AA NEW FAMILY OF TRIANGULATIONS OF R P d LORENZO VENTURELLO AND HAILUN ZHENG
Abstract.
We construct a family of PL triangulations of the d -dimensional real projectivespace R P d on Θ (( +√ ) d + ) vertices for every d ≥
1. This improves a construction due toK¨uhnel on 2 d + − Introduction and main results
Triangulations of topological spaces play an important role in many areas of mathematics,from the more theoretical to the applied ones. A classical problem in PL topology asks for theminimum number of vertices that a simplicial complex with a certain geometric realizationcan have. This invariant depends on the underlying topological space: triangulable spaceswith complicated homology or homotopy groups tend to need more vertices in their vertex-minimal triangulations. However, to determine this number is in general very hard, even ifwe restrict our discussion to manifolds or
PL manifolds . One of the few families for whichthis number is known is that of sphere bundles over the circle. K¨uhnel [K¨uh86] showedthat the boundary complex of the ( d + ) -vertex stacked ( d + ) -manifold whose facet-ridgegraph is a cycle is a PL triangulation of S d − × S for even d , and a PL triangulation ofthe twisted bundle S d − " S for odd d . Furthermore, K¨uhnel’s triangulations are vertex-minimal. In the remaining cases, namely when d is odd and the bundle is orientable, orwhen d is even and the bundle is non-orientable, the minimum number of vertices is 2 d + d [BD08, CSS08]. Novik and Swartz [NS09] (in the orientablecase) and later Murai [Mur15] proved that the number of vertices f ( ∆ ) of a k -homology d -manifold ∆ must satisfy ( f ( ∆ )− d − ) ≥ ( d + ) ̃ β ( ∆; k ) , for d ≥ k . Equalityin this formula is attained precisely by members in the Walkup class which are moreover2-neighborly, i.e, their graphs are complete. In low dimensions computational methods area precious source. Using the program BISTELLAR [BL00] Bj¨orner and Lutz constructedseveral vertex-minimal triangulations of 3- and 4-dimensional manifolds, which are collectedin [Lut99].For a manifold whose homology (computed with coefficients in Z ) has a nontrivial torsionpart, the Novik-Swartz-Murai lower bound is far from being tight. In this article we focuson the triangulations of the real projective space R P d . Arnoux and Marin [AM91] gave alower bound on the number of vertices of a triangulation of R P d . Date : July 6, 2020.2010
Mathematics Subject Classification.
Key words and phrases.
Real projective space, triangulations, PL manifolds. a r X i v : . [ m a t h . C O ] J u l heorem 1.1. [AM91] Let ∆ be a triangulation of R P d , with d ≥ . Then f ( ∆ ) ≥ ( d + ) + . It is well known that there is a unique vertex-minimal triangulation of R P on 6 vertices,and Walkup [Wal70] proved that the number of vertices needed to triangulate R P is atleast 11, and constructed one such complex. More recently, Sulanke and Lutz obtained anenumeration of manifolds on 11 vertices, revealing 30 non-isomorphic such triangulations,exhibiting 5 different f -vectors (see [SL09, Table 10]). This also shows that Walkup’s con-struction is the unique f -vectorwise minimal triangulation of R P . Computer search revealeda vertex-minimal triangulation of R P on 16 vertices. This highly symmetric simplicial com-plex was studied by Balagopalan [Bal17] who described three different ways to construct it.Even though the 3- and 4-dimensional cases suggest the formula in Theorem 1.1 is tight in d = ,
4, for d = R P on less than 24vertices. It is now very tempting to conjecture a tight lower bound of ( d + ) + ⌊ d − ⌋ , whichfits all the known cases and reflects the fact that the number of nontrivial integral homologygroups of R P d depends on the parity of d . Unfortunately, in higher dimensions we don’tknow any triangulation of R P d on O ( d ) or even O ( d i ) vertices, for any i . The current recordis due to K¨uhnel [K¨uh87]. He observed that the barycentric subdivision of the boundary ofthe ( d + ) -simplex possesses a free involution , and the quotient w.r.t. the involution is PLhomeomorphic to the d -dimensional real projective space. This construction provides a PLtriangulation of R P d on 2 d + − R P d is to find a centrally symmetric (or cs, for short) triangulation of S d , the double coverof R P d , with an additional combinatorial condition. Let ∆ be a cs simplicial complex ∆with free involution σ . We say an n -cycle C in ∆ is an induced cs cycle if C = σ ( C ) and anyface of ∆ with vertices in C must be a face of C . Lemma 1.2 ([Wal70, Proposition (8.1)]) . Let ∆ be a cs PL d -sphere with free involution σ and with no induced cs -cycle. Then ∆ / σ is a PL triangulation of R P d . In this article, we construct a family of PL d -spheres as in Lemma 1.2 for every d ≥
