Balanced triangulations on few vertices and an implementation of crossflips
Abstract.
A dimensional simplicial complex is balanced if the underlying graph is colorable. We present an implementation of crossflips, a set of local moves introduced by Izmestiev, Klee and Novik which connect any two PLhomeomorphic balanced combinatorial manifolds. As a result we exhibit a vertex minimal balanced triangulation of the real projective plane, of the dunce hat and of the real projective space, as well as several balanced triangulations of surfaces and 3manifolds on few vertices. In particular we construct small balanced triangulations of the 3sphere that are nonshellable and shellable but not vertex decomposable.
Key words and phrases:
simplicial complex, combinatorial manifolds, balanced, crossflips2010 Mathematics Subject Classification:
05E45, 57Q15, 52B051. Introduction
The study of the number of faces in each dimension that a triangulation of a manifold can have is a very classical and hard problem in combinatorial topology. Even for the case of triangulated spheres the characterization of these numbers is a celebrated unsolved problem, known as the conjecture. On an apparently simpler level one can ask the following question: what is the minimum number of vertices needed to triangulate a manifold ? Again the picture is far from being complete. An interesting tool to approach these and other kind of problems are bistellar flips, a finite set of local moves which preserves the PLhomeomorphism type, and suffices to connect any two combinatorial triangulations of a given manifold (equivalently, triangulations of PLmanifolds).
Björner and Lutz [BL00] designed a computer program called BISTELLAR, employing bistellar flips in order to obtain triangulations on few vertices and heuristically recognize the homeomorphism type. This tool led to a significant number of small or even vertexminimal triangulations which are listed in The Manifold Page [Lut], along with many other interesting examples (see also [BL14] and [BL00]).
In this article we focus on the family of balanced simplicial complexes, i.e., dimensional complexes whose underlying graph is colorable, in the classical graph theoretic sense. Many questions and results in face enumeration have balanced analogs (see for instance [IKN17, JKM18, JKMNS18, KN16]. In particular we can ask the following question: what is the minimum number of vertices that a balanced triangulation of a manifold can have?
Izmestiev, Klee and Novik introduced a finite set of local moves called crossflips, which preserves balancedness, the PLhomeomorphism type, and suffices to connect any two balanced combinatorial triangulations of a manifold. We provide primitive computer program implemented in Sage [The17] to search the set of balanced triangulations of a manifold and we obtain the following results:

We find balanced triangulations of surfaces on few vertices. In particular we describe the unique vertex minimal balanced triangulation of on 9 vertices.

We find a balanced triangulation of the dunce hat on vertices. Section 4.2 is devoted to the proof of its vertexminimality.

In Section 5 we discuss balanced triangulations of manifolds on few vertices. In particular we exhibit a vertex minimal balanced triangulation of on 16 vertices with interesting symmetries, and triangulations of the connected sums and that belong to the balanced Walkup class.

