H -sphere of dimension − n ≥
1, we define by induction • An N H -manifold of dimension n is a complex M of dimension n such that lk ( v, M )is an N H -ball of dimension 0 ≤ k ≤ n − N H -sphere of dimension − ≤ k ≤ n − v ∈ V M . • An N H -ball of dimension n is a collapsible N H -manifold of dimension n . • An N H -sphere of dimension n and homotopy dimension k is an N H -manifold S of dimension n such that there exist a top generated N H -ball B of dimension n and a top generated combinatorial k -ball L such that B + L = S and B ∩ L = ∂L .We say that S = B + L is a decomposition of S .Note that the definition of N H -ball is motivated by Whitehead’s theorem on regularneighborhoods and the definition of
N H -sphere by that of Newman’s (see [4] and [7]).
Remark . An N H -ball of dimension 1 is the same as a 1-ball. An
N H -sphere ofdimension 1 is either a 1-sphere (if the homotopy dimension is 1) or the disjoint union ofa point and a combinatorial 1-ball (if the homotopy dimension is 0). In general, an
N H -sphere of homotopy dimension 0 consists of a disjoint union of a point and an
N H -ball.These are the only
N H -spheres which are not connected.
Examples 3.4.
Figure 2 shows some examples of
N H -manifolds.
Figure 2. NH -manifolds. Remark . Note that the decomposition of an
N H -sphere need not be unique. Howeverthe homotopy dimension of the
N H -sphere is well defined since the geometric realizationof an
N H -sphere of homotopy dimension k is a homotopy k -sphere. ON-HOMOGENEOUS COMBINATORIAL MANIFOLDS 5
We show now that the notion of
N H -manifold is in fact an extension of the concept ofcombinatorial manifold to the non-homogeneous context.
Theorem 3.6.
A complex K is a homogeneous N H -manifold (resp.
N H -ball,
N H -sphere) of dimension n if and only if it is a combinatorial n -manifold (resp. n -ball, n -sphere).Proof. Let n ≥
1. It is easy to see that the result holds for
N H -manifolds of dimension n provided that it holds for N H -balls and
N H -spheres of dimension less than n . Then itsuffices to prove that the result holds for N H -balls and
N H -spheres of dimension n if itholds for N H -manifolds of dimension n .For N H -balls the result is clear by the theorem of Whitehead [4, Corollary III.17].Suppose now that S = B + L is a homogeneous N H -sphere of dimension n . It follows that B and L are combinatorial n -balls. Take σ ∈ ∂L a maximal simplex. Since lk ( σ, S ) = { v } + lk ( σ, B ) for some vertex v ∈ L and S is an n -pseudo manifold, then lk ( σ, B ) isalso a single vertex. It follows that σ ∈ ∂B . Since both ∂L and ∂B are combinatorial( n − ∂L = ∂B . This proves that S is a combinatorial n -sphere. Conversely, any n -simplex of a combinatorial n -sphere can play the role of L in itsdecomposition as an N H -sphere. The result then follows from Newman’s Theorem. (cid:3)
Following the same reasoning of [4, Theorem II.2] for combinatorial manifolds, one canshow that the links of all simplices in an
N H -manifold behave nicely. Concretely:
Proposition 3.7.
Let M be an N H -manifold of dimension n and let σ ∈ M be a k -simplex. Then lk ( σ, M ) is an N H -ball or an
N H -sphere of dimension less than n − k . The property stated in the preceding proposition is often called regularity .In order to show that the class of
N H -manifolds is PL-closed, we will need the fol-lowing lemma, which is somehow an analogue of [4, Proposition II.1]. This result will begeneralized in Corollary 3.10 and in Theorem 3.13.
Lemma 3.8.
Let K be an N H -ball or an
N H -sphere and let σ be a simplex disjoint from K . Then, (1) σ ∗ K is an N H -ball. (2) ∂σ ∗ K is an N H -ball (if K is an N H -ball) or an
N H -sphere (if K is an N H -sphere).Proof.
For the first part of the lemma, we proceed by double induction. Suppose first thatdim σ = 0, i.e. σ is a vertex v , and that the result holds for N H -balls and
N H -spheres K of dimension less than n . Note that v ∗ K (cid:38)
0, so we only have to verify that v ∗ K is an N H -manifold. Take w ∈ V K . Since lk ( w, v ∗ K ) = v ∗ lk ( w, K ), by induction applied to lk ( w, K ), it follows that lk ( w, v ∗ K ) is an N H -ball. On the other hand, lk ( v, v ∗ K ) = K ,which is an N H -ball or an
N H -sphere by hypothesis. This shows that v ∗ K is an N H -manifold and proves the case dim σ = 0. Suppose now that dim σ = k ≥
1. Write σ = τ ∗ v for some v ∈ σ . Since σ ∗ K = τ ∗ ( v ∗ K ), the results follows by induction applied to v and τ .For the second part of the lemma, suppose that dim σ = k ≥ K be an N H -ballor an
N H -sphere of dimension n . It is easy to see that the result is valid if n = 0. Supposenow that n ≥ t < n . For any vertex v ∈ ∂σ ∗ K , we have lk ( v, ∂σ ∗ K ) = (cid:26) ∂σ ∗ lk ( v, K ) v / ∈ ∂σlk ( v, ∂σ ) ∗ K v ∈ ∂σ N.A. CAPITELLI AND E.G. MINIAN
In the first case, by induction on n , it follows that lk ( v, ∂σ ∗ K ) is an N H -ball or sphere.In the second case, we use induction on k (note that lk ( v, ∂σ ) = ∂lk ( v, σ )). This provesthat ∂σ ∗ K is an N H -manifold. Now, if K is an N H -ball then ∂σ ∗ K (cid:38) ∂σ ∗ K is again an N H -ball. If K is an N H -sphere write K = B + L with B an N H -ball, L acombinatorial ball and B ∩ L = ∂L . Since ∂ ( ∂σ ∗ L ) = ∂σ ∗ ∂L = ∂σ ∗ B ∩ ∂σ ∗ L , then ∂σ ∗ K = ∂σ ∗ B + ∂σ ∗ L is an N H -sphere by the previous case. This concludes theproof. (cid:3)
In particular, from Lemma 3.8 we deduce that M is an N H -manifold if and only if st ( v, M ) is an N H -ball for all v ∈ V M . Theorem 3.9.
The classes of
N H -manifolds,
N H -balls and
N H -spheres are PL-closed.Proof.
It suffices to prove that K is an N H -manifold (resp.
N H -ball,
N H -sphere) ifand only if any starring ( τ, a ) K is an N H -manifold (resp.