0. Let F i be the i -th Fibonacci number, i.e., F = F = F n = F n − + F n − for every n ≥ Theorem 1.3.
There exists a cs PL d -sphere S d with no cs induced -cycles and f ( S d ) = F d + + F d + F d − − . Consequently, there exists a PL triangulation ∆ d of R P d with f ( ∆ d ) = F d + + F d + F d − − . The second statement is a direct consequence of Lemma 1.2. Observe that F d + + F d + F d − − < d + − d ≥
3, hence improving K¨uhnel’s construction in anydimension. Moreover the improvement of the bound is asymptotically significant, since F d + ≤ √ ( +√ ) d + ∼ √ ( . . . . ) d + .It is worth noting that in combinatorial and computational topology there are many inter-esting decompositions of topological spaces and manifolds, such as CW complexes, simplicialposets, graph encoded manifolds and more. Consequently, the related literature is vast. Eventhe expression “triangulated manifold” may refer to objects other than abstract simplicial omplexes as discussed in this paper. It is important to stress that in this article we considerexclusively objects which are simplicial complexes. For results on the minimal triangula-tions of R P (or more generally, lens spaces and other 3-manifolds) in other settings, see,for example, [BD14, CC15, JRT09, Swa13].2. Definitions A simplicial complex ∆ with vertex set V = V ( ∆ ) is a collection of subsets of V that isclosed under inclusion. The elements of ∆ are called faces . For brevity, we usually denote { v } as v and with f ( ∆ ) the number ∣ V ( ∆ )∣ . The dimension of a face F ∈ ∆ is dim F ∶= ∣ F ∣ − dimension of ∆, dim ∆, is the maximum dimension of its faces. We let f i ( ∆ ) be thenumber of i -dimensional faces of ∆, and record the numbers ( f ( ∆ ) , f ( ∆ ) , . . . , f dim ∆ ( ∆ )) in the so called f -vector of ∆.If F is a face of ∆, then the star of F and the link of F in ∆ are the simplicial complexesst ∆ ( F ) ∶= { σ ∈ ∆ ∶ σ ∪ F ∈ ∆ } and lk ∆ ( F ) ∶= { σ ∈ st ∆ ( F ) ∶ σ ∩ F = ∅} . If ∆ and Γ are simplicial complexes on disjoint vertex sets, then the join of ∆ and Γ is thesimplicial complex ∆ ∗ Γ = { σ ∪ τ ∶ σ ∈ ∆ and τ ∈ Γ } . In particular, if ∆ = {∅ , { v }} , then∆ ∗ ∆ is called the cone over ∆ with apex v .Let ∆ be a simplicial complex and W ⊆ V ( ∆ ) . The induced subcomplex of ∆ on W is∆ W = { F ∈ ∆ ∶ F ⊆ W } . If Γ is a subcomplex of ∆, define∆ ∖ Γ = ∆ V ( ∆ )∖ V ( Γ ) = { F ∈ ∆ ∶ F ∩ V ( Γ ) = ∅} . The complement of Γ in ∆, denoted by ∆ − Γ, is the set of faces in ∆ but not in Γ. The closure∆ − Γ is the subcomplex of ∆ generated by the facets of ∆ − Γ. If f is an automorphism of∆, then the quotient of ∆ w.r.t. f , ∆ / f , is the simplicial complex obtained identifying thevertices in the same orbit and all the faces with the same vertex set. Observe that in general ∣ ∆ / f ∣ ≇ ∣ ∆ ∣/ ˜ f , where ˜ f ∶ ∣ ∆ ∣ → ∣ ∆ ∣ is the continuous map induced by f . For instance, if ∆ isa square and f maps every vertex v to the unique vertex not adjacent to v then ∆ / f is the1-dimensional simplex, while ∣ ∆ ∣/ ˜ f ≅ S .We say two simplicial complexes ∆ and ∆ are PL homeomorphic , denoted as ∆ ≅ ∆ ,if there exist subdivisions ∆ ′ of ∆ and ∆ ′ of ∆ that are simplicially isomorphic. A PL d -ball is a simplicial complex PL homeomorphic to a d -simplex. Similarly, a PL d -sphere isa simplicial complex PL homeomorphic to the boundary complex of a ( d + ) -simplex. A d -dimensional simplicial complex ∆ is called a PL d -manifold if the link of every non-emptyface F of ∆ is a ( d − ∣ F ∣) -dimensional PL ball or sphere; in the former case, we say F is a boundary face while in the latter case F is an interior face . The boundary complex ∂ ∆ is thesubcomplex of ∆ that consists of all boundary faces of ∆. A PL manifold whose geometricrealization is homeomorphic to a closed manifold M is called a PL triangulation of M . Inthe literature PL manifolds as defined above are sometimes called combinatorial manifolds or combinatorial triangulations of manifolds . Remark 2.1.
For d ≥ d -manifolds is strictly contained in that of trian-gulated d -manifolds. In particular, the double suspension of any homology 3-sphere with anon-trivial fundamental group is a non-PL simplicial 5-sphere (see e.g., [RS82]). L manifolds have the following nice properties, see for instance the work of Alexander[Ale30] and Lickorish [Lic90]:
Lemma 2.2.
