Ear Decomposition and Balanced Neighborly Simplicial Manifolds
EEar Decomposition and Balanced 2-neighborly Simplicial Manifolds
Hailun Zheng
Department of MathematicsUniversity of WashingtonSeattle, WA 98195-4350, USA [email protected]
July 3, 2018
Abstract
We find the first non-octahedral balanced 2-neighborly 3-sphere and the balanced 2-neighborlytriangulation of the lens space L (3 , A simplicial complex is called k -neighborly if every subset of vertices of size at most k is the set ofvertices of one of its faces. Neighborly complexes, especially neighborly polytopes and spheres, areinteresting objects to study. In the seminal work of McMullen [10] and Stanley [17], it was shownthat in the class of polytopes and simplicial spheres of a fixed dimension and with a fixed numberof vertices, the cyclic polytope simultaneously maximizes all the face numbers. The d -dimensionalcyclic polytope is (cid:98) d (cid:99) -neighborly. Since then, many other classes of neighborly polytopes have beendiscovered. We refer to [4], [16] and [14] for examples and constructions of neighborly polytopes.Meanwhile, the notion of neighborliness was extended to other classes of objects: for instance,neighborly cubical polytopes were defined and studied in [8], [7], and [15], and neighborly centrallysymmetric polytopes and spheres were studied in [1], [6], [12], and [3].In this paper we discuss a similar notion for balanced simplicial complexes. Balanced complexeswere defined by Stanley in [18], where they were called completely balanced. A ( d − d -colorable. For instance, the barycentric sub-division of regular CW complexes and order complexes are balanced. We say that a balancedsimplicial complex is balanced k -neighborly if every set of k or fewer vertices with distinct colorsforms a face. The joins of balanced neighborly spheres give balanced neighborly spheres. How-ever, apart from the cross-polytopes, it is not known whether “join-indecomposable” balanced k -neighborly polytopes or spheres exist. To the best of our knowledge, no examples of such objectsappear in the current literature, even for k = 2. As for balanced 2-neighborly manifolds, one suchconstruction that triangulates the sphere bundle is given in [9]; it is also a minimal triangulationof the underlying topological space.This more or less explains why so far there is even no plausible sharp upper bound conjecturesfor balanced spheres or manifolds. The goal of this paper is to partially remedy this situation bysearching for balanced neighborly spheres and manifolds of lower dimensions. It turns out that1 a r X i v : . [ m a t h . C O ] N ov ven the lower dimensional cases are rather involved: we show that the octahedral spheres are theonly balanced k -neighborly (2 k − k vertices for k = 2 ,
3. However,we find two constructions of balanced 2-neighborly 3-manifolds with 16 vertices; one triangulatesthe sphere, and the other triangulates the lens space L (3 , f , such spheres cannot exist. In Section 3, we construct a balanced 2-neighborly 3-spherewith 16 vertices. In Section 4, we present the balanced 2-neighborly triangulation of L (3 ,