Finally in Section 5.4 we construct balanced 3spheres on few vertices that are nonshellable and shellable but not vertex decomposable, using results in knot theory.
The source code and the list of facets of all simplicial complexes appearing in this paper are made available in [Ven].
2. Preliminaries
An abstract simplicial complex on is a collection of elements of that is closed under inclusion. The elements are called faces, and those that are maximal w.r.t. inclusion are called facets. A simplicial complex is uniquely determined by its facets: for elements we define the complex generated by as
A subcomplex of is any simplicial complex , and such a subcomplex on is induced if for every subset such that then . The dimension of a face is the integer , and the dimension of a simplicial complex is the maximal dimension of its facets. dimensional and dimensional faces are called vertices and edges, and faces that are maximal w.r.t. inclusion are called facets. A complex whose facets all have the same dimension is called pure. We denote with the number of faces of of dimension , and we collect them together in the vector . Given two simplicial complexes and , their join is defined to be
In particular, for two vertices the operations and are respectively the cone and the suspension over . To every face we associate the two simplicial complexes
and
called the link and the star of at , which describe the ”local” properties of . There is a canonical way to associate to an abstract simplicial complex a topological space, denoted by , and via this correspondence the terminology and the operations defined above represent discrete analogs of well known tools from classical topology. On the other hand, given a topological space , a triangulation of is any simplicial complex such that . For example the complex is a standard triangulation of the sphere . More specifically we define the following.
Definition 2.1.
A pure dimensional simplicial complex is a combinatorial sphere if is PLhomeomorphic to . A pure connected dimensional simplicial complex is a (closed) combinatorial manifold if the link of each vertex is a combinatorial sphere.
Even though there is a subtle difference between the class of combinatorial manifolds and that of simplicial complexes homeomorphic to a manifold (often called simplicial manifolds), they are known to coincide for (see Section 5.2 for more details in the case ). For a fixed field a relaxation of the above definitions is given by the class of homology manifolds, that is pure dimensional simplicial complexes such that holds for every nonempty face and for every (i.e., is a homology sphere).
In this paper we study a family of complexes with an additional combinatorial property, introduced by Stanley in [Sta79].
Definition 2.2.
A dimensional simplicial complex on is balanced if there is a map , such that for every .
In words is balanced if the graph given by its vertices and edges is colorable in the classical graph theoretic sense, therefore we often refer to the elements in as colors, and to the preimage of a color as color class. Although a priori the map is part of the data defining a balanced complex, in all the examples considered in this paper is unique up to permutations of the colors. We turn our attention to balanced triangulations of interesting topological spaces. As a guide example we consider the dimensional complex
(2.1) 
on the set . This is indeed a balanced vertex minimal triangulation of , and it is in particular isomorphic to the boundary of the dimensional crosspolytope. In general it is possible to turn any triangulation of a topological space into a balanced one by considering its barycentric subdivision , defined as
Indeed more generally for the order complex of a ranked poset, the rank function gives a coloring which provides balancedness (see [Sta79]). Among the many results on face enumeration that have been recently proved to have balanced analogs we focus on a work of Izmestiev, Klee and Novik [IKN17], which specializes the theory of bistellar flips to the balanced setting. In their work the following operation preserving balancedness is defined.
Definition 2.3.
Let be a pure dimensional simplicial complexes and let be an induced subcomplex that is a ball and that is isomorphic to a subcomplex of . The operation
is called a crossflip on .
In [IKN17] the authors require the subcomplexes and to be shellable (see Section 5.4 for a definition), but since all the subcomplexes we consider in our implementation satisfy this condition we do not include it in Definition 2.3. We now describe an interesting family of subcomplexes of : for define
and let , for every . It is not hard to see that those complexes are indeed shellable subcomplexes of the boundary of the dimensional crosspolytope in 2.1. A crossflip replacing a subcomplex with its complement (note that this family is closed under taking complement w.r.t. ) is called a basic crossflip. The basic crossflip replacing with is referred to as trivial flip, because it clearly does not affect the combinatorics, and every nontrivial basic crossflips either increases or decreases the number of vertices. We refer to the former as upflips and to the latter as downflips. Moreover two distinct sets might lead to isomorphic subcomplexes , and certain basic crossflips can be generated (i.e. written as combination) by some others. As an example, the flips in the second line of Figure 1 (we count the arrows separately) can be obtained via a combination of the four moves in the first row. These issues, as well as a description of the possible vectors of the complexes , have been studied in [JV18].
Theorem 2.4 ([Jv18]).
There are precisely non isomorphic nontrivial basic crossflips in dimension . Moreover of them suffice to generate them all.
The case of surfaces has been also studied in [MS18]. The interest in crossflips, and in particular in basic crossflips, is due to the following result.
Theorem 2.5 ([Ikn17]).
Let and be balanced combinatorial manifolds. Then the following conditions are equivalent:

and are PLhomeomorphic;

