Triangulations with few vertices of manifolds with non-free fundamental group
aa r X i v : . [ m a t h . G T ] J a n TRIANGULATIONS WITH FEW VERTICES OF MANIFOLDSWITH NON-FREE FUNDAMENTAL GROUP
PETAR PAVEˇSI´C
Abstract.
We study lower bounds for the number of vertices in a PL-trian-gulation of a given manifold M . While most of the previous estimates arebased on the dimension and the connectivity of M , we show that furtherinformation can be extracted by studying the structure of the fundamentalgroup of M and applying techniques from the Lusternik-Schnirelmann categorytheory. In particular, we prove that every PL-triangulation of a d -dimensionalmanifold ( d ≥
3) whose fundamental group is not free has at least 3 d + 1vertices. As a corollary, every d -dimensional ( Z p -)homology sphere that admitsa PL-triangulation with less than 3 d vertices is homeomorphic to S d . Anotherimportant consequence is that every triangulation with small links of M iscombinatorial. Keywords : minimal triangulation, PL-manifold, homology sphere, good cover,Lusternik-Schnirelmann category
AMS classification: 57Q15, 52B70 Introduction and results A triangulation of a topological space M is a simplicial complex K together with ahomeomorphism M ≈ | K | between M and the geometric realization of K . If M is aclosed d -dimensional manifold then we are particularly interested in combinatorial triangulations, where we require that the link (see bellow) of every simplex in K ishomeomorphic to a sphere. A manifold admitting a combinatorial triangulation iscalled a PL-manifold .Given a PL-manifold M , what is the minimal number of vertices in a combinatorialtriangulation of M ? This is a difficult question, because there are no standardconstructions for triangulations with few vertices of a given manifold, nor thereare sufficiently general methods to prove that some specific triangulation is in factminimal. Apart from classical results on minimal triangulations of spheres andclosed surfaces, and a special family of minimal triangulations for certain spherebundles over a circle (so called Cs´asz´ar tori - see [11]), there exists only a handfulof examples for which the minimal triangulations are known. An exhaustive surveyof the results and the existing literature on this problem can be found in [12]. Seealso the recent article [10] which discusses a more general question of the numberof faces in triangulations of manifolds and polytopes.Generally speaking, one may expect that the minimal number of vertices in a tri-angulation of a space increases with its complexity. Most results that can be found The author was supported by the Slovenian Research Agency research grant P1-0292 andresearch project J1-7025. in the literature use dimension, connectivity or Betti numbers of M to expresslower bounds for the number of vertices in a triangulation of M . In this paper wehave been able to exploit the fundamental group and the Lusternik-Schnirelmanncategory to improve several estimates of the minimal number of vertices in a trian-gulations of a manifold.In the rest of this section we state our main results. In Section 2 we introduce andexplain prerequisites on triangulations, Lusternik-Schnirelmann category and thecovering type. Finally, in Section 3 we give the proofs of the theorems presentedbellow.Let us begin with a slight improvement of the theorem first proved by Brehmand K¨uhnel [3]. Our approach is based on the notion of covering type [7] and ismuch simpler than the original one. Recall that Poincar´e duality together withthe positive answer to the Poincar´e conjecture imply that every simply-connectedclosed d -manifold, whose homology is trivial in dimensions less or equal to d/ d -sphere. For the remaining cases the minimal number ofvertices in a triangulation can be estimated as follows. Theorem 1.1.