1) with16 vertices. In Section 5 we provide a way to construct balanced spheres whose rank-selectedsubcomplex does not have an ear decomposition. A simplicial complex ∆ with vertex set V is a collection of subsets σ ⊆ V , called faces , that isclosed under inclusion, and such that for every v ∈ V , { v } ∈ ∆. For σ ∈ ∆, let dim σ := | σ | − dimension of ∆, dim ∆, as the maximum dimension of the faces of ∆. A facet is amaximal face under inclusion. We say that a simplicial complex ∆ is pure if all of its facets havethe same dimension.If ∆ is a simplicial complex and σ is a face of ∆, the star of σ in ∆ is st ∆ σ := { τ ∈ ∆ : σ ∪ τ ∈ ∆ } .We also define the link of σ in ∆ as lk ∆ σ := { τ − σ ∈ ∆ : σ ⊆ τ ∈ ∆ } , and the deletion of a subsetof vertices W from ∆ as ∆ \ W := { σ ∈ ∆ : σ ∩ W = ∅} . If ∆ and ∆ are simplicial complexes ondisjoint vertex sets, then the join of ∆ and ∆ , denoted ∆ ∗ ∆ , is the simplicial complex withvertex set V (∆ ) ∪ V (∆ ) whose faces are { σ ∪ σ : σ ∈ ∆ , σ ∈ ∆ } .If ∆ is a pure ( d − d − boundary complex of ∆ consists of all ( d − simplicial sphere (resp. simplicial ball ) if the geometric realization of ∆ is homeomorphic to asphere (resp. ball). The boundary complex of a simplicial d -ball is a simplicial ( d − polytopal if it is the boundary complex of a convex polytope. For instance,the boundary complex of an octahedron is a polytopal sphere; we will refer to it as an octahedralsphere.For a fixed field k , we say that ∆ is a ( d − k -homology sphere if ˜ H i (lk ∆ σ ; k ) ∼ =˜ H i ( S d − −| σ | ; k ) for every face σ ∈ ∆ (including the empty face) and i ≥ −
1. A homology d -ball (over a field k ) is a d -dimensional simplicial complex ∆ such that (i) ∆ has the same homology asthe d -dimensional ball, (ii) for every face F , the link of F has the same homology as the ( d − | F | )-dimensional ball or sphere, and (iii) the boundary complex, ∂ ∆ := { F ∈ ∆ | ˜ H i (lk ∆ F ) = 0 , ∀ i } , isa homology ( d − d − d − d ≤
3. From now on we fix k and omit it from our notation.Next we define a special structure that exists in some pure simplicial complexes.2 efinition 2.1. An ear decomposition of a pure ( d − , ∆ , · · · , ∆ m of pure ( d − is a simplicial ( d − j = 2 , , · · · , m , ∆ j is a simplicial ( d − ≤ j ≤ m , ∆ j ∩ ( ∪ j − i =1 ∆ i ) = ∂ ∆ j .3. ∪ mi =1 ∆ i = ∆ . We call ∆ the initial complex , and each ∆ j , j ≥
2, an ear of this decompostion . Notice thatthis definition is more general than Chari’s original definition of a convex ear decomposition , see[2, Section 3.2], where the ∆ i ’s are required to be subcomplexes of the boundary complexes ofpolytopes. In particular, if a complex has no ear decomposition, then it has no convex ear decom-position. However, by the Steinitz theorem, all simplicial 2-spheres are polytopal, and hence alsoall simplicial 2-balls can be realized as subcomplexes of the boundary complexes of 3-dimensionalpolytopes. So for 2-dimensional simplicial complexes, the notion of an ear decomposition coincideswith that of a convex ear decomposition.A ( d − balanced if the graph of ∆ is d -colorable,or equivalently, there is a coloring map κ : V → [ d ] such that κ ( x ) (cid:54) = κ ( y ) for any edge { x, y } ∈ ∆.Here [ d ] = { , , · · · , d } is the set of colors. We denote by V i the set of vertices of color i . Abalanced simplicial complex is called balanced k -neighborly if every set of k or fewer vertices withdistinct colors forms a face. For S ⊆ [ d ], the subcomplex ∆ S := { F ∈ ∆ : κ ( F ) ⊆ S } is calledthe rank-selected subcomplex of ∆. We also define the flag f -vector ( f S (∆) : S ⊆ [ d ]) and the flag h -vector ( h S (∆) : S ⊆ [ d ]) of ∆, respectively, by letting f S (∆) := { F ∈ ∆ : κ ( F ) = S } , where f ∅ (∆) = 1, and h S (∆) := (cid:80) T ⊆ S ( − S − T f S (∆). The usual f -numbers and h -numbers can berecovered from the relations f i − (∆) = (cid:80) S = i f S (∆) and h i (∆) = (cid:80) S = i h S (∆).In the reminder of this section, we establish some restrictions on the possible size of color setsof balanced neighborly spheres. Lemma 2.2.