Let ∆ and ∆ be PL d - and d -manifolds, respectively. (1) If ∆ and ∆ are PL balls, so is ∆ ∗ ∆ . (2) If d = d = d and Γ ∶= ∆ ∩ ∆ = ∂ ∆ ∩ ∂ ∆ is a PL ( d − ) -manifold, then ∆ ∪ ∆ is a PL d -manifold. If furthermore ∆ and Γ are PL balls, then ∆ ∪ ∆ ≅ ∆ . Lemma 2.3. (Newman’s theorem) Let ∆ be a PL d -sphere and let Ψ ⊂ ∆ be a PL d -ball.Then the closure of the complement of Ψ in ∆ is a PL d -ball. Let ∆ be a PL manifold without boundary. The prism over ∆ is the pure polyhedralcomplex ∆ × [− , ] , whose cells are of the form F × { } , F × {− } or F × [− , ] , for every F ∈ ∆. Clearly ∣ ∆ × [− , ]∣ ≅ ∣ ∆ ∣ × [− , ] . The boundary of the prism consists of the cells F × { } and F × {− } , where F ∈ ∆.A simplicial complex ∆ is centrally symmetric or cs if its vertex set is endowed with a free involution σ ∶ V ( ∆ ) → V ( ∆ ) that induces a free involution on the set of all non-emptyfaces of ∆. Let ∆ be a centrally symmetric PL d -sphere and let σ be the free involution on∆. An induced cs -cycle in ∆ is an induced subcomplex C ⊆ ∆ isomorphic to a 4-cycle,with σ ( C ) = C . In particular, the vertices of C are v, w, σ ( v ) , σ ( w ) , for some v, w ∈ V ( ∆ ) .The complex ∆ has no induced cs 4-cycle if and only if st ∆ ( v ) ∩ st ∆ ( σ ( v )) = {∅} for every v ∈ V ( ∆ ) .In what follows we define a special property in the structure of a cs PL d -sphere. We use ⊎ to denote either the disjoint union of sets, or the union of two simplicial complexes definedon disjoint vertex sets. Definition 2.4 (Property P d ) . We say that a cs PL d -sphere S with free involution σ satisfies Property P d if there exists a sequence of subcomplexes S ⊂ S ⊂ ⋅ ⋅ ⋅ ⊂ S d = S such that each S i is a cs PL i -sphere with free involution σ ∣ S i that satisfies the following properties:i. S i = B i ∪ σ ( B i ) , where B i are PL i -balls and B ∩ σ ( B ) = ∂B = S i − . In words, S i − bounds two PL i -balls in S i .ii. S i ∖ S i − = D i ⊎ σ ( D i ) , where D i is a PL i -ball. In words, the complement of S i − in S i is the disjoint union of two PL i -balls.iii. There exists v i ∈ D i such that V ( st S i ( v i )) ∪ V ( st S i ( σ ( v i ))) = V ( S i ) .Property P d describes a family of cs spheres which is the key step of our construction.We shall see in Proposition 4.3 that these spheres do not have induced cs 4-cycles. In thefollowing examples we construct a 1-sphere and 2-sphere that satisfy Property P and P ,respectively. Example 2.5.
Referring to the top left complex in Figure 1, we let S be the cs 0-dimensionalsphere with vertices 3 and σ ( ) . We also let v = D = { , } , and B be the path σ ( ) − − − S . Then the cs 6-cycle S satisfies Property P w.r.t. ( B , D , v ) . Example 2.6.
The boundary complex S of the icosahedron is a cs 2-sphere on 12 verticeswhich satisfies Property P : indeed there is an induced cs 6-cycle S in S that divides S nto two antipodal 2-balls. In this case S ∖ S is the disjoint union of two triangles D , σ ( D ) and any vertex v ∈ D satisfies the third condition in Definition 2.4.3. From a triangulation of S d − to a triangulation of S d Given a cs PL ( d − ) -sphere S d − with 2 n vertices without any induced cs 4-cycles, onemay build a cs PL d -sphere with 4 n + S d − ;then triangulate the prism in such a way that no interior vertex is created and the centralsymmetry is preserved; finally cone over the boundaries of the prism (as two disjoint copies of S d − ) with two new vertices. The PL d -sphere obtained via our construction has no induced4-cycles. In this section, we will modify the above approach to reduce the number of verticesin the construction.In what follows, assume that S d − is a cs PL ( d − ) -sphere with involution σ and itsatisfies Property P d − . In particular, there exists a triple ( B d − , D d − , v d − ) satisfying thethree conditions in Definition 2.4, and S d − ∶= ∂B d − is a cs PL ( d − ) -sphere that satisfiesProperty P d − under the triple ( B d − , D d − , v d − ) . To make the notation more compact, forthe rest of this section we will denote by S the cs ( d − ) -sphere S d − and by ( B, D, v ) thetriple ( B d − , D d − , v d − ) . We will first give a triangulation Σ of S d − × [− , ] such that ● Σ is centrally symmetric; ● there exists a cs subcomplex Γ ⊆ Σ with Γ ≅ S and D × { } , σ ( D ) × {− } ⊆ Γ.This will be done in two steps. First we construct a polyhedral complex that satisfiesthe above conditions, see Proposition 3.1. Then we triangulate it while preserving theseproperties, see Corollary 3.10. Finally, we build a cs PL d -sphere from Σ, see Corollary 3.15.3.1. The prism over S . The main point of this section is Proposition 3.1. By refinement of a polyhedral complex P we mean a polyhedral complex ∆ with ∣ ∆ ∣ ≅ ∣ P ∣ obtained bysubdividing P . Proposition 3.1.
Let S be a cs PL ( d − ) -sphere satisfying Property P d − . There exists arefinement Σ ′′ of S × [− , ] such that: ● Σ ′′ is centrally symmetric. ● There exists a cs subcomplex Γ ⊆ Σ ′′ with Γ ≅ S and D × { } , σ ( D ) × {− } ⊆ Γ . After a few intermediate steps, we will prove Proposition 3.1 as a corollary of Lemma 3.4.We first refine S × [− , ] to a three-layered prism over S . Let Σ ′ be the polyhedral complexΣ ′ ∶= ( S × [− , ]) ∪ ( S × [ , ]) . In other words, Σ ′ is a prism over S with three layers S × {− } , S × { } and S × { } . Notethat Σ ′ is centrally symmetric with the free involution σ ′ induced by ( u, i ) ↦ ( σ ( u ) , − i ) , for i = − , ,
1. Since S satisfies Property P d − , from conditions ii and iii of Definition 2.4 itfollows that V ( S ) = V ( D ) ⊎ V ( σ ( D )) ⊎ V ( S d − )= V ( D ) ⊎ V ( st S d − ( v d − )) ⊎ V ( σ ( D )) ⊎ V ( σ ( st S d − ( v d − ))) . Hence the following map is well-defined: ∶ V ( Σ ′ ) → V ( Σ ′ )( u, i ) ↦ ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩( u, i ) if i ∈ {− , }( u, ) if i = u ∈ V ( D ) ⊎ V ( st S d − ( v d − ))( u, − ) if i = u ∈ V ( σ ( D )) ⊎ V ( st S d − ( σ ( v d − ))) , (3.1)Observe that the map ψ induces a map from the polyhedral complex Σ ′ to itself. Definition 3.2.