N H -ball,
N H -sphere). Wesuppose first that the result is valid for
N H -manifolds of dimension n and prove that itis valid for N H -balls and
N H -spheres of the same dimension. If ( τ, a ) K is an N H -ball ofdimension n then K is also an N H -ball since it is an
N H -manifold with α (( τ, a ) K ) (cid:38) α . On the other hand, if K is an N H -manifold of dimension n with αK (cid:38)
0, by [4, Theorem I.2] we can find a stellar subdivision δ and an arbitrarysubdivision β such that β (( τ, a ) K ) = δ ( αK ). Since stellar subdivisions preserve collapses,( τ, a ) K is collapsible and hence an N H -ball. Now, if K is an N H -sphere of dimension n with decomposition B + L then the result holds by the previous case and the followingidentities.( τ, a ) K = ( τ, a ) B + L , with ( τ, a ) B ∩ L = ∂L a ∈ B − LB + ( τ, a ) L , with B ∩ ( τ, a ) L = ∂L a ∈ L − B ( τ, a ) B + ( τ, a ) L , with ( τ, a ) B ∩ ( τ, a ) L = ( τ, a ) ∂L a ∈ B ∩ L = ∂L Note that ( τ, a ) ∂L = ∂ ( τ, a ) L . The converse follows by replacing ( τ, a ) with ( τ, a ) − .We assume now that the result is valid for N H -balls and
N H -spheres of dimension n and prove that it is valid for N H -manifolds of dimension n + 1. Suppose K is an N H -manifold of dimension n + 1 and take v ∈ ( τ, a ) K . If v (cid:54) = a then lk ( v, ( τ, a ) K ) isPL-homeomorphic to an elementary starring of lk ( v, K ) . The inductive hypothesis on lk ( v, K ) shows that lk ( v, ( τ, a ) K ) is also an N H -ball or
N H -sphere. On the other hand, lk ( a, ( τ, a ) K ) = ∂τ ∗ lk ( τ, K ), which is an N H -ball or an
N H -sphere by Lemma 3.8. Onceagain, the converse follows by replacing ( τ, a ) with ( τ, a ) − . (cid:3) Corollary 3.10.
Let B be a combinatorial n -ball, S a combinatorial n -sphere and K an N H -ball or
N H -sphere. Then, (1) B ∗ K is an N H -ball. (2) S ∗ K is an N H -ball (if K is an N H -ball) or an
N H -sphere (if K is an N H -sphere).Proof.
Follows from Lemma 3.8 and Theorem 3.9. (cid:3)
Proposition 3.11.
Let K be an n -dimensional complex and let B be a combinatorial r -ball. Suppose K + B is an N H -manifold such that (1) K ∩ B ⊂ ∂B is homogeneous of dimension r − and ON-HOMOGENEOUS COMBINATORIAL MANIFOLDS 7 (2) lk ( σ, K ) is collapsible for all σ ∈ K ∩ B Then, K is an N H -manifold.Proof.
We show first that
K, B ⊂ K + B are top generated. Clearly, B is top generatedsince it intersects K in dimension r −
1. On the other hand, a principal simplex in K which is not principal in K + B must lie in K ∩ B . Then, by hypothesis, it has a collapsiblelink in K . But this contradicts the fact that it is principal in K . Therefore K, B ⊂ K + B are top generated and, in particular, r ≤ n .We prove the result by induction on r . For r = 0 the result is trivial. Let r ≥ v ∈ K . If v / ∈ B then lk ( v, K ) = lk ( v, K + B ), which is an N H -ball or
N H -sphere byhypothesis. Suppose now that v ∈ K ∩ B . If r = 1, then lk ( v, K + B ) = lk ( v, K ) + ∗ .It follows that lk ( v, K ) is an N H -ball. Suppose r ≥ n ≥ lk ( v, K ) , lk ( v, B ) also satisfies the conditions of the theorem. Note that lk ( v, K ) + lk ( v, B ) = lk ( v, K + B ) is an N H -manifold by hypothesis and lk ( v, K ) ∩ lk ( v, B ) = lk ( v, K ∩ B ) is homogeneous of dimension r −
2. On the other hand, if η ∈ lk ( v, K ) ∩ lk ( v, B ) then, v ∗ η ∈ K ∩ B , so lk ( η, lk ( v, K )) = lk ( v ∗ η, K ) is collapsible. Byinduction, it follows that lk ( v, K ) is an N H -manifold, and, since it is also collapsible, itis an
N H -ball. This shows that K is an N H -manifold. (cid:3)
Lemma 3.12.
Suppose S = G + L and S = G + L are two disjoint N H -spheres.Then, G ∗ S + L ∗ G is collapsible.Proof. Since G and G are collapsible, there exist subdivisions (cid:15) , (cid:15) such that (cid:15) G (cid:38) (cid:15) G (cid:38)
0. We can extend these subdivisions to S and S and then suppose withoutloss of generality that G (cid:38) G (cid:38)
0. Note that G ∗ S ∩ L ∗ G = ∂L ∗ G . We will show that some subdivision of L ∗ G collapses to (the induced subdivision of) ∂L ∗ G . Let α be an arbitrary subdivision of L and δ a derived subdivision of ∆ r suchthat αL = δ ∆ r . Then, α ( L ∗ G ) = δ (∆ r ∗ G ). Since G (cid:38)
0, then ∆ r ∗ G (cid:38) ∂ ∆ r ∗ G ([4, Corollary III.4]). Therefore α ( L ∗ G ) = δ (∆ r ∗ G ) (cid:38) δ ( ∂ ∆ r ∗ G ) = α ( ∂L ∗ G ) . We extend α to ( G ∗ S + L ∗ G ) and then α ( G ∗ S + L ∗ G ) = α ( G ∗ S ) + α ( L ∗ G ) (cid:38) α ( G ∗ S ) + α ( ∂L ∗ G ) = α ( G ∗ S ) . By [4, Theorem III.6] there is a stellar subdivision s such that sαG (cid:38) sα ( G ∗ S + L ∗ G ) (cid:38) sα ( G ∗ S ) = sαG ∗ sαS (cid:38) . (cid:3) Theorem 3.13.
Let B , B be N H -balls and S , S be N H -spheres. Then, (1) B ∗ B and B ∗ S are N H -balls. (2) S ∗ S is an N H -sphere.Proof.