Let ∆ be a balanced k -neighborly homology (2 k − -sphere. Then ∆ has the samenumber of vertices of each color. In particular, f (∆) = 2 kl for some l ≥ .Proof: Let W ⊆ [2 k ] be an arbitrary subset of the set of the colors with | W | = k . Since ∆is balanced k -neighborly, ∆ W is also balanced k -neighborly, and hence ∆ W is the join of k colorsets of colors in W , each considered as a 0-dimensional complex. By the fact that h k (∆ ∗ ∆ ) = (cid:80) kj =0 h j (∆ ) h k − j (∆ ), we obtain that (cid:89) i ∈ W ( | V i | −
1) = (cid:89) i ∈ W h (∆ { i } ) = h | W | (∆) = h k −| W | (∆) = (cid:89) i ∈ [2 k ] \ W h (∆ { i } ) = (cid:89) i ∈ [2 k ] \ W ( | V i | − . Since W ⊆ [2 k ] can be chosen arbitrarily, it follows that each color set in ∆ must have the samesize. (cid:3) The above lemma is not sufficient to tell whether a balanced k -neighborly homology (2 k − kl vertices can exist for given k, l ≥
2. In the first non-trivial case l = 3 we proposethe following conjecture. Conjecture 2.3.
For an arbitrary k ≥ , there does not exist a balanced k -neighborly homology (2 k − -sphere with 6k vertices.
3n the remaining of this section, we prove this conjecture for k ≤ Lemma 2.4.
Let d ≥ . If ∆ is a balanced homology ( d − -sphere and V d = { v , v , v } is theset of vertices of color d , then lk ∆ v i ∩ lk ∆ v j is a homology ( d − -ball for any ≤ i < j ≤ , and ∩ k =1 lk ∆ v k is a homology ( d − -sphere.Proof: Let Σ = lk ∆ v i ∩ lk ∆ v j and Γ = ∩ k =1 lk ∆ v k . Every facet σ of Γ is a ( d − v , v , v . Hence σ is contained in exactly one facet σ ∪ { w } of Σ, where w is the unique vertex adjacent to both v i , v j in lk ∆ σ . We conclude that Γ isthe boundary complex of Σ and it is pure.We first prove that Σ and Γ have the same homology as a ( d − d − d − ∆ v i ∪ lk ∆ v j = ∆ [ d − . By the Mayer-Vietoris sequence, for any n ≥ · · · → H n +1 (∆ [ d − ) → H n (Σ) → H n (lk ∆ v i ) ⊕ H n (lk ∆ v j ) → H n (∆ [ d − ) → · · · . (2.1)Note that ∆ [ d − is a deformation retract of ∆ minus three points, hence β d − (∆ [ d − ) = 2 and β k (∆ [ d − ) = 0 for 0 ≤ k ≤ d −
3. We conclude from (2.1) that β k (Σ) = 0 for all k ≥
0. Again bythe Mayer-Vietoris sequence and the fact that lk ∆ v [3] −{ i,j } ∪ Σ = ∆ [ d − , we obtain · · · → H n +1 (∆ [ d − ) → H n (Γ) → H n (lk ∆ v [3] −{ i,j } ) ⊕ H n (Σ) → H n (∆ [ d − ) → · · · . Hence β d − (Γ) = 1 and β k (Γ) = 0 for 0 ≤ k ≤ d − τ ∈ Γ, we have lk Σ τ = lk lk ∆ τ v i ∩ lk lk ∆ τ v j and lk Γ τ = ∩ i =1 lk lk ∆ τ v i . Sincelk ∆ τ is a balanced homology ( d − − | τ | )-sphere, using the same argument as above, we mayshow that lk Σ τ and lk Γ τ has the same homology as a ( d − − | τ | )-ball and ( d − − | τ | )-sphererespectively. Therefore Γ is a homology ( d − σ of Σ,lk Σ σ = lk lk ∆ v i σ = lk lk ∆ v j σ , and hence lk Σ σ is a homology sphere. By definition we conclude thatΣ is a homology ( d − (cid:3) Remark 2.5.
The complex Γ in Lemma 2.4 is not balanced, since Γ is ( d − d − Proposition 2.6.