and are connected by a sequence of crossflips;

and are connected by a sequence of basic crossflips.
Essentially Theorem 2.5 states that any two balanced PLhomeomorphic combinatorial manifolds can be transformed one into the other by a sequence of a finite number of flips. This serves as a motivation to develop an implementation of this moves, as it was done in the setting of bistellar flips by Björner and Lutz in [BL00] with BISTELLAR. In particular our goal is to find balanced triangulations of a given manifold on few vertices, since taking barycentric subdivision typically leads to large complexes.
Remark 2.6.
We conclude this section by offering a way to visualize the results above. Consider a graph whose vertices are all the balanced combinatorial triangulation of a certain manifold , and whose edges are basic crossflips. We call this graph the crossflip graph of , and observe that Theorem 2.5 states that this graph is connected. Furthermore we can associate to another connected graph on the same vertex set, but with edges the sufficient flips guaranteed in Theorem 2.4. We call this graph the reduced crossflip graph. Figure 2 shows a plot of a subgraph of the reduced crossflip graph, displayed by ranking the (all nonisomorphic) complexes according to the number of vertices (see the numbers on the left). Here we stop performing upflips on a sphere if . Note that there is no guarantee of enumerating all the balanced spheres in this way: as an example all the spheres in Figure 2 with satisfy , whereas in [Zhe16a] a balanced sphere on vertices with edges is constructed.
3. The implementation
The main purpose of our implementation is to obtain small, possibly vertex minimal, balanced triangulations of surfaces and 3manifolds. To achieve this we start from the barycentric subdivision of a nonbalanced triangulation, many of which can be found in [Lut], and reduce them using crossflips. We first establish some notations: a vertex is called removable if there exists a down flip such that . A balanced simplicial complex without removable vertices is called irreducible. In Figure 2 irreducible triangulations of can be visualized as vertices not connected with any lower vertex.
Remark 3.1.
While a vertex minimal balanced triangulation is clearly irreducible, the converse is not true. Indeed irreducible triangulations are quite many, and they can have a large vertex set, as shown in Corollary 3.3.
Lemma 3.2.
Let be a pure dimensional balanced simplicial complex. If a vertex is removable then .
Proof.
If the vertex is removable then there exists an induced subcomplex that is isomorphic to a subcomplex of , such that is a ball and is in the interior of , because vertices on the boundary are preserved. Since the link of a vertex in the interior of a balanced ball is a balanced sphere, and the only such subcomplex of is isomorphic to it follows that , hence . ∎
Corollary 3.3.
Let be a combinatorial manifold, with . Then the barycentric subdivision is irreducible.
Proof.
For every vertex , corresponding to a face we have . Moreover, since is a combinatorial sphere, it has at least faces. Hence
For a fixed the last expression is minimized when , and in that case we have , which is strictly larger than , for . ∎
Note that Corollary 3.3 holds for more general classes, e.g., homology manifolds, for which the inequality still holds for every face and for every . The computation above also shows that for every vertex arising from the subdivision of an edge has degree , hence it is potentially removable. Since the barycentric subdivision is the standard way of turning any triangulation into a balanced triangulation of the same space, for the result above represents a bad news. Indeed Corollary 3.3 states that to reduce such a subdivision we are forced to start with some upflips and to increase the number of vertices, which for the case of barycentric subdivisions is typically quite large. Our code presents two main challenges:

List all the flippable subcomplexes of any type;