Let K represent B or S and let K represent B or S . We must show that K ∗ K is an N H -ball or an
N H -sphere. We proceed by induction on s = dim K + dim K . If s = 0 , s ≥
2. We show first that K ∗ K is an N H -manifold. Let v ∈ K ∗ K be a vertex. Then, lk ( v, K ∗ K ) = (cid:26) lk ( v, K ) ∗ K v ∈ K K ∗ lk ( v, K ) v ∈ K Since dim lk ( v, K )+dim K = dim K +dim lk ( v, K ) = s −
1, then by induction, lk ( v, K ∗ K ) is an N H -ball or an
N H -sphere. It follows that K ∗ K is an N H -manifold. Now,if K = B or K = B , then K ∗ K (cid:38) K ∗ K is an N H -ball.We prove now that S ∗ S is an N H -sphere. Decompose S = G + L and S = G + L .Note that S ∗ S = ( G ∗ S + L ∗ G ) + L ∗ L and that ( G ∗ S + L ∗ G ) ∩ ( L ∗ L ) = ∂ ( L ∗ L ), then it suffices to show that ( G ∗ S + L ∗ G ) is an N H -ball. By Lemma3.12 it is collapsible, so we only need to check that ( G ∗ S + L ∗ G ) is an N H -manifold.In order to prove this, we apply Proposition 3.11 to the complex G ∗ S + L ∗ G andthe combinatorial ball L ∗ L . The only non-trivial fact is that lk ( σ, G ∗ S + L ∗ G )is collapsible for σ ∈ ∂ ( L ∗ L ). To see this, take η ∈ ∂ ( L ∗ L ) = ∂L ∗ L + L ∗ ∂L and write η = l ∗ l with l ∈ L , l ∈ L . Then, lk ( η, G ∗ S + L ∗ G ) = lk ( l , G ) ∗ lk ( l , S ) + lk ( l , L ) ∗ lk ( l , G ) . Now, if l ∈ L − ∂L then lk ( l ∗ l , G ∗ S ) = ∅ and lk ( η, G ∗ S + L ∗ G ) = lk ( l , L ) ∗ lk ( l , G ) (cid:38)
0. By a similar argument, the same holds if l ∈ L − ∂L . If l ∈ ∂L and l ∈ ∂L then lk ( l , S ) = lk ( l , G ) + lk ( l , L ) and lk ( l , S ) = lk ( l , G ) + lk ( l , L ) are N H -spheres (by Lemma 4.8). By Lemma 3.12, it follows that lk ( η, G ∗ S + L ∗ G ) is alsocollapsible. By Proposition 3.11, we conclude that G ∗ S + L ∗ G is an N H -manifold. (cid:3)
The following result will be used in the next section. First we need a definition.
Definition 3.14.
Two principal simplices σ, τ ∈ M are said to be adjacent if the inter-section τ ∩ σ is an immediate face of σ or τ . Lemma 3.15.
Let M be a connected N H -manifold. Then (1)
For each ridge σ ∈ M , lk ( σ, M ) is either a point or an N H -sphere of homotopydimension . (2) Given any two principal simplices σ, τ ∈ M , there exists a sequence σ = E , . . . , E s = τ of principal simplices of M such that E i is adjacent to E i +1 for every ≤ i ≤ s − . By analogy with the homogeneous case, a complex K satisfying properties (1) and (2)of this lemma will be called an N H -pseudo manifold . For more details on (homogeneous)pseudo manifolds we refer the reader to [9] (see also [13]). The proof of Lemma 3.15 willfollow from the next result.
Lemma 3.16. If K is a connected complex such that st ( v, K ) is an N H -pseudo manifoldfor all v ∈ V K then K is an N H -pseudo manifold.Proof.
We will show that K satisfies properties (1) and (2) of Lemma 3.15. Let σ ∈ K be a ridge and let v ∈ σ be any vertex. Then σ is also a ridge in st ( v, K ) and lk ( σ, K ) = lk ( σ, st ( v, K )). Therefore K satisfies property (1).Let ν, τ ∈ K be maximal simplices and let v ∈ ν, w ∈ τ . Take an edge path from v to w . We will prove that K satisfies property (2) by induction on the length r of theedge path. If r = 0, then v = w . In this case, ν, τ ∈ st ( v, K ) and the results follows byhypothesis. Suppose now that ψ , . . . , ψ r is an edge path from v to w of length r ≥
1. Takemaximal simplices E i such that ψ i ≤ E i . Note that E ∩ E contains the vertex ψ ∩ ψ .By hypothesis, st ( ψ ∩ ψ , K ) satisfies property (2) and therefore we can join E with E by a sequence of adjacent maximal simplices. Now the result follows by induction. (cid:3) ON-HOMOGENEOUS COMBINATORIAL MANIFOLDS 9
Proof of Lemma 3.15.
We proceed by induction on the dimension n of M . By Lemma3.16, it suffices to prove that st ( v, M ) is an N H -pseudo manifold for every vertex v . Thecase n = 0 is trivial. Suppose that n ≥ k ≤ n −
1. Now, if lk ( v, M ) is an N H -ball or a connected
N H -sphere then, by induction, it is an
N H -pseudomanifold. It follows that st ( v, M ) is also an N H -pseudo manifold since it is a cone of an
N H -pseudo manifold. In the other case, lk ( v, M ) is an N H -sphere of homotopy dimension0 of the form B + ∗ , for some N H -ball B . Since vB is an N H -pseudo manifold, it followsthat st ( v, M ) is also an N H -pseudo manifold. (cid:3) Boundary, pseudo boundary and the Anomaly Complex
The concept of boundary is not defined in the non-homogeneous setting and, in fact, itis not clear what a boundary of a general complex could be. However, the characterizationof the boundary of combinatorial manifolds allows us to extend this notion to the class of
N H -manifolds.
Definition 4.1.
Let M be an N H -manifold. The pseudo boundary of M is the set ofsimplices ˜ ∂M whose links are N H -balls. The boundary of M is the subcomplex ∂M spanned by ˜ ∂M . In other words, ∂M is the closure ˜ ∂M . M M~ M
Figure 3.
Boundary and pseudo boundary.
It is clear that ˜ ∂M = ∂M for any combinatorial manifold M . We will see that, in fact,this is the only case where this happens. The result will follow from the next lemma. Lemma 4.2.
Let M be an N H -manifold and let σ ∈ M . If σ is a face of two principalsimplices of different dimensions then σ ∈ ∂M .Proof. Let τ = σ ∗ η and τ = σ ∗ η be principal simplices such that dim τ (cid:54) = dim τ .By Lemma 3.15 we may assume that τ and τ are adjacent. Let ρ = τ ∩ τ and suppose ρ ≺ τ . Then, lk ( ρ, M ) is an N H -sphere of homotopy dimension 0 with decomposition lk ( ρ, M − τ ) + ∗ . Since dim lk ( ρ, M − τ ) ≥ ∂lk ( ρ, M − τ ) = ˜ ∂lk ( ρ, M ) is non-empty. For any simplex ν in ˜ ∂lk ( ρ, M ), ν ∗ ρ ∈ ˜ ∂M . Thus σ ∈ ∂M . (cid:3) Proposition 4.3. If M is a connected N H -manifold such that ˜ ∂M = ∂M then M is acombinatorial manifold. In particular, N H -manifolds without boundary (or pseudo bound-ary) are combinatorial manifolds.