We define Σ ′′ ∶= Σ ′ / ∼ , where u ∼ w if and only if w = ψ ( u ) .Informally speaking, we identify two disjoint subcomplexes of the middle layer of Σ ′ withthe upper and lower layers respectively. In this way we obtain a polyhedral complex con-taining an induced subcomplex isomorphic to S . Remark 3.3.
We illustrate these maps in Figure 1, in which case d =
2. As in Example 2.5,We let v = D = { , } , B be the path σ ( ) − − −
3, and S be the cs 6-cycle. Following(3.1), the map ψ acts as the identity on the vertices in S × { } and S × {− } , and mapsthe vertices of S × { } to either the upper or lower corresponding vertices. Hence, to obtainthe complex Σ ′′ from Σ ′ in Figure 1, we identify all pairs of boxed vertices (resp. crossedvertices) lying on the same vertical segments in S × [ , ] (resp. S × [− , ] ). S × { } S × { } S × {− } Σ (cid:48) Σ (cid:48)(cid:48) Σ S = S σ (2)2 σ (1)1 σ (3) 3 Figure 1.
The complexes Σ ′ and Σ ′′ , together with a cs triangulation Σ. emma 3.4. The complex Σ ′′ in Definition 3.2 has the following properties: i. ∣ Σ ′′ ∣ ≅ S d − × [− , ] . ii. ∂ Σ ′′ consists of two copies of S . iii. There exists an induced cs subcomplex Γ ⊆ Σ ′′ with Γ ≅ S and both D × { } and σ ( D ) × {− } are subcomplexes of Γ . iv. Σ ′′ is centrally symmetric under the involution σ ′′ induced by σ ′ .Proof: Part i is clear, and for ii it suffices to observe that the boundary of ∂ Σ ′′ is exactlythe disjoint union of S × { } and S × {− } .Let Γ be the image of S × { } in Σ ′′ . Since the restriction of ψ on S × { } is injective,Γ = ψ ( S ×{ }) ≅ S . Moreover D ×{ } = ψ ( D ×{ }) ⊆ Γ and σ ( D )×{− } = ψ ( σ ( D )×{ }) ⊆ Γ.This proves iii.Finally, we observe that if F = G in Σ ′′ then σ ( F ) = σ ( G ) . Equivalently, the map thatassigns the equivalence class of the vertex ( u, i ) to the class of σ ′ (( u, i )) is well defined.Therefore, it induces a free involution of σ ′′ ∶ Σ ′′ → Σ ′′ . ◻ The cs triangulation Σ of Σ ′′ . In this subsection we construct a simplicial complexΣ which refines Σ ′′ such that ● Σ is a PL manifold with V ( Σ ) = V ( Σ ′′ ) , i.e., no new vertex is introduced. ● Σ is centrally symmetric with a free involution τ , such that τ ∣ Σ ′′ = σ ′′ .Our construction is based on certain orientations of the graph of a simplicial complex calledlocally acyclic orientations. We refer to [JDLS10, Section 7.2] for a more detailed treatmentof the subject. Definition 3.5. A locally acyclic orientation (l.a.o.) of ∆ is an orientation of the edges ofits graph such that none of the 2-simplices of ∆ contains an oriented cycle. Lemma 3.6 ([JDLS10, Lemma 7.2.9]) . Let ∆ be a simplicial complex. The simplicial re-finements of ∆ × [− , ] are in bijection with locally acyclic orientations of ∆ . Every simplicial complex has a locally acyclic orientation, obtained for example from a(globally) acyclic orientations of its graph. The bijection in Lemma 3.6 is easy to describe: if { i, j } is an edge of ∆ with i → j , then {( i, ) , ( j, − )} is an edge of ∆ × [− , ] and vice versa.This induces a triangulation of F × [− , ] for every face F ∈ ∆. Furthermore, the locallyacyclicity guarantees that the union of triangulations of individual cells can be coherentlycompleted to a triangulation of ∆ × [− , ] .Given an l.a.o. (cid:96) of ∆, we denote by ( ∆ × [− , ]) (cid:96) the refinement of ∆ × [− , ] inducedby (cid:96) . The following lemma shows that if ∆ is a PL sphere, then ( ∆ × [− , ]) (cid:96) is also in thePL category. Lemma 3.7.