Decide which type of move to apply and which subcomplex to flip.
Already in dimension 1 the problem of deciding if a fixed complex has a subcomplex isomorphic to a given one, known as the subgraph isomorphism problem, is NPcomplete. However since graphs are computationally well studied it is convenient to reduce the problem to the one dimensional case, to employ structures and algorithms designed for graphs. We say that a pure strongly connected dimensional simplicial complex is a pseudomanifold if every face is contained in exactly two facets.
Definition 3.4.
For a pure dimensional pseudomanifold the dual graph is the graph on the vertex set and with edge set .
Given a dimensional pseudomanifold and a pure subcomplex that is a ball, we list all subgraphs of that are isomorphic to using an algorithm such as the VF2 algorithm [CFSV04], from which we only keep those that correspond to induced subcomplexes. Moreover once a flip is performed we do not need to rerun the check on the entire complex to list all the flippable subcomplexes of , but it suffices to update the list locally, by considering only the induced subcomplexes of that are not induced subcomplexes of . Even though this idea allows to deal with relatively large dimensional complexes, higher dimensions appear to be still out of reach.
For the second problem, namely to decide which subcomplex to flip, we propose and combine two very naive strategies:
given a balanced pseudomanifold we choose a subcomplex among those which

maximize ,

maximize .
With the first condition we simply maximize the number of potentially removable vertices, while maximizing the sum of squares of the vertex degrees forces the new triangulation to have an inhomogeneous degree distribution, and hence some very poorly connected vertices.
Remark 3.5.
Typically, starting from a large triangulation, we cannot hope to reduce drastically the number of vertices through a sequence consisting only of downflips, because irreducible triangulations are quite frequent. Even a restricted example like Figure 2 reveals several irreducible triangulations of on few vertices. We can overcome this inconvenience by interposing a certain number of random upflips to avoid local minima.
Remark 3.6.
We make no claim of efficiency, and we do not take into account the time needed to reduce a triangulation. Undoubtedly many details in the implementation can be improved, and the strategies refined. So far we were able to obtain small balanced vectors of all the dimensional examples considered.
4. Surfaces and the Dunce Hat
The first complexes we consider are triangulations of compact manifolds. In this case the number of vertices uniquely determines the remaining entries of the vector. In Table 2 we display a list of minimal known vectors of several surfaces, as well as the vectors of the corresponding barycentric subdivisions, which is always the starting input for our procedure. Finally in the fourth column we report the smallest known vectors of balanced triangulations found via the program.
4.1. Real projective plane
We found a unique vertex minimal balanced triangulation of the real projective plane, which is depicted in two ways in Figure 3. The vector is . The non balanced minimal one has 6 vertices.
Lemma 4.1.
The simplicial complex is a vertex (hence every ) minimal balanced triangulation of the projective plane.
Proof.
The claim follows from a result of Klee and Novik ([[KN16], Proposition 6.1]) which states that any balanced triangulation of an homology manifold that is not an homology sphere has at least three vertices in each color class. We briefly recall their argument. Assume there exists a color (say 1) whose class contains less than three vertices: if there is only one vertex colored with 1 then is a cone, since every facet contains one vertex per color, and hence contractible. In the same way, since every face colored with is connected to exactly 2 vertices of color 1, if has precisely two of color 1 is 2, say and , then is a suspension over the homology sphere . Hence is an homology sphere. ∎
Remark 4.2.
Since triangulated surfaces on 9 vertices are listed in [Lut], we can check that is indeed the unique balanced triangulation of on 9 vertices.
4.2. Dunce hat