Proof. If M is non-homogeneous, by Lemma 3.15 there exist two adjacent principal sim-plices τ , τ of different dimensions. By Lemma 4.2, ρ = τ ∩ τ ∈ ∂M − ˜ ∂M . (cid:3) The following result will be used in the next sections. It is the non-homogeneous versionof the well-known fact that any n -homogeneous subcomplex of an n -combinatorial manifoldwith non-empty boundary has also a non-empty boundary (see [4]). Lemma 4.4.
Let M be a connected N H -manifold with non-empty boundary and let L ⊆ M be a top generated N H -submanifold. Then, ∂L (cid:54) = ∅ .Proof. We may assume L (cid:54) = M . We proceed by induction on n = dim M . The 1-dimensional case is clear. Let n ≥
2. Take adjacent principal simplices σ ∈ L and τ ∈ M − L and let ρ = σ ∩ τ . If dim σ = dim τ then lk ( ρ, M ) = S and therefore, ρ ∈ ∂L .If dim σ (cid:54) = dim τ then lk ( ρ, M ) = B + ∗ is a non-homogeneous N H -sphere of homotopydimension 0. We analyze both cases: ρ ≺ σ and ρ ≺ τ . If ρ ≺ σ then lk ( ρ, L ) is either a0-ball, which implies ρ ∈ ˜ ∂L , or a non-homogeneous N H -sphere of homotopy dimension0. In this case, by Proposition 4.3 ˜ ∂lk ( ρ, L ) (cid:54) = ∅ . If ρ ≺ τ then ˜ ∂lk ( ρ, L ) (cid:54) = ∅ by inductionapplied to lk ( ρ, L ) ⊂ B . In any case, if η ∈ ˜ ∂lk ( ρ, L ) then η ∗ ρ ∈ ˜ ∂L . (cid:3) Corollary 4.5. If M is a connected N H -manifold of dimension n ≥ containing a topgenerated combinatorial manifold L without boundary then M = L . Note that if S = B + ∗ is a non-homogeneous N H -sphere of homotopy dimension 0 and M is a non-trivial top generated combinatorial n -manifold contained in S , then M ⊆ B .This implies that ∂M (cid:54) = ∅ by Corollary 4.5. We state this fact in the following Corollary 4.6.
A non-homogeneous
N H -sphere S = B + ∗ of homotopy dimension cannot contain a non-trivial top generated combinatorial manifold without boundary. In contrast to the classical situation, the boundary of an
N H -manifold is not in generalan
N H -manifold (see Figure 3). However, similarly as in the homogeneous setting, if M is an N H -manifold and η ∈ M is any simplex, then lk ( η, ∂M ) = ∂lk ( η, M ). Moreover,it is well-known that the boundary of a combinatorial manifold has no boundary. Thefollowing result generalizes this fact to the non-homogeneous setting. Proposition 4.7.
The boundary of an
N H -manifold M has no collapsible simplices.Proof. Let σ be a ridge in ∂M . Since lk ( σ, ∂M ) = ∂lk ( σ, M ), it suffices to show thatthe boundary of any N H -ball or
N H -sphere cannot be a singleton. This is clear forclassical balls and spheres and, by Proposition 4.3, the same is true for
N H -balls and
N H -spheres. (cid:3)
A simplex σ ∈ M will be called internal if lk ( σ, M ) is an N H -sphere, i.e. if σ / ∈ ˜ ∂M .We denote by ◦ M the relative interior of M , which is the set of its internal simplices. Lemma 4.8.
Let S be an N H -sphere with decomposition B + L . Then, every σ ∈ L isinternal in S . In particular, ˜ ∂S = ˜ ∂B − L .Proof. This is a particular case of Lemma 5.7. (cid:3)
Definition 4.9.
Let M be an N H -manifold. The anomaly complex of M is the subcom-plex A ( M ) = { σ ∈ M : lk ( σ, M ) is not homogeneous } . ON-HOMOGENEOUS COMBINATORIAL MANIFOLDS 11
M A(M) M A(M)
Figure 4.
Anomaly complex.
The fact that A ( M ) is a simplicial complex follows from the equation lk ( σ ∗ η, M ) = lk ( σ, lk ( η, M )). Figure 4 shows examples of anomaly complexes. Proposition 4.10.
For any
N H -manifold M , ∂M = ˜ ∂M + A ( M ) .Proof. If σ ∈ A ( M ) then σ is face of two principal simplices of M of different dimensions.Therefore σ ∈ ∂M by Lemma 4.2. For the other inclusion, let σ ∈ ∂M − ˜ ∂M . Then lk ( σ, M ) is an N H -sphere and σ < τ with τ ∈ ˜ ∂M . Write τ = σ ∗ η , thus lk ( τ, M ) = lk ( η, lk ( σ, M )). If σ / ∈ A ( M ) then lk ( σ, M ) is a combinatorial sphere and so is lk ( τ, M ),contradicting the fact that τ ∈ ˜ ∂M . (cid:3) NH-bouquets and shellability
Recall that, similarly as in the homogeneous setting, an
N H -sphere is obtained by“gluing” a combinatorial ball to an
N H -ball along its entire boundary. In the homogeneouscase one can no longer glue another ball to a sphere for it would produce a complex whichis not a manifold (not even a pseudo manifold). The existence of boundary in non-homogeneous
N H -spheres allows us to glue balls and obtain again an
N H -manifold. Thisis the idea behind the notion of
N H -bouquet. This concept arises naturally when studyingshellability of non-homogeneous manifolds.
Definition 5.1.
We define an
N H -bouquet G of dimension n and index k by inductionon k . • If k = 0 then G is an N H -ball of dimension n . • If k ≥ G is an N H -manifold of dimension n such that there exist a topgenerated N H -bouquet S of dimension n and index k − L , such that G = S + L and S ∩ L = ∂L .We will show below that the index k is well defined since an N H -bouquet of index k ishomotopy equivalent to a bouquet of k spheres (of different dimensions). In fact, the indexis the number of balls that are glued to an N H -ball. A decomposition G = B + L + · · · + L k of an N H -bouquet G consists of top generated subcomplexes of G such that B is an N H -ball, L i is a combinatorial ball for each i = 1 , . . . , k and ( B + · · · + L i ) ∩ L i +1 = ∂L i +1 .Of course, a decomposition is not unique. Examples 5.2.
Figure 5 shows some examples of
N H -bouquets of low dimensions.
Remark . Clearly an
N H -bouquet of index 1 is an
N H -sphere. Note also that for every n ≥ k ≥ N H -bouquet G of dimension n and index k .Similarly as in Theorem 3.9, it can be proved that the class of N H -bouquets is PL-closed.
