Betti splitting from a topological point of view

A Betti splitting $I=J+K$ of a monomial ideal $I$ ensures the recovery of the graded Betti numbers of $I$ starting from those of $J,K$ and $J \cap K$. In this paper, we introduce this condition for simplicial complexes, and, by using Alexander duality, we prove that it is equivalent to a recursive splitting conditions on links of some vertices. The adopted point of view enables for relating the existence of a Betti splitting for a simplicial complex $\Delta$ to the topological properties of $\Delta$. Among other results, we prove that orientability for a manifold without boundary is equivalent to admit a Betti splitting induced by the removal of a single facet. Taking advantage of this topological approach, we provide the first example in literature admitting Betti splitting but with characteristic-dependent resolution. Moreover, we introduce the notion of splitting probability, useful to deal with results concerning existence of Betti splitting.


Introduction
A fundamental tool to describe the structure of a homogeneous ideal I in a polynomial ring is given by the minimal graded free resolution of I and, in particular, by its graded Betti numbers β i,j (I). Dealing with ideals of large size, the retrieval of these algebraic invariants can be hard from a computational point of view, also in the case of monomial ideals. A common strategy to obtain the information on I is to decompose it into smaller ideals, in order to recover the invariants of I using the invariants of its pieces. Following this idea, originally introduced in [6] and developed in [8], a Betti splitting of a monomial ideal I consists of a suitable decomposition I = J + K of I ensuring the complete retrieval of the graded Betti numbers of I from the ones of J, K and J ∩ K. The decomposition I = J + K is called a Betti splitting of I if β i,j (I) = β i,j (J) + β i,j (K) + β i−1,j (J ∩ K), for all i, j ∈ N.
In this paper, we formalize this condition for simplicial complexes. Combinatorial structures arise in several concrete applications (for instance computational topology, data analysis and shape recognition). Our aim is to give a strong computational support for researches focused on Betti splitting: we implemented several algorithms in Macaulay2 [9] and Python to check, for instance, if a given decomposition of a simplicial complex is a Betti splitting, see https://github.com/fugacci/Check-Decompositions.git. We think that this can be extremely useful for further developments on this subject.
In several application domains, it is interesting to describe the topology of a geometric realization of a simplicial complex ∆, in particular its homology. The remarkable Hochster's formula relates the graded Betti numbers of the Alexander dual ideal I * ∆ of ∆ and the reduced homology of suitable subcomplexes of ∆; in this way Betti splittings are related to the classical Mayer-Vietoris approach to solve homological problems. Reading the Betti splitting condition for I * ∆ from a purely topological point of view, we investigate topological properties and features of a geometric realization of ∆ inducing a suitable Betti splitting for I * ∆ . In this framework, we introduce the notion of homological splitting for a simplicial complex ∆, see Definition 3.5. It corresponds to a decomposition of I * ∆ for which the previous equation on graded Betti numbers holds for j = n and for every i ∈ N. Algorithm 1 checks if a given decomposition of ∆ is a homological splitting. Using this notion, we are able to prove a complete characterization of Betti splitting for simplicial complexes, pointing out also the intrinsic recursive nature of this tool, see Theorem 3.10. Algorithm 2 checks if a given decomposition of ∆ is a Betti splitting, taking advantage of the characterization given in this result.
Inspired by the case of shellable simplicial complexes, we study homological splittings induced by the removal of a single monomial of I * ∆ , introducing the notion of essential facet of a simplicial complex, see Definition 4.3. If there is at least an essential facet in ∆, then a suitable homological splitting is ensured (Theorem 4.5) and the existence of essential facets of dimension equal to dim(∆) is related to the non-vanishing of top homology (Theorem 4.8). As consequence, a simplicial complex with top-homology always admits a suitable Betti splitting (Corollary 4.11).