Let ∆ be a PL ( d − ) -sphere and let (cid:96) be any locally acyclic orientation on ∆ .Then, the refinement ( ∆ × [− , ]) (cid:96) is a PL d -manifold with boundary and without interiorvertices.Proof: Since ∆ is a PL ( d − ) -sphere it can be subdivided to a complex ∆ ′ all whosesimplices can be linearly embedded in R N for some N . Moreover, for every locally acyclicorientation (cid:96) of ∆, consider a linear ordering of the vertices of ∆ ′ such that v > w for every ∈ V ( ∆ ′ ) ∖ V ( ∆ ) and w ∈ V ( ∆ ) . Orient all the edges { u, v } ∉ ∆ with u → v if u > v . Thisextends to a locally acyclic orientation (cid:96) ′ of ∆ ′ that agrees with (cid:96) when restricted to the edgesof ∆. The corresponding refinement of ∆ ′ × [− , ] is a subdivision of ∆ × [− , ] in whichevery simplex is linearly embedded in R N + . Therefore, ( ∆ × [− , ]) (cid:96) is a PL manifold withboundary. Furthermore V (( ∆ × [− , ]) (cid:96) ) = V ( ∆ × {− }) ∪ V ( ∆ × { }) = V ( ∂ ( ∆ × [− , ]) (cid:96) ) and hence ∆ (cid:96) has no interior vertices. ◻ In what follows, we show that if ∆ is cs, then there exists an l.a.o. (cid:96) such that ( ∆ ×[− , ]) (cid:96) is also cs. Lemma 3.8.
Let ∆ be a cs simplicial complex with free involution σ and consider a refine-ment of ∆ ×[− , ] which is cs with free involution induced by ̃ σ ∶ ( v, − ) ↦ ( σ ( v ) , ) , ( v, ) ↦( σ ( v ) , − ) for any vertex v ∈ ∆ . Then the corresponding locally acyclic orientation of ∆ isorder reversing w.r.t. the symmetry, i.e., v → w if and only if σ ( w ) → σ ( v ) .Proof: In any cs triangulation of ∆ × [− , ] the set {( v, − ) , ( w, )} is an edge if andonly if {̃ σ (( v, − )) , ̃ σ (( w, ))} = {( σ ( v ) , ) , ( σ ( w ) , − )} is an edge, which implies that on thecorresponding l.a.o. we have that v → w if and only if σ ( w ) → σ ( v ) . ◻ Lemma 3.9.
Let S be a cs P L ( d − ) -sphere that satisfies property P d − w.r.t. ( B, D, v ) .Let W ∶= V ( D ) ⊎ V ( st S d − ( v d − )) and let A ∶= {{ u, w } ∈ S ∶ u ∈ W and w ∈ σ ( W )} . Thereexists an l.a.o. of S such that: ● u → w for every edge { u, w } in A . ● For every edge { u, w } ∈ S , u → w if and only if σ ( w ) → σ ( u ) .Proof: First we choose any l.a.o. of the induced subcomplex of S on W . FollowingLemma 3.8 we impose on the induced subcomplex on σ ( W ) a reverse orientation. Since S satisfies Property P d − , V ( S ) = V ( W ) ⊎ V ( σ ( W )) and hence the edges of S with a vertexin V ( W ) and one in V ( σ ( W )) belong to A . We orient all edges { u, w } in A as u → w .This orientation is by definition acyclic on every 2-simplex not containing edges in A . Itsuffices to check that no edge in A is contained in an oriented cycle. Since every 2-simplexcontaining an edge in A also contains another edge in A , it contains a vertex u or a vertex w such that either w ← u → w or u → w ← u . This proves the claim. ◻ It follows from Lemmas 3.7 and 3.9 that ( S ×[− , ]) (cid:96) is cs PL triangulation of S d − ×[− , ] for some l.a.o. (cid:96) of S . It is left to prove that this triangulation refines Σ ′′ . Figure 2.
Two different locally acyclic orientations of the cs 6-cycle S thatsatisfy Lemma 3.9 and induced triangulations of S × [− , ] . orollary 3.10. Let S be a cs ( d − ) -sphere that satisfies property P d − w.r.t. ( B, D, v ) .There exists an l.a.o. (cid:96) on S such that i. ( S × [− , ]) (cid:96) is a cs PL manifold with boundary that refines Σ ′′ ; ii. ( S × [− , ]) (cid:96) contains an induced cs subcomplex Γ isomorphic to S which contains ( D × { }) ∪ ( σ ( D ) × {− }) ; iii. V (( S × [− , ]) (cid:96) ) = V ( Σ ′′ ) .Proof: Choose any l.a.o. (cid:96) on S that satisfies the conditions in Lemma 3.9. The orientationon any edge { u, w } ∈ A (as defined in Lemma 3.9) generates a new edge {( u, ) , ( w, − )} ∈ Σ.This coincides with those edges in Σ ′′ but not in S × [− , ] . Hence ( S × [− , ]) (cid:96) is arefinement of Σ ′′ . By Lemma 3.7, ( S × [− , ]) (cid:96) is a PL manifold. Part ii and iii follow fromLemmas 3.4 and 3.7. ◻ Remark 3.11.
As we see from Figure 2, the l.a.o. of S that satisfies the conditions inLemma 3.9 is not unique. In what follows, we denote by Σ any PL manifold ( S × [− , ]) (cid:96) obtained in Corollary 3.10. Although Σ can be defined directly from the l.a.o., Corollary3.10 ii (which follows from subsection 3.1) will play a key role in our inductive constructionin Section 4.3.3. From Σ to a PL triangulation of S d . In this subsection we complete the cs trian-gulation of the prism over a PL ( d − ) -sphere as in Corollary 3.10 to a PL d -sphere. Weintroduce two pairs of antipodal vertices ( v + , v − ) , ( w + , w − ) and define(3.2) Φ ′ ∶= Σ ∪ ( v + ∗ K + ) ∪ ( v − ∗ K − ) ∪ ( w + ∗ L + ) ∪ ( w − ∗ L − ) , where K + ∶= B × { } , K − ∶= σ ( B ) × {− } , L + ∶= ( σ ( B ) × { }) ∪ ∂B ×{ } ( v + ∗ ∂B × { }) , and L − ∶= ( B ×{− })∪ ∂B ×{− } ( v + ∗ ∂B ×{− }) . The free involution σ on Σ extends to an involutionon Φ ′ by additionally letting σ ( v + ) = v − and σ ( w + ) = w − . The second complex in Figure 3offers a visualization of Φ ′ in the 2-dimensional case. Proposition 3.12.