Figure 5. NH -bouquets. Lemma 5.4. If G = B + L + · · · + L k is a decomposition of an N H -bouquet of index k ≥ , then L i ∩ L j = ∂L i ∩ ∂L j for all ≤ j < i ≤ k .Proof. L i ∩ L j ⊆ ∂L i by definition. Suppose that L i ∩ L j (cid:42) ∂L j . Then there exists a simplex σ ∈ L i ∩ L j such that lk ( σ, L j ) is a sphere. By Corollaries 4.5 and 4.6, lk ( σ, L j ) = lk ( σ, G ).In particular lk ( σ, L i ) ⊆ lk ( σ, L j ), but if ν ∈ lk ( σ, L i ) is maximal, then σ ∗ ν is a maximalsimplex in G and it is contained in L i ∩ L j ⊆ ∂L i which is a contradiction. (cid:3) Proposition 5.5. If G = B + L + · · · + L k is a decomposition of an N H -bouquet, then ∂L i ⊆ B for every i = 1 , . . . , k . In particular, an N H -bouquet of index k is homotopyequivalent to a bouquet of spheres of dimensions dim L i , for ≤ i ≤ k .Proof. ∂L ⊆ B by definition. For i ≥ ∂L i ⊆ B for every i , G is homotopy equivalentto a CW-complex obtained by attaching cells of dimensions dim L i to a point. (cid:3) Remark . It is not hard to see that a homogeneous
N H -bouquet of dimension n ≥ n -ball or n -sphere. This follows from Theorem 3.6 and Corollary 4.5.The following result extends Lemma 4.8 and will be used below. Lemma 5.7.
Let G = B + L + · · · + L k be a decomposition of an N H -bouquet. Thenevery simplex in each L i is internal in G . Furthermore, if σ ∈ ∂L i then lk ( σ, G ) is an N H -sphere with decomposition lk ( σ, B ) + lk ( σ, L i ) . In particular, ˜ ∂G = ˜ ∂B − ∪ i L i .Proof. It is clear that every simplex internal in L i is internal in G . Given σ ∈ ∂L i , byProposition 5.5 lk ( σ, G ) = lk ( σ, B ) + lk ( σ, L i ). Also lk ( σ, L i ) ∩ lk ( σ, B ) = ∂lk ( σ, L i ). (cid:3) Shellings are structure-preserving moves that transform a combinatorial manifold intoanother one. They were first studied by Newman [10] (see also [7, 12, 14]) and they turnedout to be central in the theory. At the beginning of the 90’s Pachner [11] showed that two(connected) combinatorial manifolds with boundary are PL homeomorphic if and only ifone can obtain one from the other by a sequence of elementary shellings, inverse shellingsand simplicial isomorphisms (see also [7]).An elementary shelling on a combinatorial n -manifold M is the move M sh → M (cid:48) = M − τ , where τ = σ ∗ η is an n -simplex of M with σ ∈ ◦ M and ∂σ ∗ η ⊂ ∂M . The oppositemove is called an inverse shelling . It is not hard to see that these moves are specialcases of regular collapses and expansions and therefore, they preserve the structure of themanifold.A combinatorial n -manifold which can be transformed into a single n -simplex by asequence of elementary shellings is said to be shellable . Shellable combinatorial n -manifolds ON-HOMOGENEOUS COMBINATORIAL MANIFOLDS 13 are collapsible and, hence, combinatorial n -balls. The definition of shellability can also beextended to combinatorial n -spheres by declaring S to be shellable if for some n -simplex σ , S − σ is a shellable n -ball.The alternative, and more constructive, definition of shellability by means of inverseshellings requires the existence of a linear order F , . . . , F t of all the n -simplices such that F k ∩ ( F + · · · + F k − ) is ( n − ≤ k ≤ t . This formulation canbe used to define the concept of shellability in arbitrary n -homogeneous complexes. It isnot difficult to see that shellable pseudo manifolds are necessarily combinatorial balls ([2,Proposition 4.7.22]). It is also known that every ball of dimension less than or equal to2 is shellable. Examples of non-shellable 3-balls abound in the bibliography, the first onewas discovered by Furch in 1924 (see [15] for a survey of non-shellable 3-balls). A way forconstructing non-shellable balls for every n ≥ F , . . . , F t of its maximal simplices such that F k ∩ ( F + · · · + F k − )is (dim F k − ≤ k ≤ t . A simplex F k is said to be a spanningsimplex if F k ∩ ( F + · · · + F k − ) = ∂F k . It is not hard to see that the spanning simplicesmay be moved to any later position in the shelling order (see [6]). It is known that ashellable complex is homotopy equivalent to a wedge of spheres, which are indexed bythe spanning simplices (see [6, Theorem 12.3]). In particular, shellable N H -balls cannothave spanning simplices and shellable
N H -spheres have exactly one spanning simplex. Ingeneral, a shellable
N H -bouquet of index k must have exactly k spanning simplices. Theorem 5.8.
Let M be a shellable N H -manifold. Then, for every shelling order F , . . . , F t of M and every ≤ l ≤ t , F l ( M ) = F + · · · + F l is an N H -manifold. Moreover, F l ( M ) is an N H -bouquet of index (cid:93) { F j ∈ T | j ≤ l } , where T is the set of spanning simplices. Inparticular, M is an N H -bouquet of index (cid:93) T .Proof. We proceed by induction on n = dim M . Suppose n ≥ F , . . . , F t . Let 1 ≤ l ≤ t and let v ∈ M be a vertex. Since lk ( v, M ) is a shellable N H -ball or
N H -sphere with shelling order lk ( v, F ) , . . . , lk ( v, F t ) (some of them possiblyempty), then by induction F j ( lk ( v, M )) is an N H -bouquet of index at most 1 for all1 ≤ j ≤ l . Since lk ( v, F l ( M )) = F l ( lk ( v, M )) then F l ( M ) is an N H -manifold. To seethat F l ( M ) is actually an N H -bouquet, reorder F , . . . , F l so that the spanning simplicesare placed at the end of the order. If F p +1 is the first spanning simplex in the order,then F p ( M ) is a collapsible N H -manifold (see [6, Theorem 12.3]) and hence an
N H -ball.Then, F l ( M ) = F p ( M ) + F p +1 + · · · + F l is an N H -bouquet of index (cid:93) { F j ∈ T | j ≤ l } bydefinition. (cid:3) Regular collapses, elementary shellings and Pachner moves
Recall that a regular expansion in an n -combinatorial manifold M is a geometricalexpansion M → N = M + B n such that M ∩ B n ⊂ ∂M . As we mentioned before, thismove produces a new combinatorial n -manifold. In this section we prove a general versionof this result for N H -manifolds. We start with some preliminary results.
Lemma 6.1.
Let B be a combinatorial n -ball and let L ⊂ ∂B be a combinatorial ( n − )-ball. Then, there exists a stellar subdivision s such that sB (cid:38) sL . Proof.