No balanced 2-neighborly homology 3-spheres with 12 vertices exist.Proof:
Assume that ∆ is such a sphere. By Lemma 2.2, each color set of ∆ has three vertices.We let V = { v , v , v } be the set of vertices of color 4. Since ∆ is balanced 2-neighborly, eachlk ∆ v i is a 2-sphere with 9 vertices, its f -vector is (1,9,21,14). Furthermore, the balancedness of ∆implies that every vertex v ∈ lk ∆ v i has deg lk ∆ v i v = 4 or 6. If x is the number of vertices of degree6 in lk ∆ v i , then 4(9 − x ) + 6 x = (cid:88) u ∈ lk ∆ v i deg(lk lk ∆ v i u ) = 2 f (lk ∆ v i ) = 42 , and hence x = 3. A balanced 2-sphere with 9 vertices, 3 of which have degree 6, is unique up toisomorphism, as shown in Figure 1. It is immediate that the missing edges between vertices ofdifferent colors in this sphere form a 6-cycle.On the other hand, Σ := lk ∆ v ∩ lk ∆ v is a triangulated 2-ball by Lemma 2.4. If we delete allof the boundary edges from Σ, the resulting complex Σ (cid:48) is still contractible. However, Σ does not4 z u w z u z w u w u z w u z Figure 1: Left: triangulation of the vertex link lk ∆ v i for v i ∈ V , where { u , u , u } , { w , w , w } and { z , z , z } are the three other color sets. Right: the missing edges between vertices of differentcolor of lk ∆ v i .have interior vertices. (An interior vertex of Σ would not be in V (lk ∆ v ), which would contradictthe 2-neighborliness of ∆.) Hence the missing edges of lk ∆ v that form a 6-cycle form the onlyinterior edges of Σ, i.e., Σ (cid:48) is a 6-cycle. This contradicts that Σ (cid:48) is contractible, so no such sphereexists. (cid:3) In fact, a stronger result holds.
Lemma 2.7.
Up to an isomorphism, there are three triangulations of balanced 3-spheres with eachcolor set of size 3 and no more than 50 edges.Proof:
Let ∆ be such a sphere and let V = { v , v , v } . Each vertex link of ∆ is a balanced2-sphere with at most 9 vertices, hence it is either the octahedral sphere, the suspension of a 6-cycle,or the connected sum of two octahedral spheres. We denote these three 2-spheres as Σ , Σ andΣ respectively. By Lemma 2.4, ∆ [3] is the union of three triangulated 2-balls B i = lk ∆ v j ∩ lk ∆ v k ,where { i, j, k } = [3], glued along their common boundary complex c . Assume that f (lk ∆ v i ) ≤ f (lk ∆ v j ) when i ≤ j . An easy counting leads to f (∆ [3] ) = f ( c ) + (cid:88) i =1 f ( B i \ c ) = 9 , f (lk ∆ z i ) = f ( c ) + f ( B j \ c ) + f ( B k \ c ) ∈ { , , } , where f ( B i \ c ) counts the number of interior vertices of B i . We enumerate all possible values ofthe triple ( f (lk ∆ v ) , f (lk ∆ v ) , f (lk ∆ v )) under the condition f (∆) = 2 (cid:80) i =1 f (lk ∆ v i ) ≤ f (lk ∆ v ) , f (lk ∆ v ) , f (lk ∆ v )) = (6 , ,
9) or (6 , , ∆ v is combinatorially equiv-alent to the octahedral sphere, it follows that lk ∆ v is obtained from lk ∆ v by a cross flip(see [5] for a reference). So in the former case lk ∆ v ∼ = Σ , lk ∆ v ∼ = Σ , and the cross flipreplaces a 2-face of lk ∆ v with its complement in the octahedral sphere. In the latter caselk ∆ v ∼ = Σ , lk ∆ v ∼ = Σ , and the cross flip replaces the union of three 2-faces of lk ∆ v withits complement in the octahedral sphere.2. ( f (lk ∆ v ) , f (lk ∆ v ) , f (lk ∆ v )) = (8 , , c is a 6-cycle and ∆ [3] \ c consists of threedisjoint vertices. It is easy to see that at least one of these vertices has degree 6. Then sincelk ∆ v ∼ = lk ∆ v ∼ = Σ , the other two vertices must be of degree 6 as well, and hence ∆ [3] isthe join of c and three disjoint vertices. 5. ( f (lk ∆ v ) , f (lk ∆ v ) , f (lk ∆ v )) = (8 , , ∆ z form a3-cycle, the two disjoint vertices in ∆ [3] \ c cannot both have degree 6 or 4. However, if onevertex of ∆ [3] \ c is of degree 6, then since lk ∆ z and lk ∆ z are combinatorially equivalent toΣ and c is a 7-cycle, B must be the join of one vertex u and a path of length 6. Then u isnot connected to any vertex of ∆ [3] − c , a contradiction.In sum, we obtain three balanced 3-spheres with 12 vertices: S , the connected sum of two octahe-dral 3-spheres; S , the join of two 6-cycles, and S , with lk ∆ v i ∼ = Σ i for 1 ≤ i ≤ (cid:3) The above lemma implies that all balanced 3-spheres with each color set of set 3 can only have f = 42 , , ,
52. (The first three numbers are attained by S i .) Proposition 2.8.