The simplicial complex Φ ′ in (3.2) is a cs PL triangulation of S d .Proof: By Corollary 3.10, Σ is a PL manifold with boundary. By Property P d − , K ± is aPL ( d − ) -ball and by Lemma 2.2, L ± is a PL ( d − ) -sphere. Again by Lemma 2.2, Φ ′ is aPL manifold. Finally it is clear that Φ ′ is centrally symmetric and ∣ Φ ′ ∣ is homeomorphic to S d . ◻ We next contract certain edges of Φ ′ in order to reduce the number of vertices. A simplicialcomplex is obtained from ∆ via an edge contraction of { i, j } ∈ ∆ if it is the image of ∆ w.r.t.the simplicial map that identifies i with j . The edges we will contract are those of theform {( u, ) , ( u, − )} , where u ∈ V ( D ) ∪ V ( σ ( D )) . In other words, we identify D × { } and σ ( D ) × {− } with D × {− } and σ ( D ) × { } . It is easy to see that in this case the proceduredoes not depend on the order in which contractions are applied. To prove that the resultingcomplex is still a PL sphere we use a result of Nevo [Nev07]. Theorem 3.13 ([Nev07, Theorem 1.4]) . Let ∆ be a PL manifold and let ∆ ′ be the contractionof ∆ at the edge { i, j } . Then ∆ ′ is PL homeomorphic to ∆ if and only if lk ∆ ( i ) ∩ lk ∆ ( j ) = lk ∆ ({ i, j }) . emma 3.14. Let ∆ ′ be a simplicial complex and let ∆ be a simplicial refinement of ∆ ′ ×[− , ] induced by an l.a.o. of ∆ ′ . Then for any vertex u ∈ ∆ ′ , lk ∆ (( u, )) ∩ lk ∆ (( u, − )) = lk ∆ ({( u, ) , ( u, − )}) . Proof:
The triangulation of the prism induced by an l.a.o. has the following key property:for every F ∈ ∆ ′ , a subset G = {( w , t ) , . . . , ( w k , t k )} of V ( F × [− , ]) is a face of ∆ if andonly if any two vertices ( w i , t i ) , ( w j , t j ) form an edge of ∆. Therefore if both G ∪ {( u, − )} and G ∪ {( u, )} are faces of ∆, then G ∪ {( u, − ) , ( u, )} is also a face of ∆. This proves thatlk ∆ (( u, )) ∩ lk ∆ (( u, − )) ⊆ lk ∆ ({( u, ) , ( u, − )}) . Since the other inclusion holds in general,the claim follows. ◻ We underline that Lemma 3.14 clearly does not imply that the simplicial complex ob-tained from Σ by contracting {( u, ) , ( u, − )} , u ∈ S , is PL homeomorphic to Σ. Indeed,Theorem 3.13 holds for PL manifolds without boundary . Proposition 3.15.
The simplicial complex Φ obtained from Φ ′ by contracting every edge ofthe form {( u, ) , ( u, − )} , u ∈ V ( D ) ∪ V ( σ ( D )) , is a cs PL d -sphere.Proof: Let V ( D ) = { v , . . . , v k } , e i = {( v i , ) , ( v i , − )} for i = , . . . , k . LetΦ ′ = Φ e ,σ ( e ) —→ Φ e ,σ ( e ) —→ . . . —→ Φ k − e k ,σ ( e k ) —→ Φ k = Φbe the sequence of complexes obtained from Φ ′ by contracting a pair of antipodal edges e i , σ ( e i ) at a time. Since every vertex in V ( Φ ′ ) ∖ V ( Σ ) is connected to at most one vertexfrom {( w, ) , ( w, − )} , where w ∈ S , we have that for every vertex v ∈ S ,lk Φ ′ (( v, )) ∩ lk Φ ′ (( v, − )) = lk Σ (( v, )) ∩ lk Σ (( v, − )) and lk Φ ′ ({( v, ) , ( v, − )}) = lk Σ ({( v, ) , ( v, − )}) . By Theorem 3.13, Lemma 3.14, and the fact that the links of e and σ ( e ) in Σ are antipodaland disjoint, it follows that Φ is a cs PL d -sphere. Note that e i ∪ e j is not a face in Φ or Φ , . . . , Φ i − for any distinct i and j ; in other words, at least one edge between ( v j , ± ) and ( v i , ± ) is missing. Hence e j ∉ lk Φ k (( v i , )) ∩ lk Φ k (( v i , − )) for any 0 ≤ k ≤ i − Φ i − (( v i , ))∩ lk Φ i − (( v i , − )) = lk Φ i − (( v i , ))∩ lk Φ i − (( v i , − )) = ⋅ ⋅ ⋅ = lk Φ (( v i , ))∩ lk Φ (( v i , − )) . Similarly, e j is not an edge in lk Φ i − ( e i ) , lk Φ i − ( e i ) , . . . , lk Φ ( e i ) for any i ≠ j and hencelk Φ i − ( e i ) = lk Φ ( e i ) . This fact implies the first and third equalities in the following equation:lk Φ i − (( v i , )) ∩ lk Φ i − (( v i , − )) = lk Φ (( v i , )) ∩ lk Φ (( v i , − ))= lk Φ ( e i )= lk Φ i − ( e i ) , while the second equality follows from Theorem 3.13. Again by Theorem 3.13 and the factthat lk Φ i − ( e i ) and lk Φ i − ( σ ( e i )) are antipodal and disjoint, we conclude that every Φ i is acs PL d -sphere. ◻ emark 3.16. In fact, we can contract more edges {( u, − ) , ( u, )} and their antipodes with u ∈ V ( S ) and still obtain a cs PL d -sphere. However, contracting too many edges wouldcreate induced cs 4-cycles in the resulting complex. Remark 3.17.