In the relevant case of simplicial manifolds without boundary, the homological splitting condition for the removal of a single facet induces a Betti splitting (Theorem 5.7). This yields to the existence of Betti splittings for orientable simplicial manifolds (Corollary 5.8) and for all simplicial manifolds if char(F) = 2 (Corollary 5.9). Moreover we are able to characterize orientability of a manifold in terms of Betti splittings (Remark 5.11).
In Section 6, we consider pathological simplicial complexes that do not admit Betti splitting, depending on the field. We investigate obstructions that prevent such kind of decomposition for these complexes (Proposition 6.2), introducing Algorithm 3. In the Example 6.3, we present the first example in literature of an ideal with characteristicdependent resolution with a Betti splitting over every field. All our result are independent on the chosen triangulation of the considered manifolds. To formalize this statement, we introduce the notion of Betti splitting probability, see Definition 6.4. Several results of the paper can be given in terms of this new notion and it can be the starting point for future investigations, using Algorithm 4.

Preliminaries
Let F be a field, R = F[x 1 , . . . , x n ] be the polynomial ring on n variables with coefficients in F and m = (x 1 , . . . , x n ) its maximal homogeneous ideal. Let M be a finitely generated graded R-module. The minimal graded free resolution of M as R-module is a free resolution of M of the form If M is graded over Z n , we can consider its multigraded resolution and its multigraded Betti numbers β i,a (M ), where a ∈ Z n . Typical examples of multigraded modules are given by monomial ideals. Denote by supp(a) the set {i : a i = 0}.
Given a monomial ideal I ⊆ R, we denote by G(I) the minimal system of monomial generators of I. Definition 2.1. Let I, J and K be monomial ideals such that I = J + K and G(I) is the disjoint union of G(J) and G(K). We say that J + K is a Betti splitting of I over F if This condition is equivalent to the vanishing of some maps between Tor modules, see [8,Proposition 2.1]. In particular, the induced maps Tor i (J ∩ K; F) j → Tor i (J; F) j ⊕ Tor i (K; F) j must be zero, for all i, j ∈ N. Using this condition, it is straightforward to prove that the Betti splitting condition for degrees is equivalent to the Betti splitting condition for multidegrees.
. It is not difficult to see that β 1,4 (J) = 0, since J is a complete intersection. But β 1,4 (I) = 0, since I is an ideal with linear resolution. Then, the considered splitting is not a Betti splitting. Instead, for instance, In some cases, a special kind of splitting is considered for a monomial ideal I. Assume I, J and K are monomial ideals as in Definition 2.1 and K = (m) is generated by a single monomial m. We call the decomposition I = J + K a facet splitting if it is a Betti splitting.
In this paper, we study splitting properties of ideals associated to simplicial complexes. An abstract simplicial complex ∆ on n vertices is a collection of subsets of [n] = {1, . . . , n}, called faces, such that if F ∈ ∆, G ⊆ F , then G ∈ ∆. A simplicial complex ∆ is completely determined by the collection F(∆) of its facets, the maximal faces with respect to inclusion. We denote by F 1 , . . . , F r the simplicial complex determined by the facets F 1 , . . . , F r .
A simplex of dimension k or k-simplex is the convex hull of k + 1 points in general position in R n . Every abstract simplicial complex can be seen as a topological space |∆|: in fact the facets of ∆ can be realized as simplices in a suitable Euclidean real space in such a way that two simplices must intersect in a common subface (possibly the empty face). The dimension dim(∆) of ∆ is the largest dimension of its simplices.
Let k ∈ N. Denote by H k (∆; F) the k th simplicial reduced homology group of ∆ with coefficients in F. It can be proved that the simplicial homology of ∆ is isomorphic to the singular homology of a geometric realization |∆|, see [11]. We denote by β k (∆; F) the dimension as F-vector space of H k (∆; F). Recall that β −1 (∆; F) = 0 if and only if ∆ = {∅}; in this case β −1 (∆; F) = 1.