The dunce hat is a topological space which exhibits interesting properties: it is contractible but noncollapsible, and its triangulations are CohenMacaulay over any field but not shellable. For the definition of shellability we refer to Section 5.4, while we avoid defining CohenMacaulay simplicial complexes here, since they go beyond the aim of this article. The interested reader can find an extensive treatment of this topic in [Sta96]. The dunce hat can be visualized as a triangular disk whose edges are identified via a noncoherent orientation. Surprisingly, even if it is possible to construct a balanced triangulation by allowing only 3 vertices to be on the boundary of such disk, the vertex minimal one, which is depicted in Figure 4, is achieved when the singularity contains 4 vertices. Its vector is . In the rest of this section we prove that this is indeed the least number of vertices that a balanced triangulation of the dunce hat can have. We call the singularity of a triangulation of the dunce hat the dimensional subcomplex of faces whose link is not a sphere. Note that the link of edges in the singularity consists of three isolated vertices. The set of faces whose link is a sphere is called the interior of the dunce hat, and it coincides with the interior of the triangular disk.
Since the dunce hat is not a manifold the number of vertices of a triangulation does not uniquely determine the other face numbers, but the number of vertices involved in the singularity also plays a role. If we let be this number, then the vector of any triangulation of the dunce hat satisfies the following equations:
(4.1) 
In particular it holds that . We proceed now with a sequence of lemmas leading to Proposition 4.9, proving that the triangulation in Figure 4 is indeed a balanced vertex minimal triangulation of the dunce hat.
Lemma 4.3.
Let be a balanced dimensional CohenMacaulay complex that is not shellable. Then each color class contains at least two vertices.
Proof.
Assume there exists a color class containing only one vertex . As discussed in Lemma 4.1 it follows that is a cone over the dimensional subcomplex . It is well known that the links of a CohenMacaulay simplicial complex are CohenMacaulay (see e.g., [Sta96]). But since every dimensional CohenMacaulay complex is shellable, and coning preserves shellability this implies that is shellable. ∎
Lemma 4.4.
Let be a balanced dimensional CohenMacaulay complex that is not shellable. Assume moreover that every edge of is contained in at least two triangles. Then each color class contains at least three vertices.
Proof.
By Lemma 4.3 we can assume that there are only two vertices of color 1, say and . Let be the subcomplex generated by all faces of not containing color . Since we assumed that every edge of is contained in at least two triangles it follows that every edge of is contained in the two triangles and . This implies that is obtained taking the suspension of , and hence . In particular is CohenMacaulay and dimensional, hence shellable. We deduce that is the suspension over the shellable complex and hence shellable, since suspension preserves shellability. ∎
Lemma 4.5.
If then .
Proof.
Observe that by Lemma 4.4 we need at least 9 vertices to triangulate the dunce hat in a balanced way, and the only possible configuration is , where is the number of vertices of color . Moreover note that in this case the singularity consists of one vertex per color class. Let us consider the edge in the singularity containing the colors and . Since there are three copies of in the boundary of the disk, each of which needs to be completed to a triangle using an interior vertex of color , we infer that , because another vertex of color already lies in the singularity. ∎
Remark 4.6.
By bichromatic missing edge we mean a pair of vertices , such that , and .
Lemma 4.7.
Let be a balanced triangulation of the dunce hat. Then . Let be the number of bichromatic missing edges. If then .
Proof.
For any balanced dimensional simplicial complex with vertices of color , for , the number of edges is clearly bounded by above by the number of edges of the complete partite graph , which equals . Combined with Lemma 4.5, which allows us to assume , this yields
(4.2) 
where the last inequality follows by maximizing the function , under the constraint . Solving the inequality for we obtain
By Lemma 4.5 we can assume , from which it follows that . The second statement follows in the same way by imposing in the first inequality, which yields
Under the assumption we obtain . ∎
In order to show that the minimum number of vertices for a balanced triangulation of the dunce hat is 11 it remains to show that no such simplicial complex exists with and .
Lemma 4.8.
No balanced triangulation of the dunce hat on 10 vertices exists.
Proof.
Since any triangulation of the dunce hat is CohenMacaulay, nonshellable, and has the property that every edge is contained in two or three triangles, Lemma 4.3 and Lemma 4.4 imply that every color class of contains at least three vertices. Assume a balanced triangulation on 10 vertices exists. There is a unique way to partition 10 vertices in three classes, such that every class contains at least three, namely , where is the number of vertices of color . Moreover, due to Lemma 4.7, we can assume that . In what follows we denote with the number of vertices in the singularity of color . Note the following facts:

If then there are at least four missing bichromatic edges.