No balanced 2-neighborly homology -spheres with each color set of size 3 exist.Proof: Let ∆ be such a sphere and let its color set V = { v , v , v } . By Alexander Du-ality, ˜ H i (∆ { , } ) ∼ = ˜ H − i (∆ [3] ). In particular, since ∆ { , } is balanced 2-neighborly, β (∆ [3] ) = β (∆ { , } ) = 4 and β (∆ [3] ) = 0. Hence f (∆ [3] ) = ( f − f + χ )(∆ [3] ) = 9 · − . By double counting, (cid:80) i =1 f (lk ∆ v i ) = (cid:80) W = { i,j, }⊆ [5] f (∆ W ) = (cid:0) (cid:1) f (∆ [3] ) = 138. But f (lk ∆ v i ) ∈{ , , , , , } , it follows that either 138 = 42+48 ∗
3, that is, lk ∆ v ∼ = S and lk ∆ v , lk ∆ v ∼ = S ; or 138 = 46 ∗ ∆ v i ∼ = S for all i .Consider the first case above. It can be checked that for any W = { i, j } , f ((lk ∆ v ) W ) = 7 and f ((lk ∆ v ) W ) = 6 or 9, depending on whether (lk ∆ v ) W is a 6-cycle or not. Hence f (∆ W ∪{ } ) = (cid:80) i =1 f ((lk ∆ v i ) W ) (cid:54) = 23, a contradiction. As for the second case, since lk ∆ v ∩ lk ∆ v is a homol-ogy 3-ball with 12 vertices on the boundary, by Lemma 2.7 there is a unique balanced 3-spherecombinatorially equivalent to S that contains lk ∆ v ∩ lk ∆ v as a subcomplex. It follows thatlk ∆ v = lk ∆ v , a contradiction. Hence no balanced 2-neighborly homology 4-spheres with 15vertices exist. (cid:3) Corollary 2.9.
The only balanced 3-neighborly homology 5-sphere with ≤ vertices is the octa-hedral 5-sphere.Proof: Let ∆ be such a sphere. The vertex links of ∆ are balanced 2-neighborly 4-spheres with ≤
16 vertices. By Proposition 2.8, each link must be the suspension of a balanced 2-neighborly3-sphere with ≤
14 vertices. Then the result follows from Lemma 2.2 and Proposition 2.6. (cid:3)
In this section we provide a balanced 2-neighborly triangulation of the 3-sphere.
Construction 3.1.
Assume that V = { u , u , u , u } , V = { v , v , v , v } , V = { w , w , w , w } and V = { z , z , z , z } are the four color sets of a balanced 3-sphere Γ. We let lk Γ z = A ∪ ∂A ∼ ∂C C and lk Γ z = B ∪ ∂B ∼ ∂C C , where A , B and C are triangulated 2-balls sharing the same boundaryas shown in Figure 2. All possible edges that do not appear in A , B and C are shown in Figure6 u v w u v u w v u w v w u v w u v u w v u w v w u v w u v u w v u w v Figure 2: Discs A , B and C (from left to right) w u v w u v u w v u w v w u v w u v u w v u w v Figure 3: Left: disc D (cid:48) . Right: disc D obtained after rearranging the boundary of D (cid:48) .3 as solid red edges in disc D (cid:48) . Notice that the dashed edges in D (cid:48) are edges in discs A and B ,so we may rearrange the boundary of D by switching the positions of vertices v and v , and thenreplacing the edges containing v or v in ∂D (cid:48) by the dashed edges. In this way, we obtain atriangulation of a 12-gon D as shown in Figure 3. Furthermore, ∂D ⊆ A ∪ B , and ∂D divides thesphere = A ∪ ∂A ∼ ∂B B into two discs A (cid:48) and B (cid:48) as shown in Figure 6.We let lk Γ z = A (cid:48) ∪ ∂A (cid:48) ∼ ∂D D and lk Γ z = B (cid:48) ∪ ∂B (cid:48) ∼ ∂D D . Since both st ∆ z ∩ st ∆ z = C andst ∆ z ∩ (st ∆ z ∪ st ∆ z ) = A (cid:48) are simplicial 2-balls, it follows that Σ = ∪ i =1 st ∆ z i is a simplicial3-ball. Furthermore, the boundary of Σ is exactly lk ∆ z . Hence Γ = Σ ∪ st ∆ z is indeed a balanced2-neighborly 3-sphere. Remark 3.2.