By [4, Lemma III.8] there exists a derived subdivision δ and a subdivision α suchthat δB = α ∆ n and δL = α ∆ n − , where ∆ n − is an ( n − n . Now, by [4,Lemma III.7] there exists a stellar subdivision ˜ s such that ˜ sα ∆ n (cid:38) ˜ sα ∆ n − and therefore˜ sδB (cid:38) ˜ sδL . (cid:3) Corollary 6.2.
Let B be a combinatorial n -ball and let K ⊂ ∂B be a collapsible complex.Then, there exists a stellar subdivision s such that sB (cid:38) sK .Proof. Subdivide B baricentrically twice and consider a regular neighborhood N of K (cid:48)(cid:48) in ∂B (cid:48)(cid:48) (see [4, Corollary III.17]). Since K (cid:48)(cid:48) is collapsible, then N is an ( n − N ⊂ ∂B (cid:48)(cid:48) , by the previous lemma, there is a stellar subdivision ˜ s such that ˜ sB (cid:48)(cid:48) (cid:38) ˜ sN .We conclude that ˜ sB (cid:48)(cid:48) (cid:38) ˜ sN (cid:38) ˜ sK (cid:48)(cid:48) . (cid:3) Theorem 6.3.
Let M be an N H -manifold and B r a combinatorial r -ball. Suppose M ∩ B r ⊆ ∂B r is an N H -ball or an
N H -sphere generated by ridges of M or B r and that ( M ∩ B r ) ◦ ⊆ ˜ ∂M . Then M + B r is an N H -manifold. Moreover, if M is an N H -bouquetof index k and M ∩ B r (cid:54) = ∅ for r (cid:54) = 0 , then M + B r is an N H -bouquet of index k (if M ∩ B r is an N H -ball) or k + 1 (if M ∩ B r is an N H -sphere).Proof.
We note first that
M, B r ⊂ M + B r are top generated. Since M ∩ B r ⊆ ∂B r then B r is top generated. On the other hand, if σ is a principal simplex in M which is notprincipal in M + B r then σ must be in M ∩ B r . Since σ / ∈ ˜ ∂M then σ / ∈ ( M ∩ B r ) ◦ .Hence, σ is not principal in M ∩ B r , which contradicts the maximality of σ in M .We shall prove the result by induction on r . The case M ∩ B r = ∅ is clear, so let r ≥ M ∩ B r (cid:54) = ∅ . We need to prove that every vertex in M + B r isregular. It is clear that the vertices in ( M − B r ) + ( B r − M ) are regular since B r and M are N H -manifolds. Consider then a vertex v ∈ M ∩ B r . We claim that the pair lk ( v, M ) , lk ( v, B r ) fulfills the hypotheses of the theorem. Note that lk ( v, M ) is an N H -ball or
N H -sphere, lk ( v, B r ) is a combinatorial ball, since v ∈ M ∩ B r ⊆ ∂B r , and lk ( v, M ∩ B r ) is an N H -ball or
N H -sphere contained in ∂lk ( v, B r ). Note also that theinclusion ( M ∩ B r ) ◦ ⊆ ˜ ∂M implies that lk ( v, M ∩ B r ) ◦ ⊆ ˜ ∂lk ( v, M ). We now checkthat lk ( v, M ∩ B r ) is generated by ridges of lk ( v, M ) or lk ( v, B r ). This is easily seen if lk ( v, M ∩ B r ) (cid:54) = ∅ . For the case lk ( v, M ∩ B r ) = ∅ we need to show that there is a principal0-simplex in lk ( v, M ) or lk ( v, B r ). Now, lk ( v, M ∩ B r ) = ∅ implies that v is principal in M ∩ B r , so v ∈ ( M ∩ B r ) ◦ ⊆ ˜ ∂M and lk ( v, M ) is an N H -ball (and hence, collapsible).And since v ∈ M ∩ B r ⊆ ∂B r then lk ( v, B r ) is a ball. Now, if v is a ridge in B r then r = 1 and, hence, lk ( v, B ) = ∗ . If, on the other hand, v is a ridge of M then there existsa principal 1-simplex σ with v ≺ σ . Since σ is principal in M , ∗ = lk ( v, σ ) is principal in lk ( v, M ). Since lk ( v, M ) is collapsible, then lk ( v, M ) = ∗ .Therefore, by induction, lk ( v, M + B r ) is an N H -manifold. Now, if lk ( v, M ∩ B r ) (cid:54) = ∅ ,then lk ( v, M + B r ) is an N H -ball or an
N H -sphere if lk ( v, M ) is an N H -ball and it isan
N H -sphere if lk ( v, M ) is an N H -sphere. If lk ( v, M ∩ B r ) = ∅ , we showed above that lk ( v, M ) = ∗ and lk ( v, B r ) is a ball or lk ( v, B r ) = ∗ and lk ( v, M ) is an N H -ball. In eithercase, lk ( v, M + B r ) is an N H -sphere of homotopy dimension 0. This proves that M + B r is an N H -manifold.We prove now the second part of the statement. We proceed by induction on the index k . Suppose first that k = 0, i.e. M is an N H -ball. Let α be a subdivision such that αM (cid:38)
0, and extend α to all M + B r . If M ∩ B r is an N H -ball we can apply Corollary6.2 to α ( M ∩ B r ) ⊂ α∂B r and find a stellar subdivision s such that sαB r (cid:38) sα ( M ∩ B r ). ON-HOMOGENEOUS COMBINATORIAL MANIFOLDS 15
This implies that sα ( M + B r ) (cid:38) sαM (cid:38) M + B r is an N H -ball. If M ∩ B r is an N H -sphere S with decomposition S = G + L , take any maximal simplex τ ∈ L with an immediate face σ in ∂L and consider the starring ( τ, ˆ τ ) S of S (see Figure6). Let ρ = ˆ τ ∗ σ ∈ ( τ, ˆ τ ) S . We claim that ( τ, ˆ τ ) S − { ρ } is an N H -ball. On one hand, it isclear that (( τ, ˆ τ ) S − { ρ } ) ∩ ρ = ∂ρ . On the other hand, ( τ, ˆ τ ) L − { ρ, σ } is a combinatorialball because it is PL-homeomorphic to L . Since G is an N H -ball, ( τ, ˆ τ ) L − { ρ, σ } is acombinatorial ball and G ∩ (( τ, ˆ τ ) L − { ρ, σ } ) = ∂L − { σ } , which is a combinatorial ball byNewman’s Theorem, it follows that ( τ, ˆ τ ) S − { ρ } is an N H -ball, as claimed. Now, since τ ∈ L ⊂ M ∩ B r is principal then it must be a ridge of M or of B r . We analyze bothcases. Suppose τ is a ridge of B r and let τ ≺ η ∈ B r . Write η = w ∗ τ (see Figure 6).Note that the starring ( τ, ˆ τ ) S performed earlier also subdivides η and the simplex ρ liesin the boundary of ( τ, ˆ τ ) η . Consider the simplex ν = w ∗ ρ , which is one of the principalsimplices in which η has been subdivided. Now make the starring ( ν, ˆ ν ) in ( τ, ˆ τ ) η (seeFigure 6). By removing the simplex ˆ ν ∗ ρ from ( ν, ˆ ν )( τ, ˆ τ ) B r , we obtain a complex whichis PL-homeomorphic to B r . Then( ν, ˆ ν )( τ, ˆ τ ) B r − { ˆ ν ∗ ρ } is a combinatorial ball and it intersects M in ( τ, ˆ τ ) S − { ρ } , which is an N H -ball. It followsthat ( ν, ˆ ν )( τ, ˆ τ )( M + B r ) − { ˆ ν ∗ ρ } = ( τ, ˆ τ ) M + ( ν, ˆ ν )( τ, ˆ τ ) B r − { ˆ ν ∗ ρ } is again an N H -ball. If we now plug the simplex ˆ ν ∗ ρ , ( ν, ˆ ν )( τ, ˆ τ )( M + B r ) is an N H -sphere by definition. This completes the case where τ is a ridge of B r . The case that τ isa ridge of M is analogous. LB w w ( , ) ( , )( , ) ( , ) Figure 6.