There are several ways to associate a squarefree monomial ideal to a simplicial complex ∆. In the literature, the most studied ideal is the so-called Stanley-Reisner ideal. In this paper, we are interested in the Alexander dual ideal of ∆, that can be viewed as the Stanley-Reisner ideal of a suitable complex ∆ * (for more details see for instance [12]).
Remark 2.4. Given a squarefree monomial ideal I ⊆ R and fixing the number of variables of R, there is a unique simplicial complex ∆ such that I = I * ∆ .
In [13], M. Hochster proved a remarkable formula that is the main bridge between Combinatorics and Commutative Algebra: it gives a relation between the homology of suitable subcomplexes of a simplicial complex and the graded Betti numbers of its associated ideals. The version of Hochster's formula that we recall here is due to Eagon and Reiner [5].
Given a simplicial complex ∆, we define the link of a face F in ∆ as Notice that link ∆ ∅ = ∆.
Remark 2.6. As mentioned above, Hochster's formula allows to relate explicitly the homology of a simplicial complex ∆ with the graded Betti numbers of I * ∆ . For j = n, we obtain β i,n (I * ∆ ) = β i−1 (∆; F).

Betti splitting for simplicial complexes
Using Alexander dual ideals, it is possible to define the Betti splitting condition for a simplicial complex ∆. The main result of this section (Theorem 3.10) describe this condition recursively, in terms of Betti splitting of suitable links. Before proving it, we need some definitions.
The previous definition is the combinatorial counterpart of the assumption on disjoint minimal systems of generators. Definition 2.1 for squarefree monomial ideals yields the following natural version for simplicial complexes.
Remark 3.4. In the previous definition and in the rest of the paper, the Alexander duals ideals are computed with respect to all the n variables of F[x 1 , · · · , x n ].
The following key definition is useful to state Theorem 3.10.
Remark 3.6. By Theorem 2.5 and Remark 3.2, a homological splitting of ∆ is equivalent to a Betti splitting of I * ∆ for j = n and every i ∈ N. In this way, if ∆ = ∆ 1 ∪ ∆ 2 is a Betti splitting of ∆, then ∆ = ∆ 1 ∪ ∆ 2 is a homological splitting of ∆. In general, the converse does not hold.
In the following example, we show that a homological splitting is not in general a Betti splitting. We will describe a class of simplicial complexes and standard decompositions for which the two notions are equivalent (see Theorem 5.7).
Example 3.7. Let ∆ be the simplicial complex defined by 123, 234, 245, 345 . Clearly ∆ = 123, 245 ∪ 234, 345 is a standard decomposition of ∆. It is easy to see that this is a homological splitting over every field F. But it is not a Betti splitting, since I * ∆ is the ideal considered in Example 2.2 and the splitting above is the algebraic counterpart of the splitting here.
For sake of completeness, in the next result we give a straightforward equivalent condition to homological splitting condition.
Proposition 3.8. Let ∆ be a simplicial complex and ∆ = ∆ 1 ∪ ∆ 2 be a standard decomposition of ∆. Then, the following are equivalent: Proof. Consider the Mayer-Vietoris exact sequence in homology arising from the decomposition ∆ = ∆ 1 ∪ ∆ 2 (for sake of simplicity we omit the field F in the notations): From the long exact sequence above we get To prove that 1. implies 2. we proceed by induction on k ≥ 0.
Assume k ≥ 1 and φ k−1 = 0; we prove that φ k = 0 using an argument similar to the previous one: The next algorithm checks if a given standard decomposition of a simplicial complex is a homological splitting over a given field F. It can be given another version of it using Proposition 3.8.
Finally, we are able to prove the main theorem of this section: a Betti splitting for I * ∆ can be characterized in terms of homological splittings or recursively in terms of Betti splitting of suitable links.