Here we provide some properties of Γ in Construction 3.1.1. ( A ∪ B, C, D ) is an ear decomposition of Γ [3] .2. The automorphism group of Γ has two generators( u u u u )( v z v z )( v z v z )( w w w w ) , ( z v )( z v )( z v )( z v )( u w )( u w )( u w )( u w ) . (The second generator is given by switching vertices of color 1 and 3, and color 2 and 4, butwith the same subscript.) Hence Aut(∆) has 8 elements.3. The complex Γ given in Construction 3.1 is shellable. For lk Γ z = A ∪ ∂A ∼ ∂C C , there existtwo shellings c , . . . , c , a , . . . , a and a (cid:48) , . . . , a (cid:48) , c (cid:48) , . . . , c (cid:48) such that for any 1 ≤ i ≤ u v w u v u w v u w v w u v w u v u w v u w v Figure 4: Left: disc A (cid:48) . Right: disc B (cid:48) . Notice that ∂A (cid:48) = ∂B (cid:48) = ∂D . c i , c (cid:48) i are facets from C and a i , a (cid:48) i are facets from A . Similarly, there exist two shellings c , . . . , c , b , . . . , b and b (cid:48) , . . . , b (cid:48) , c (cid:48) , . . . , c (cid:48) for lk Γ z = B ∪ ∂B ∼ ∂C C , where b i , b (cid:48) i arefacets from B . Then a (cid:48) ∗ z , . . . , a (cid:48) ∗ z , c (cid:48) ∗ z , . . . , c (cid:48) ∗ z , c ∗ z , . . . , c ∗ z , , b ∗ z . . . , b ∗ z gives a shelling of st Γ z ∪ st Γ z . We may extend this shelling into a complete shelling of Γby constructing two similar shellings of lk Γ z and lk Γ z . However, we tried some computertests and failed to prove either polytopality or non-polytopality. Remark 3.3.
It is easy to see that if ∆ is a balanced 2-neighborly ( d − is abalanced 2-neighborly ( d − ∗ ∆ is a balanced 2-neighborly ( d + d − k − k vertices for any k ≥ Question 3.4.
Let d ≥ and m ≥ be arbitrary integers. Is there a balanced 2-neighborlysimplicial ( d − -sphere all of whose color sets have the same size m ? Is there a polytopal spherewith these properties? In this section we present our first construction of a balanced 2-neighborly lens space L (3 ,
1) with16 vertices. We denote it by ∆. Each color set of ∆ has four vertices.
Construction 4.1.