The starrings of Theorem 6.3.
Suppose now that M is an N H -bouquet of index k ≥
1. Write M = G + L with G an N H -bouquet of index k − L a combinatorial ball glued to G along its entireboundary. If r = 0 we obtain an N H -bouquet. Suppose then that M ∩ B r (cid:54) = ∅ . Weclaim that B r ∩ L ⊆ ∂L . Suppose ( L − ∂L ) ∩ B r (cid:54) = ∅ and let η ∈ ( L − ∂L ) ∩ B r .Now, lk ( η, M ) = lk ( η, L ) is a combinatorial sphere and Corollaries 4.5 and 4.6 imply that lk ( η, B r ) ⊂ lk ( η, M ). But if τ ∈ B r is a principal simplex containing η then lk ( η, τ ) ∈ lk ( η, M ) and τ ∈ M ∩ B r ⊆ ∂B r , contradicting the maximality of τ in B r . This proves that B r ∩ L ⊆ ∂L and, therefore M ∩ B r = G ∩ B r . Also, ( G ∩ B r ) ◦ ⊆ ˜ ∂M = ˜ ∂G − L ⊂ ˜ ∂G . Byinduction, G + B r is an N H -bouquet of index k − G ∩ B r = M ∩ B r is an N H -ball) or k (if G ∩ B r = M ∩ B r is an N H -sphere). In either case, M + B r = G + L + B r = ( G + B r )+ L with ( G + B r ) ∩ L = G ∩ L + B r ∩ L = ∂L . Thus, M + B r is an N H -bouquet of index k or k + 1. This completes the proof. (cid:3) Note that the previous theorem generalizes Alexander’s Theorem on regular expansions([7, Theorem 3.9]) to the non-homogeneous setting. The condition ( M ∩ B ) ◦ ⊂ ˜ ∂M corresponds to M ∩ B ⊂ ∂M in the homogeneous case. We next extend the notion ofregular expansion to the non-homogeneous context. This will be used to characterize thenotion of shelling on N H -manifolds similarly as in the case of manifolds.
Definition 6.4. A regular expansion on an N H -manifold M is a geometrical expansion M → M + B (i.e. B is a ball and M ∩ B ⊂ ∂B is a ball of dimension dim B −
1) suchthat ( M ∩ B ) ◦ ⊂ ˜ ∂M .Recall that an inverse shelling in a combinatorial n -manifold M corresponds to a (clas-sical) regular expansion M → M + σ involving a single n -simplex σ . An elementaryshelling is the inverse move [14]. We investigate now shellable N H -balls. First we needthe following result.
Proposition 6.5.
Let M → M + B be a geometrical expansion in an N H -manifold M .If M + B is an N H -manifold and
M, B ⊂ M + B are top generated then ( M ∩ B ) ◦ ⊂ ˜ ∂M (i.e. M → M + B is a regular expansion).Proof. Take ρ ∈ ( M ∩ B ) ◦ . Since lk ( ρ, M ∩ B ) is a sphere contained in the sphere ∂lk ( ρ, B ),then lk ( ρ, M ∩ B ) = ∂lk ( ρ, B ). Suppose ρ / ∈ ˜ ∂M . Then lk ( ρ, M + B ) = lk ( ρ, M ) + lk ( ρ, B )is an N H -bouquet of index 2 since lk ( ρ, M ) , lk ( ρ, B ) ⊂ lk ( ρ, M + B ) are top generated byhypothesis. This contradicts the fact that M + B is an N H -manifold. (cid:3)
Definition 6.6.
Let M be an N H -manifold. An inverse shelling is a regular expansion M → M + σ where σ is a single simplex. An elementary shelling is the inverse move.By Proposition 6.5 and Theorem 5.8, we obtain the following characterization of shellable N H -balls in terms of elementary shellings.
Corollary 6.7. An N H -ball B is shellable if and only if B can be transformed into asingle maximal simplex by a sequence of elementary shellings. A stellar exchange κ ( σ, τ ) is the move that transforms a complex M into a new complex κ ( σ, τ ) M by replacing st ( σ, M ) = σ ∗ ∂τ ∗ L with ∂σ ∗ τ ∗ L , for σ ∈ M and τ / ∈ M (see[7, 11]). Note that elementary starrings and welds are particular cases of stellar exchanges(when τ or σ is a vertex). When L = ∅ the stellar exchange is called a bistellar move . Also,since κ ( σ, τ ) = ( τ, b ) − ( σ, a ), two simplicial complexes are PL-homeomorphic if and onlyif they are related by a sequence of stellar exchanges. In the case of PL-homeomorphiccombinatorial manifolds without boundary, all the moves in this sequence can be taken tobe bistellar moves (see [7, 11] for more details). This discussion motivates the followingdefinition. Definition 6.8.
Let M be a combinatorial n -manifold and let σ ∈ M be a simplex suchthat lk ( σ, M ) = ∂τ ∗ L with τ / ∈ M . An N H -factorization is the move M → M + σ ∗ τ ∗ L .We write F ( σ, τ ) M = M + σ ∗ τ ∗ L . When L = ∅ , we call it a bistellar factorization .Note that, in fact, N H -factorizations can be defined for arbitrary complexes. When τ isa single vertex b / ∈ M , we will denote M + σ = F ( σ, b ) M . Note that M + σ is the simplicial coneof the inclusion st ( σ, M ) ⊆ M . Note also that, since st ( σ, M ) is collapsible, M + σ (cid:38) M .By definition, the following diagram commutes (this justifies the term “factorization”). ON-HOMOGENEOUS COMBINATORIAL MANIFOLDS 17 M κ ( σ,τ ) (cid:47) (cid:47) F ( σ,τ ) (cid:38) (cid:38) (cid:76)(cid:76)(cid:76)(cid:76)(cid:76)(cid:76)(cid:76)(cid:76)(cid:76)(cid:76)(cid:76)(cid:76) κ ( σ, τ ) M F ( τ,σ ) (cid:119) (cid:119) (cid:110)(cid:110)(cid:110)(cid:110)(cid:110)(cid:110)(cid:110)(cid:110)(cid:110)(cid:110)(cid:110)(cid:110) M + σ ∗ τ ∗ L .