Theorem 3.10. Let ∆ = ∆ 1 ∪ ∆ 2 be a standard decomposition of a simplicial complex ∆. Then, the following statements are equivalent: Since all the maps Tor i (I * Now we prove 2. implies 3. Consider F = ∅. By assumption, we have that ∆ = ∆ 1 ∪ ∆ 2 is a homological splitting of ∆ over F. If ∆ 1 ∩ ∆ 2 = {∅}, we are done. Let v ∈ ∆ 1 ∩ ∆ 2 be a vertex and a ∈ Z n such that supp(a) = [n] \ G for a suitable face G ∈ link ∆ v. If G / ∈ link ∆ 1 v (the same for link ∆ 2 v), it follows that link ∆ v = link ∆ 2 v and we have nothing to prove. We may assume Summing over all a ∈ Z n such that | supp(a)| = j, we obtain the desired splitting formula for j and for every i ∈ N. Finally, we prove 3. implies 1. Since ∆ = ∆ 1 ∪ ∆ 2 is a homological splitting of ∆, for j = n the splitting formula holds. Let j = n and a ∈ Z n , such that supp(a) = [n] \ F , for some face F ∈ ∆ with |F | = n − j. If F ∈ ∆ \ ∆ 1 (the same for ∆ 2 ) we have link ∆ F = link ∆ 2 F . Then, β i,a (I * ∆ ) = β i,a (I * ∆ 2 ) and β i,a (I * ∆ 1 ) = β i−1,a (I * ∆ 1 ∩ I * ∆ 2 ) = 0. Then, we may assume F ∈ ∆ 1 ∩∆ 2 . Let v ∈ F and b ∈ Z n such that supp(b) = [n]\(F \v). By assumption, for every i ∈ N. By Hochster's formula, Applying the previous argument to ∆ 1 , ∆ 2 and ∆ 1 ∩ ∆ 2 , we get Summing over all a ∈ Z n such that | supp(a)| = j we conclude the proof.
Using Theorem 3.10 we are able to present the following algorithm: it checks if a given standard decomposition of a simplicial complex is a Betti splitting over a given field F. It is possible to give another formulation of the algorithm taking advantage of the third condition of the theorem.

Essential facets
In several cases, a nice Betti splitting for a simplicial complex ∆ is given by a standard decomposition obtained removing a single facet from ∆. By Theorem 3.10(3.), we start studying situations for which the standard decomposition ∆ = G ∈ F(∆) : G = F ∪ F is a homological splitting for ∆.
The following result is very useful for our purpose. Given a simplicial complex ∆, we call essential each facet of dimension d of ∆ whose removal only affects the d th homology group of ∆.
The next lemma shows that an essential facet induces a standard decomposition of ∆. Finally, in the next result, we prove that an essential face over F induces a homological splitting over F. Proof. For simplicity, in the proof we omit the field F from the notations. Let F ∈ F(∆) be a d-dimensional essential facet of ∆ with respect to F. Consider the complexes By Lemma 4.4, ∆ 1 = ∆ \ {F } and ∆ = ∆ 1 ∪ ∆ 2 is a standard decomposition of ∆. Since F is a d-dimensional essential facet, we have that Moreover, β k (∆ 2 ) = 0, for k ∈ N.
Remark 4.6. We remark that in the previous result, the dimension d of the essential facet F is not forced to be equal to the dimension of the simplicial complex ∆.
In the next example, we show that in general the converse of  If the dimension of the essential facet equals the dimension of ∆, a stronger statement holds.
Theorem 4.8. Let ∆ be a simplicial complex of dimension d and let F be a field. Then, the following conditions are equivalent: In this case, there exists a facet F ∈ F(∆) such that ∆ = G ∈ F(∆) : G = F ∪ F is a homological splitting of ∆ over F.
Proof. The implication from 2. to 1. is given by Theorem 4.5. Then, we have to prove only that 1. implies 2. Let (∆ i ) i=0,...,M be a collection of simplicial complexes such that: Given such a collection, let j := min{i ∈ {0, · · · , M } : β d (∆ i ; F) = 0}. We set F := F j , the face introduced in ∆ j . Since β d (∆ j ; F) = 0, F is a facet of dimension d in ∆ which belongs to a d-cycle of ∆ j . Then, by Proposition 4.1, F is an essential facet of ∆ over F.