By Corollary 3.10, the cs d -sphere Φ that we define is usually not unique,but it depends on the l.a.o. considered. Indeed, the number of combinatorial types of Φis related to the number of l.a.o. on S that satisfy the conditions in Lemma 3.9. With aslight abuse of notation we write “a sphere Φ” to indicate any PL d -sphere that could beconstructed from S as in this section.4. The induction step
In this section, we show that the sphere Φ constructed in the previous section satisfiesProperty P d w.r.t. a certain flag of spheres S ⊂ S ⊂ ⋅ ⋅ ⋅ ⊂ S d − = S ⊂ S d = Φ. Finally, weshow that if S d − does not have induced cs 4-cycles then the same holds for S d . Using thisfact, together with the initial cases, i.e., the 0-dimensional sphere S and the cs 6-cycle S ,we prove Theorem 1.3. Assume that inductively we’ve constructed a sequence of cs i -spheres S i , 0 ≤ i ≤ d −
1, that satisfies Property P i under the triple ( B i , D i , v i ) , that is ● B i is a PL i -ball in S i with boundary S i − . ● S i ∖ S i − = D i ⊎ σ ( D i ) for some PL i -ball D . ● V ( st S i ( v i )) ∪ V ( st S i ( σ ( v i ))) = V ( S i ) .We will now show that the PL d -sphere Φ constructed in the previous section by setting Γto be the image of S d − × { } w.r.t. the map ψ in (3.1) (consequently, Γ satisfies PropertyP d ). With the notation introduced earlier we define: ● D d ∶= { v + , w + } ∗ st S d − ×{ } (( σ ( v d − ) , )) . ● B d is the closure of one of the two connected components of Φ − Γ ∶= { F ∈ Φ ∣ F ∩ Γ = ∅} . ● v d ∶= v + . Remark 4.1.
By Jordan’s theorem, the geometric realization of Φ − Γ consists of two con-nected components. Since Φ and Γ are PL spheres, it is known that B d is a simplicial ball[New60, Theorem 6]. However, it is an open problem in PL topology (known as PL Schoen-flies problem) to decide whether B d is also a PL ball. The following lemma gives a positiveanswer in the special case of our construction. Σ v + v − w + w − v + v − w + w − Dσ ( D ) v + w + v − w − Φ (cid:48) ΦΓ Figure 3.
An illustration of Theorem 4.4 in the case d = emma 4.2. The triple ( B d , D d , v d ) satisfies the following properties: ● B d is a PL d -ball in Φ with boundary Γ ≅ S d − . ● D d is isomorphic to { v d , w + } ∗ st S d − ( v d − ) . ● D d ⊆ st Φ ( v d ) ⊆ B d and V ( Φ ) = V ( st Φ ( v d )) ⊎ V ( st Φ ( σ ( v d ))) .In particular, Φ satisfies Property P d under the triple ( B d , D d , v d ) .Proof: We only need to verify the first and third bullet points. By the inductive hypothesisand the definition of D d , we obtain V ( st Φ ( v d )) = V ( B d − × { }) ∪ { v d , w + }= ( V ( D d − × { }) ∪ V ( st S d − ×{ } ( v d − , ))) ∪ ( V ( st S d − ×{ } ( σ ( v d − ) , )) ∪ { v d , w + })= V ( st Γ ( σ ( v d − ))) ∪ V ( D d ) . Therefore, V ( st Φ ( v d )) ⊎ V ( st Φ ( σ ( v d ))) = V ( st Γ ( v d − )) ⊎ V ( st Γ ( σ ( v d − ))) ⊎ V ( D d ) ⊎ V ( σ ( D d ))= V ( Γ ) ⊎ V ( D d ) ⊎ V ( σ ( D d ))= V ( Φ ) , where the second equality follows from Definition 2.4 iii on Γ.To see that B d is a PL d -ball, we consider the simplicial complex Φ ∗ obtained from Φ bycontracting all the edges of the form {( v, − ) , ( v, )} with v ∈ V ( st Φ ( v d )) . By Lemma 3.14and Theorem 3.13, Φ ∗ is a PL d -sphere. The image of Σ ∪ ( v + ∗ K + ) ∪ ( w + ∗ L + ) is a PL d -ball, since K + and L + are a PL ( d − ) -ball and a ( d − ) -sphere respectively. It followsfrom Lemma 2.3 that B d is a PL d -ball, since it can be decomposed as the (closure of) thecomplement of the PL d -ball Σ ∪ ( v + ∗ K + ) ∪ ( w + ∗ L + ) w.r.t. the PL d -sphere Φ ∗ .Finally, by the definition, we have that B d ∖ Γ = D d and D d ⊆ st Φ ( v d ) ⊆ B d . Furthermore, V ( st Φ ( v d ))∪ V ( st Φ ( σ ( v d )) = V ( Φ ) follows from the inductive assumption that V ( st Γ ( v d − ))∪ V ( st Γ ( σ ( v d − ))) = V ( Γ ) . ◻ The property which motivates our construction is the content of the following lemma.
Proposition 4.3.