Since the singularity is a triangle it must be colored using all three colors. This implies that , and hence there are at least four interior vertices of color or . Denote with one of these vertices and assume . Since the link of is a polygon with an even number of vertices, but there are 7 remaining vertices of color and ( and respectively), then there exists a vertex of color such that . We obtain in this way a bichromatic missing edge for each of the four interior vertices of color and .


If then there are at least three missing bichromatic edges.

If then there are at least three vertices of color and in the interior of the disk and the link of these vertices is a polygon with an even number of vertices. Let be one of these vertices, and assume w.l.o.g. that , where is the coloring map of . Since only three of the vertices colored with can appear in the link of . Hence there is at least one missing bichromatic edge for each of the three vertices.

If then there are exactly two vertices of color and (say and ) in the interior of and their link is an even polygon. Again since each of these two vertices avoids at least one vertex of color 3 and there is at least one missing bichromatic edge for each of the two vertices. Let us denote with and these missing edges. If then since is in the interior and since the link of contains at most two vertices of color , it can only contain two vertices of color . Hence there is at least a third bichromatic missing edge . If then the link of is the whole singularity (a square) and, in particular, there exists an edge in the interior of the disk whose endpoints are in the singularity. This yields a contradiction, because in the case considered two vertices in the singularity are either connected by an edge in the boundary of the disk, or they have the same color.


If then there are at least two missing bichromatic edges.

Since the singularity is a gon it must be colored using all the three color classes. This implies that , and hence there are at least two interior vertices of color or . As in the previous paragraph each of these vertices must avoid at least one vertex of color , giving rise to two bichromatic missing edges.


If then there is at least one missing bichromatic edge.

If then the link of every interior vertex is the whole singularity, hence is the join of a triangulation of with isolated vertices. This is clearly a contradiction.

If then there is at least one interior vertex of color or . Once more its link cannot contain all the remaining vertices of a different color, so it must miss at least one vertex from the color class . This produces a bichromatic missing edge.