The last claim is an immediate consequence of Theorem 4.5. It is a homological splitting over every field F, by Corollary 3.9, but β 2 (∆; F) = 0. The following corollary is an immediate consequence of Theorem 4.8.
Corollary 4.11. Let ∆ be a simplicial complex of dimension d. If β d (∆; F) = 0, then ∆ admits a homological splitting over F.

Betti splitting of manifolds
In this section, we apply our results to triangulations of manifolds, a large class of simplicial complexes with nice properties. Let us recall briefly some basic definitions.
Definition 5.1. A topological d-manifold M is a Hausdorff space such that every point x ∈ X has a neighbourhood which is homeomorphic to the d-dimensional Euclidean space.
All the manifolds that we consider are closed, i.e. compact and without boundary. A triangulation of a manifold M is a simplicial complex ∆ such that |∆| ∼ = M . In this case, from now on we say that ∆ itself is a d-manifold. We summarize some properties of a d-manifold ∆: • ∆ is pure and strongly connected (i.e. all the facets have the same dimension d and for every two facets F, G there exists a path H 0 = F,

a topological manifold with
the same homology of the sphere, for every field F).
The graded Betti numbers of I * ∆ are essentially known by Hochster's formula, see Remark 5.2. For the case of a manifold ∆, it is interesting to investigate if nice topological properties on ∆ induce nice splitting properties on I * ∆ and how the topology of ∆ may be reflected on the structure of the resolution of I * ∆ .
) is the f-vector of ∆ and f i (∆) denotes the number of i-dimensional faces of ∆. In this case, I * ∆ is generated in degree n − d − 1. The equality above follows immediately from Hochster's formula, since link ∆ (F ) is a (j − 1)dimensional homology sphere if F = ∅ is a face of ∆ of cardinality n − j.
A d-manifold M is called orientable if it has a global consistent choice of orientation. For the formal definition and other details on orientability, we refer to [11,Chapter 3.3]. Examples of orientable manifolds are (homology) spheres, see [11,Corollary 3.28]. The Möbius strip and the Klein bottle are examples of non-orientable spaces.
Poincaré duality Theorem is a fundamental result on the topology of orientable dmanifold. We state it in a classical fashion (for a modern treatment see [11,Theorem 3.30]). Using Poincaré duality, we prove that the facets of an orientable manifold are all essential.
Proposition 5.5. Let F be a field and ∆ be an orientable manifold. Then, any facet F ∈ F(∆) is an essential facet over F. Moreover ∆ = G|G ∈ F(∆), G = F ∪ F is a homological splitting over F.
Proof. Poincaré duality Theorem 5.3 ensures that H d (∆; F) ∼ = H 0 (∆; F) = 0. Furthermore, the geometrical realizations of the d-cycles of H d (∆; F) coincide with the connected components of ∆ itself. So, given any facet F in ∆, F belongs to a d-cycle of ∆. Then, by Theorem 4.8 and Remark 4.10, F is an essential facet of ∆ over F. By Remark 5.4, we can state the following result, the proof of which is similar to the previous one.
Proposition 5.6. Let ∆ be a manifold. Then, any facet F ∈ F(∆) is an essential facet over Z 2 . Moreover ∆ = G|G ∈ F(∆), G = F ∪ F is a homological splitting over Z 2 .
For manifolds, the Betti splitting condition is equivalent to the homological splitting condition.