Proposition 6.9.
Let M be a combinatorial n -manifold and let M −→ N = F ( σ, τ ) M bean N H -factorization. Then N is an N H -manifold.Proof.
Let N = M + σ ∗ τ ∗ L with τ / ∈ M . Since ( τ, b ) N = M + b ∗ ∂τ ∗ σ ∗ L = M + σ ,by Theorem 3.9 it suffices to prove that M + σ is an N H -manifold. We prove by inductionon n that the simplicial cone M + B of the inclusion of any combinatorial ball B ⊆ M is an N H -manifold.Denote M + B = M + b ∗ B and let v be a vertex of M + B . If v / ∈ B , then lk ( v, M + B ) = lk ( v, M ). If v ∈ ◦ B then lk ( v, M + B ) = b ∗ lk ( v, M ), which is a combinatorial n -ball. If v ∈ ∂B then lk ( v, M + B ) = lk ( v, M ) + b ∗ lk ( v, B ) is an N H -manifold by induction. Since lk ( v, B ) is collapsible then lk ( v, M + B ) (cid:38) lk ( v, M ), so lk ( v, M + B ) is an N H -ball if v ∈ ∂M .If v / ∈ ∂M then lk ( v, B ) is strictly contained in lk ( v, M ). It follows that there is an n -simplex η ∈ M − B containing v . By Newman’s Theorem, lk ( v, M ) − lk ( v, η ) is an( n − lk ( v, M + B ) is an N H -sphere with decomposition( lk ( v, M − η ) + b ∗ lk ( v, B )) + lk ( v, η )since lk ( v, M − η ) + b ∗ lk ( v, B ) is an N H -ball by the previous case and( lk ( v, M − η ) + b ∗ lk ( v, B )) ∩ lk ( v, η ) = ( lk ( v, M ) − lk ( v, η )) ∩ lk ( v, η ) = ∂lk ( v, η ) . (cid:3) Lemma 6.10.
Let M , M be combinatorial n -manifolds without boundary and let B i ⊂ M i be combinatorial n -balls. Suppose M − B = M − B . Then, M (cid:39) P L M .Proof. Note that M i − B i is a combinatorial n -manifold and that ∂B i = M i − B i ∩ B i .Since ∂B = M − B ∩ B = M − B ∩ B and M = M − B + B = M − B + B ,then B ∩ M − B ⊆ ∂ ( M − B ) = ∂B . Hence, ∂B ⊆ ∂B . Analogously, ∂B ⊆ ∂B .The result now follows from the fact that every ball may be starred (see [4, TheoremII.11]). (cid:3) Theorem 6.11.
Let M, ˜ M be combinatorial n -manifolds (with or without boundary). If M and ˜ M are PL-homeomorphic then there exists a sequence M = M → N ← M → N ← M → · · · ← M r − → N r − ← M r = ˜ M where the N i ’s are N H -manifolds, the M i ’s are n -manifolds, and M i , M i +1 → N i are N H -factorizations. Moreover, if M and ˜ M are closed then the converse holds. Also, inthis case the N H -factorizations may be taken to be bistellar factorizations.Proof.
Let κ ( σ , τ ) , . . . , κ ( σ r , τ r ) be a sequence of stellar exchanges taking M to ˜ M . Thenfor each i , the sequence M i F ( σ i ,τ i ) −→ N i F ( τ i ,σ i ) ←− M i +1 = κ ( σ i , τ i ) M i is a factorization and N i is an N H -manifold by Lemma 6.9.
For the second part of the proof, assume that M F ( σ,τ ) −→ N F ( ρ,η ) ←− ˜ M are N H -factorizations,with M and ˜ M n -manifolds and M closed. Since M + σ ∗ τ ∗ L = ˜ M + ρ ∗ η ∗ T , by adimension argument and the homogeneity of M and ˜ M , it follows that σ ∗ τ ∗ L = ρ ∗ η ∗ T .Hence, M − σ ∗ ∂τ ∗ L = N − σ ∗ τ ∗ L = N − ρ ∗ η ∗ T = ˜ M − ρ ∗ ∂η ∗ T . The resultnow follows from Lemma 6.10. (cid:3) References [1] J. W. Alexander.
The combinatorial theory of complexes . Ann. of Math. 31 (1930), 294-322.[2] A. Bj¨orner, M. Las Vergnas, B. Sturmfels, N. White, G. Ziegler.
Oriented Matroids . Second Edition,Cambridge University Press (1999).[3] A. Bj¨orner, M. Wachs.
Shellable nonpure complexes and posets . Trans. Amer. Math. Soc. 348 (1996),No. 4, 1299-1327.[4] L. Glaser.
Geometrical Combinatorial Topology - Volume I . Van Nostrand Reinhold Company (1970).[5] J.F.P. Hudson.
Piecewise linear topology . W.A. Benjamin (1969).[6] D. Kozlov.
Combinatorial Algebraic Topology . Algorithms and Computation in Mathematics, Vol. 21.Springer, Berlin (2008).[7] W.B.R. Lickorish.
Simplicial moves on complexes and manifolds . Geometry & Topology Monographs2 (1999), 299-320.[8] W.B.R. Lickorish.
Unshellable triangulations of spheres . European J. Combin. 12 (1991), 527-530.[9] J.R. Munkres.
Elements of Algebraic Topology . Addison-Wesley Publishing Co. (1984).[10] M.H.A. Newman.
On the foundation of combinatorial analysis situs . Proc. Royal Acad. Amsterdan29 (1926), 610-641.[11] U. Pachner.
PL homeomorphic manifolds are equivalent by elementary shellings . European J. Combin.12 (1991), 129-145.[12] C.P. Rourke, B.J. Sanderson.
Introduction to piecewise-linear topology . Springer-Verlag (1972).[13] E. Spanier.
Algebraic Topology . Springer (1966).[14] J.H.C Whitehead.
Simplicial spaces, nuclei and m-groups . Proc. London Math. Soc. 45 (1939), 243-327.[15] G.M. Ziegler.
Shelling Polyhedral 3-Balls and 4-Polytopes . Discrete Comput. Geom. 19 (1998), 159-174.
Departamento de Matem´atica-IMAS, FCEyN, Universidad de Buenos Aires, Buenos Aires,Argentina
E-mail address : [email protected] E-mail address ::