We denote the color sets of ∆ by V = { u , u , u , u } , V = { v , v , v , v } , V = { w , w , w , w } and V = { z , z , z , z } .In Figure 2 we illustrate the construction of the vertex links lk ∆ z i for i = 1 , . . . ,
4. All theselinks are realized as cylinders. Two links lk ∆ z and lk ∆ z share the same top and bottom, whichare triangulated hexagons spanned by vertices { u i , v i , w i : i = 1 , } and { u i , v i , w i : i = 2 , } ,respectively. To construct lk ∆ z from lk ∆ z , we switch the positions of vertices u , v , w withvertices u , v , w respectively and form a new cylinder. The new top and bottom hexagons containthe 2-faces { u , v , w } and { u , v , w } . Similarly, we construct the link lk ∆ z from lk ∆ z byswitching the positions of vertices u , v , w with vertices u , v , w and letting { u , v , w } and8 v w u v w w u v w u v (a) lk ∆ z u v w u v w w u v w u v (b) lk ∆ z u v w u v w w u v w u v (c) lk ∆ z u v w u v w w u v w u v (d) lk ∆ z Figure 5: Four vertex links of ∆ { u , v , w } be the 2-faces that appear in the triangulation of the top and bottom hexagons. Itfollows that lk ∆ z and lk ∆ z also share the same top and bottom.Now since ∆ is balanced 2-neighborly, by our construction, it only remains to show that ∆triangulates the lens space L (3 , ∆ z and st ∆ z are filled cylindersthat share top and bottom. So their union A := st ∆ z ∪ st ∆ z is a filled torus (that is, a genus-1handlebody); so is the union B := st ∆ z ∪ st ∆ z . Note that these two handlebodies have identicalboundary complexes, thus they provide a Heegaard splitting of a lens space.To identify which lens space ∆ triangulates, we need to determine the homeomorphism φ : ∂A → ∂B . Consider two generators γ, δ of π ( A ∩ B ) = π ( ∂A ), where γ is the 6-cycle ( u , v , w , u , v , w )and δ is the 4-cycle ( u , w , u , w ). In particular, δ is also a generator of π ( A ). From theconstruction we see that φ ( γ ) is a loop running around the equator of ∂B thrice and the meridianof ∂B once. Also φ ( δ ) runs around the equator of ∂B twice and the meridian of ∂B once. Henceit is indeed the lens space L (3 , Remark 4.2.
Our construction ∆ has the following properties:1. All vertex links are combinatorially equivalent.2. From Figure 5 we see lk ∆ z i ∩ lk ∆ z j has two connected components when { i, j } = { , } or { , } (they are the top and bottom hexagons as shown in Figure 2); and it has three9onnected components when i ∈ { , } and j ∈ { , } (each component is the union of twofacets along the side of the cylinders). In general, the intersection of two vertex links, wherethe vertices are of the same color, always has at least two connected components.3. There are three group actions on the vertices of ∆:(a) Fix the subscript and rotate the corresponding vertices of color 1, 2 and 3 respectively.The generator is given by ( u v w )( u v w )( u v w ).(b) Rotate vertices of the same color. The generator is( u u u u )( v v v v )( w w w w )( z z z z ) . (c) Exchange lk ∆ z and lk ∆ z , lk ∆ z and lk ∆ z , by exchanging v i and w i (or u i and w i , u i and v i ) for all i ∈ [4]. The generators are ( z z )( z z )( v w )( v w )( v w )( v w ),( z z )( z z )( u w )( u w )( u w )( u w ) and ( z z )( z z )( u v )( u v )( u v )( u v ).The automorphism group of ∆ is of size 96. Proposition 4.3.
The complex ∆ is a balanced vertex minimal triangulation of L (3 , .Proof: By Proposition 6.1 in [9], each color set of ∆ is of size at least 3. If there are exactlythree vertices v , v , v of color 1 in ∆, apply the Mayer-Vietoris sequence on the triple (st ∆ v ∪ st ∆ v , st ∆ v , ∆) and we obtain that0 = H (lk ∆ v ) → H (st ∆ v ∪ st ∆ v ) ⊕ H (st ∆ v ) → H (∆) → H (lk ∆ v ) = 0 . Hence H (st ∆ v ∪ st ∆ v ) ∼ = H (∆) = Z / Z . However, this is impossible since H (st ∆ v ∪ st ∆ v ) ∼ = H (st ∆ v ∩ st ∆ v ), which cannot be Z / Z . (cid:3) The same argument as above also shows that the balanced triangulation of any lens space L ( p, q )with p > In this section our goal is to construct a balanced 3-sphere whose rank-selected subcomplexes donot have ear decompositions. The motivation is from the balanced 2-neighborly construction of L (3 ,