Theorem 5.7. Let ∆ be a d-manifold and let F be a field. Given a facet F ∈ F(∆), the following statements are equivalent: Proof. Since a Betti splitting is always a homological splitting, it is enough to prove that 1. implies 2. Consider ∆ 1 = G|G ∈ F(∆), G = F and ∆ 2 = F . We prove the claim by induction on the dimension d of the manifold ∆. For d = 0, ∆ is a set of isolated points. In this case, ∆ 1 ∩ ∆ 2 = {∅}, then we have nothing to prove. Assume the implication proved up to dimension k and we prove it for dimension d = k + 1. Let v be a vertex of ∆ 1 ∩ ∆ 2 . Recall that link ∆ v is a k-dimensional homology sphere. Notice that link ∆ 2 v is a facet of link ∆ v. By Proposition 5.5, link ∆ v = link ∆ 1 v ∪ link ∆ 2 v is a homological splitting of link ∆ v over F, since any homology sphere is an orientable manifold. By induction, it is also a Betti splitting. Then, by Theorem 3.10, we have that the homological splitting ∆ = ∆ 1 ∪ ∆ 2 is a Betti splitting of ∆.
By Proposition 5.5 and Proposition 5.6 respectively, the following results are immediate consequences of Theorem 5.7.
Corollary 5.8. Let F be a field and ∆ be an orientable manifold. Then, for each facet F ∈ F(∆), the standard decomposition ∆ = G|G ∈ F(∆), G = F ∪ F is a Betti splitting over F. Corollary 5.9. Let ∆ be a manifold. Then, for each facet F ∈ F(∆), the standard decomposition ∆ = G|G ∈ F(∆), G = F ∪ F is a Betti splitting over Z 2 .
In the following example, we show that Corollary 5.8 does not hold for manifolds with boundary.
Example 5.10. It can be proved that the standard triangulation of Rudin's ball does not admit a Betti splitting obtained removing a single facet; it admits, anyway, another kind of splitting (see [1,Example 4.7]).

Applications
The results of the previous sections describe a large class of discretized topological spaces admitting a Betti splitting. In this section, we present several interesting examples.
Corollary 5.8 ensures us that every triangulation of an orientable manifold, for instance an n-sphere S n , a torus T and a projective plane of odd dimension RP 2n+1 , admits a Betti splitting decomposition over every field F, induced by the removal of any of its top dimensional simplices. Another interesting example is provided by the triangulation ∆ given in [14] of the lens space L(3, 1) (for more details see Example 6.3).
Theorem 4.5 ensures the existence of a homological splitting also for non-manifold simplicial complexes. For instance, consider the mod 3 Moore space M [3] depicted in Figure  1(c). M is a 2-dimensional simplicial complex with β 2 (M ; Z 3 ) = 0. So, by Corollary 4.11, it admits a homological splitting over Z 3 . In this case, all the facets of M induce a homological splitting; it can be easily proved that it is also a Betti splitting. The situation is completely different if char(F) = 3.
In view of Remark 5.11, we know that every triangulation of some relevant non-orientable manifolds, such as a projective space of even dimension RP 2n and the Klein bottle K do not admit Betti splitting or homological splitting when we remove a single facet if char(F) = 2. Indeed, we are able to show that suitable triangulations of these spaces (see Figure 1(a) and Figure 1(b)) do not admit any Betti splitting over a field of characteristic different from 2. For the dunce hat D, see Figure 1(d) the pathology is the same, but over every field. The first author already proved in [1] that the given triangulation of the dunce hat does not admit Betti splitting.
The spaces considered and their topological properties are summarized in Table 1: The notion we are going to introduce allows us to detect a large class of simplicial complexes that do not admit Betti splitting, i.e. every possible standard decomposition is not a Betti splitting.
As an immediate consequence of Definition 3.5, we can state the following proposition.
Proposition 6.2. Let F be a field and let ∆ be a simplicial complex of dimension d > 1 with β d (∆; F) = 0. If ∆ is non-trivially decomposable over F, then ∆ does not admit homological splitting, and so, Betti splitting.
Thanks to the proposition above, given a simplicial complex ∆ of dimension d > 1 with β d (∆; F) = 0 and a field F, to check if ∆ is non-trivially decomposable is enough to ensure that ∆ does not admit any homological and Betti splitting. The property of being non-trivially decomposable can be algorithmically checked by performing the following algorithm.