The complex Φ as constructed above has no induced cs -cycle if Γ ≅ S d − has no induced cs -cycle.Proof: As we see from the construction, since lk Φ ( v + )∩ lk Φ ( v − ) = lk Φ ( w + )∩ lk Φ ( w − ) = {∅} , itfollows that none of the vertices v + , v − , w + , w − belong to any induced cs 4-cycle. Furthermore,any vertex ( a, ) ∈ S d − ×{ } is only adjacent to either v + , w + , or its neighbors in S d − ×{ } , orsome ( σ ( b ) , − ) ∈ S d − × {− } where { σ ( a ) , σ ( b )} ∈ S d − × {− } . By the inductive hypothesis, S d − and S d − do not contain any induced cs 4-cycle. We conclude that ( a, ) is also notin any induced cs 4-cycle in S . However, any cs 4-cycle must include two vertices outside D d ∪ σ ( D d ) . Hence Φ has no induced cs 4-cycle. ◻ Finally, we conclude with the proof of the main result.
Theorem 4.4.
There exists a family of cs PL i -spheres S ⊂ S ⊂ S . . . such that each S i satisfies Property P i with respect to ( B i , D i , v i ) , where ∂B i = S i − , and each S i has no inducedcs 4-cycles. Furthermore f ( S i + ) = f ( S i ) + f ( S i − ) + for i ≤ . roof: The first statement follows directly from Proposition 3.15 and Proposition 4.3. Thenumber of vertices in the prism over S i − equals 2 f ( S i − ) , and together with v + , v − , w + , w − sums up to 2 f ( S i − ) +
4. Identifying the vertices of D i − × {± } and σ ( D i − ) × {± } decreasesthe number of vertices by 2 f ( D i − ) . Since f ( S i − ) = f ( D i − ) + f ( S i − ) , the claim follows. ◻ Proof of Theorem 1.3:
Via Theorem 4.4 we know there exists a PL d -sphere S d whosenumber of vertices n d is given by the sequence n = n = n i + = n i + n i − +
4. Solvingthe recursion we obtain the desired formula. The second statement follows from a directapplication of Lemma 1.2. ◻ Remark 4.5.
Our inductive method produces the double cover of the minimal triangulationof R P , the boundary of the icosahedron, from the double cover of the minimal triangula-tion of R P , the 6-cycle; see Figure 3. In dimension 3, we find two non-isomorphic PLtriangulations of R P with the f -vector ( , , , ) , starting from the boundary of theicosahedron. They are vertex-minimal but not f -vectorwise minimal triangulations. For d = ,
5, our construction is not vertex-minimal. We developed a naive implementation ofour construction in the software
Sage and generated triangulations of R P d as in Theorem 4.4up to d =
7, and compute standard invariants (e.g., fundamental group, homology groups andpseudomanifold property) up to d =
6. The code and the lists of facets of these triangulationscan be found in [Ven]. In Table 1 we report their f -vectors. d f -vector1 (3,3)2 (6, 15, 10)3 (11, 52, 82, 41)4 (19, 151, 424, 485, 194)5 (32, 403, 1797, 3536, 3165, 1055)6 (53, 1022, 6811, 20545, 30919, 22701, 6486)7 (87, 2514, 24099, 104628, 235599, 286041, 177864, 44466) Table 1. f -vectors of a triangulation of R P d as in Theorem 1.3.5. Open problems
We conclude this article with a few questions. The first one is about the (asymptotic)tight lower bound on the number of vertices required for a vertex-minimal triangulation of R P d . Question 5.1.
Does there exist a PL triangulation of R P d with ( d + ) + ⌊ d − ⌋ vertices forevery d ≥ ? Does at least a construction with a number of vertices that is polynomial in d exist? We do not know if the PL spheres constructed in Theorem 4.4 are polytopal, i.e., theycan be realized as the boundary complex of a simplicial polytope. It is natural to ask thefollowing question. uestion 5.2. What is the minimum number of vertices required for a cs simplicial d -polytope with no induced cs 4-cycles? Frequently in the literature, additional combinatorial properties are imposed on a tri-angulation. We focus on two properties, namely flagness and balancedness. A simplicialcomplex is flag if all minimal subsets of the vertices which do not form a face are edges. A d -dimensional simplicial complex is balanced if there exists a simplicial projection (often calledcoloring) to the d -simplex which preserves the dimension of faces. Recently Bibby et al.[BOW +
19] and the first author in [Ven19] implemented local flips and transformations in or-der to obtain flag and balanced triangulations of manifolds which are vertex-minimal w.r.t.these properties. In particular the authors obtained a flag and balanced vertex-minimaltriangulation of R P on 11 and 9 vertices respectively, and a balanced vertex-minimal trian-gulation of R P on 16 vertices. For higher d , a flag and balanced triangulation of R P d can beobtained by considering the barycentric subdivision of the boundary complex of the ( d + ) -dimensional cross-polytope , and identifying antipodal vertices. These simplicial complexeshave d + − vertices. Problem 5.3.
Construct flag or balanced PL triangulations of R P d for every d with lessthen d + − vertices. Does a construction on a number of vertices that is polynomial in d exist? Acknowledgements
We would like to thank Basudeb Datta, Isabella Novik and Martina Juhnke-Kubitzke forhelpful comments. We also would like to express our gratitude to the anonymous referees forproviding references for results on minimal triangulated manifolds in other settings, as wellas an insightful remark on the properness of group actions on PL manifolds. Their detailedfeedback greatly helped us to improve earlier versions of the article.
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E-mail address : [email protected] Max Planck Institute for Mathematics in the Sciences, Inselstr. 22, 04103 Leipzig, GER-MANY.