If we let be the number of bichromatic missing edges, then the four claims above imply for any . We conclude using Lemma 4.7. ∎
Proposition 4.9.
The simplicial complex in Figure 4 is a vertex minimal balanced triangulation of the dunce hat.
5. 3manifolds
In this section we report some interesting and small balanced triangulations of manifolds found using our computer program.
5.1. Real projective space
We present a peculiar balanced triangulation of the real projective space on 16 vertices. An interesting feature of this complex is its strong symmetry: it is centrally symmetric (i.e., there is a free involution acting) and all the vertex links are isomorphic to the sphere in Figure 5. Since the projective space is homeomorphic to the lens space , a particular case of the following result of Zheng shows that our triangulation is vertex minimal.
Proposition 5.1.
[Zhe16a, Proposition 4.3] Any balanced triangulation of the lens space , with , has at least 16 vertices.
5.2. Small balanced triangulations of manifolds
In Table 4 we report the smallest known vectors of balanced triangulations of several manifolds. We point out that some of these triangulations were previously known, and they are referenced through this section. For instance Klee and Novik [KN16] proved the existence of a dimensional simplicial complex on vertices which provides a vertex minimal balanced triangulation of when is even, and of the twisted bundle when is odd. Moreover they construct balanced triangulations of both and on vertices. This, combined with a result of Zheng ([Zhe16b]) proving that any balanced triangulation of a manifold with and has at least vertices, shows that the second and third line in Table 4 correspond to vertex minimal balanced triangulations. As previously discussed, via Proposition 5.1 we can conclude that also and a triangulation of the lens space constructed in [Zhe16a] are balanced vertex minimal. Most notably the mentioned triangulations of ( even), ( odd) and of are balanced neighborly, that is they do not have bichromatic missing edges.
In the rest of the table we report the minimal balanced vectors achieved for different manifolds, such as several lens spaces , connected sums, two additional spherical 3manifolds called the octahedral space and the cube space, and the Poincaré homology sphere. For a more extensive treatment of this topic we refer to [Lut99]. A classical theorem in topology by Edwards and Cannon states that the fold suspension of any homology sphere is homeomorphic to , even though it is not a combinatorial sphere. Since balancedness is preserved by taking suspension we obtain a family of noncombinatorial balanced triangulations of , for . For we report the smallest vector known for a noncombinatorial balanced sphere.
Corollary 5.2.
There exists a balanced noncombinatorial sphere with vector . Moreover by taking further suspensions we obtain balanced noncombinatorial spheres on vertices, for every .
Remark 5.3.
As it was pointed out in [BL00] there exists a procedure introduced by Datta to construct the suspension by increasing the number of vertices only by one. Unfortunately this one point suspension does not preserve balancedness.
5.3. The connected sum of bundles over and the balanced Walkup class
The lower bound theorem for manifolds (actually true for normal pseudomanifolds, see [Mur15]) gives a bound for the number of edges of a triangulation of an homology manifold with a certain number of vertices, depending of . It is an interesting refinement of the lower bound theorem for homology spheres, obtained from the study of algebraic invariants of Buchsbaum graded rings. JuhnkeKubitzke, Murai, Novik and Sawaske proved a balanced analog of this bound (see [JKMNS18]), and established a conjecture of Klee and Novik [KN16, Conjecture 4.14] for the characterization of the case of equality, when the dimension is greater or equal to . Let and be pure balanced simplicial complexes of the same dimension on disjoint vertex sets, let be two facets of and respectively and let be a bijection. Then the connected sum is the simplicial complex obtained from and identifying and , for every . Let be a balanced simplicial complex, two facets of and a bijection such that and , for all . We say that the simplicial complex obtained from removing and and identifying with is a balanced handle addition. Note that this operations preserves balancedness, as well as the property of being an homology manifold. We define the balanced Walkup class as the set of all balanced simplicial complexes obtained from by successively applying the operations of connected sums with and balanced handle additions. In particular the set of balanced spheres on vertices obtained via connected sums of copies of , called crosspolytopal stacked spheres, is a subset of the balanced Walkup class.
Theorem 5.4.
[JKMNS18] Let be a connected dimensional balanced homology manifold, with . Then
(5.1) 
Moreover if equality holds if and only if is in the balanced Walkup class.
Theorem 5.4 leaves unsolved the case of equality when , which is still part of Conjecture 4.14 in [KN16].
Conjecture 5.5 ([Kn16]).
Let be a connected dimensional balanced homology manifold. Then if and only if is in the balanced Walkup class.
Using our computer program we found two balanced triangulation of and respectively with vector . Since , it is easy to see that both triangulations attain equality in Equation 5.1. In light of creftypecap 5.5 it is natural to ask if these two simplicial complexes belong to the balanced Walkup class. We answer positively by giving an explicit decomposition. In what follows we denote with the boundary of the crosspolytope on the vertex set , such that is not an edge for any .

:

As it was pointed out in the first paragraph of Section 5.3 there is a balanced simplicial complex on vertices which triangulates . It can be obtained from three copies of , namely , and , via two connected sums and one handle addition, identifying the vertices and (see [KN16]). Clearly it belongs to in .

For any facet of we can take the connected sum with and , again identifying , with and with .

Finally we can choose any other facet of such that and such that the distance from and (measured on the dual graph) is even, and perform balanced handle addition identifying the vertices with . Since the vertices in the link of are a subset of and , we conclude that the links of and do not intersect, and hence the handle addition is well defined. Note that the last choice can produce nonisomorphic triangulations of , all of which sit inside .


:

With a similar construction as in the previous case we can triangulate the orientable bundle with four copies of , namely , , and . Here we perform three connected sums and a balanced handle addition identifying vertices , and . We denote this simplicial complex on vertices with (note that it is not the vertex minimal balanced triangulation of ).

We now pick the facets and of , and we observe that the link of any vertex in (respectively