1) in Section 4. Indeed, we want to construct a balanced 3-dimensional complex ∆ so that 1)each vertex link is a 2-sphere; 2) for a fixed color set V = { v , · · · , v k } , the intersection of any twovertex links lk ∆ v i ∩ lk ∆ v j always has at least two connected components (as the property listed inRemark 4.2); and 3) ∪ i =1 st ∆ v i is 3-ball, which together with the condition 1) guarantees that ∆is a 3-sphere.In the following we take k = 5 and give such a construction. Figure 6 illustrates the linkslk ∆ v , · · · , lk ∆ v . Every label represents the color of the vertex. Also each connected componentof lk ∆ v ∩ lk ∆ v is colored in green, lk ∆ v i ∩ lk ∆ v is colored in blue for i = 1 ,
2, and lk ∆ v j ∩ lk ∆ v is colored in pink for j = 1 , ,
3. Immediately we check that all these intersections of vertex linkshave 2 or 3 connected components.Figure 7 shows how ∆ \ W is formed from these links. First we glue lk ∆ v and lk ∆ v alongtwo green triangles. The resulting complex lk ∆ v ∪ lk ∆ v is shown in Figure 7a. Then we place10 (a) lk ∆ v and lk ∆ v
33 131 3 1 3 212 12212 1 2 131 31 (b) lk ∆ v (c) lk ∆ v Figure 6: Four vertex links as triangulated 2-spheres. For simplicity’s sake, we omit some diagonaledges in the quadrilaterals in (b), and some labels of vertices in (c). (a) lk ∆ v ∪ lk ∆ v (b) ∪ i =1 lk ∆ v i (c) ∪ i =1 lk ∆ v Figure 7: how the links are glued together.lk ∆ v on top of lk ∆ v ∪ lk ∆ v . As we see from Figure 7b, the boundary complex of ∪ i =1 st ∆ v i is a triangulated torus. Finally, we place lk ∆ v on top of ∪ i =1 lk ∆ v i so that st ∆ v “covers the1-dimensional hole” in ∪ i =1 st ∆ v i , see Figure 7c. We denote the subspace of R enclosed by lk ∆ v i as S i for 1 ≤ i ≤
4, and let S := ∪ i ≤ S i . From our construction it follows that the boundarycomplex of S is a 2-sphere; we let it be lk ∆ v . Indeed ∆ is a 3-sphere since ∆ is the union of two3-balls S and st ∆ v glued along their common boundary lk ∆ v .Since each lk ∆ v i ∩ lk ∆ v j has at least two connected components for 1 ≤ i (cid:54) = j ≤
4, the Mayer-Vietoris sequence implies that S i ∪ S j is not contractible for all 1 ≤ i (cid:54) = j ≤
4. A similar inspectionof lk ∆ v i ∪ lk ∆ v j ∪ lk ∆ v k also implies that the boundary complexes of S i ∪ S j ∪ S k ’s cannot betriangulated 2-spheres for distinct 1 ≤ i, j, k ≤ Proposition 5.1.
Not all rank-selected subcomplexes of balanced simplicial spheres have ear de-compositions.Proof:
Consider the complex ∆ constructed above. We denote the union of interior faces of a11omplex τ by int τ . Suppose ∆ \ V has an ear decomposition (Γ , Γ , · · · , Γ k ). Since | V | = 5 and β (∆ \ W ) = 4, k must be 4. Notice first that ∪ i ≤ lk ∆ v i divides R into five subspaces, namely, S , · · · , S and the complement of S , each having lk ∆ v i as the boundary complex for 1 ≤ i ≤ ∪ Γ − int(Γ ∩ Γ ) must be a triangulated 2-sphere, by the Jordan theorem, itseparates R into two connected components, hence the bounded component must be either S i ∪ S j or S i ∪ S j ∪ S k for some 1 ≤ i, j, k ≤
4. (We may assume that it is not S i , since otherwise we mayconsider the 2-sphere ∪ i ≤ Γ i − ∪ ≤ i (cid:54) = j ≤ int(Γ i ∩ Γ j ) instead of Γ ∪ Γ − int(Γ ∩ Γ ), where thesubset enclosed by this sphere in R cannot be S i anymore.) This contradicts the fact that theboundaries of S i ∪ S j or S i ∪ S j ∪ S k are not 2-spheres. (cid:3) Remark 5.2.
One can think of all the figures illustrated above as projections of a subcomplex of∆ − st ∆ v onto R . However, we do not know whether the complex provided in this section canbe realized as the boundary of a 4-polytope. Acknowledgements
The author was partially supported by a graduate fellowship from NSF grant DMS-1361423. Ithank Moritz Firsching for pointing out the automorphism groups of the constructions in Section3 and 4 and running some computational tests to decide whether the constructions are polytopal.I also thank Lorenzo Venturello for pointing out mistakes in an earlier version and giving helpfulsuggestions.
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