Algorithm 3 isN onT riviallyDecomposable(∆, F)
Input: F, a field Input: ∆, a simplicial complex of dimension d Compute all the possible standard decomposition ∆ 1 ∪ ∆ 2 of ∆ for each standard decomposition A version of Algorithm 3 for 2-dimensional simplicial complexes has been developed and implemented in Python. The source code of this tool and of the other algorithms described in the paper can be found in https://github.com/fugacci/Check-Decompositions.git. In this algorithm, we take advantage of the fact that ∆ 1 ∩ ∆ 2 has dimension 1, i.e. it is a graph.
Using Proposition 6.2 and Algorithm 3, we prove that a homological splitting, and so a Betti splitting, is not available for several simplicial complexes ∆, considering fields F for which β 2 (∆; F) = 0 and proving that they are not trivially-decomposable. The considered spaces are depicted in Figure 1: the real projective plane RP 2 , the Klein bottle K, the mod 3 Moore space M and the Dunce hat D. Table 2 focus on simplicial complexes which have been detected as non-trivially decomposable. For each simplicial complex ∆, n V denotes the number of vertices of the chosen triangulation. In the third column, the number of standard decompositions that have to be checked is showed, while the last column shows the required time (in seconds) to perform the entire computation and to check if ∆ is non-trivially decomposable. Example 6.3. Consider the triangulation ∆ given in [14] of the lens space L(3, 1) (for more details on this space see [11]). It is an orientable 3-manifold. Then, by Corollary 5.8 it admits Betti splitting induced by the removal of any of its facets. The Alexander dual ideal I * ∆ is the first example in literature of an ideal with characteristic dependent resolution admitting Betti splitting over every field. Proposition 6.2 and the structure of the spaces considered, suggests that the pathology of some examples of this section does not depend on the chosen triangulation. To formalize this statement we introduce the following definition. Definition 6.4. Let ∆ be a simplicial complex. Denote by S ∆ and B ∆ the collection of standard decompositions of ∆ and the collection of these decompositions that are Betti splitting, respectively. We can define the Betti splitting probability of ∆ over F as follows: For a topological space X admitting a triangulation, we can define the best Betti splitting probability over F: P Betti (X; F) := sup{P Betti (∆) : ∆ is a triangulation of X}.
Analogously we can define the homological splitting probability P Hom (∆; F) of ∆ over F. Clearly P Betti (∆) ≤ P Hom (∆). The next algorithm computes easily the Betti splitting probability for a simplicial complex ∆ over a field F. In this paper, we proved that P Hom (∆) = P Betti (∆), if ∆ is the triangulation of a manifold and that P Betti (∆) > 0 if ∆ is orientable. Moreover we showed, for instance, that for the given triangulation of the Klein bottle, P Betti (K; F) = P Hom (K; F) = 0, if char(F) = 2.
Focusing the attention only on the standard decompositions induced by the removal of a single facet, we define the facet splitting probability of ∆: where B F (∆) is the collection of standard decompositions of ∆ induced by the removal of a single facet that are Betti splitting.
Remark 6.5. Let ∆ be a d-manifold. By Remark 5.11, we proved that ∆ is orientable if and only if P F acet (∆; F) = 0 for every field F. In this case, P F acet (∆; F) = 1. Moreover we have that if ∆ is not orientable, then P F acet (∆; F) = 0 if and only if char(F) = 2. Also in this case P F acet (∆; F) = 1.
We are now able to state precise problems: Problem 1: Let X be a non-orientable d-manifold without boundary. Is it true that P Betti (X, F) = 0, if F is a field with char(F) = 2? Is it true that every triangulation of X is non-trivially decomposable? Problem 2: Let X be the mod 3 Moore space. Is it true that P Betti (X, F) = 0, if F is a field with char(F) = 3? Problem 3: Let X be the dunce hat. Is it true that P Betti (X, F) = 0, for every field F?