Let M be a simply-connected d -dimensional closed PL-manifold,and let i be the minimal index for which e H i ( M ) = 0 .(a) If i = d then every combinatorial triangulation of M has at least d + k +2 vertices, where k is the minimal integer for which (cid:0) i + ki +1 (cid:1) ≥ rank H i ( M ) .Moreover, k can be equal to 1 only if d ∈ { , , , } .(b) If i < d then every combinatorial triangulation of M has at least d − i + 4 vertices.In particular, every combinatorial triangulation of a closed, simply-connected d -manifold with at most d + 2 vertices represents the d -dimensional sphere. The main contribution of this paper is the following theorem and its corollaries. Inparticular we obtain considerable improvements of estimates by Brehm-K¨uhnel [3]and Bagchi-Datta [1] of the number of vertices in PL-triangulations of homologyspheres. By [3, Corollary 2] every PL-triangulation of a non simply-connected d -manifold ( d ≥
3) has at least 2 d + 3 vertices. That the value cannot be improved ingeneral is shown by K¨uhnel who constructed a family of S d − -bundles over the circle S that admit PL-triangulations with 2 d + 3 vertices. However, if the fundamentalgroup of M is not free, then we obtained a better estimate: Theorem 1.2. If M is a d -dimensional ( d ≥ ) closed manifold whose fundamentalgroup is not free, then every combinatorial triangulation of M has at least d + 1 vertices. It is worth noting that closed 3-manifolds whose fundamental group is free are quitespecial, being either the 3-sphere or connected sums of tori S × S and twisted tori S × S . All the other closed 3-manifolds (in particular, all hyperbolic manifolds)satisfy the assumptions of the above theorem.An important family of examples whose fundamental group is not free are the homology spheres , i.e., manifolds, whose homology groups vanish except in the topdimension, where the homology group is Z . A simply-connected homology sphere ishomeomorphic to a sphere by the positive answer to the Poincar´e conjecture but for RIANGULATIONS WITH FEW VERTICES 3 every d ≥ d -dimensional homology spheres that are not homeomorphicto S d . As the fundamental group of a homology sphere must be a perfect group,it cannot be free (unless it is trivial), therefore Theorem 1.2 implies the followingimprovement of the estimate in [3, Corollary 4]. Corollary 1.3.
Every d -dimensional homology sphere that admits a combinatorialtriangulation with at most d vertices is a PL-sphere. Bagchi and Datta [1] obtained an estimate of the minimal number of vertices inPL-triangulations of Z -homology spheres, i.e., manifolds whose Z -homology isisomorphic to that of a sphere (non-trivial examples are 3-dimensional odd lensspaces). Their results are improved (except in dimensions 3 and 4) by the following: Corollary 1.4.
Every d -dimensional Z p -homology sphere that admits a combina-torial triangulation with at most d vertices is a PL-sphere. We conclude with a useful recognition criterion for combinatorial triangulations.
Theorem 1.5.
Let K be a triangulation of a d -dimensional manifold. If for every k ≥ the link of each simplex of codimension k + 1 in K has at most k vertices,then the triangulation K is combinatorial. Preliminaries
In this section we give recollect concepts and results that are needed in the proofsof above theorems.2.1.
Simplicial complexes and PL-triangulations.
Here we describe two spe-cial constructions and refer the reader to the article of J. Bryant [4] for the defi-nitions of triangulations, skeleta, open and closed stars, links, joins, combinatorialtriangulations and other standard concepts of PL-topology.Given a triangulation M ≈ | K | we identify the set of vertices of the triangulationwith the 0-skeleton K of the simplicial complex K . For a subset V ⊆ K , let K ( V )denote the full subcomplex of K spanned by V , i.e. the maximal subcomplex of K whose 0-skeleton is V . It is easy to check that for every vertex v ∈ K thesubcomplex K ( V ∪ { v } ) can be obtained as the union of K ( V ) and the join of v with the part of the link of v contained in K ( V ), which can be expressed by thefollowing formula: K ( V ∪ { v } ) = K ( V ) ∪ v ∗ (lk( v ) ∩ K ( V )) (1)Furthermore, let us denote by N ( V ) ⊆ | K | the union of open stars (with respectto K ) of vertices in V . Clearly, the geometric realization | K ( V ) | is a subspace of N ( V ). Lemma 2.1. N ( V ) = | K |−| K ( K − V ) | , therefore N ( V ) is an open neighbourhoodof | K ( V ) | in | K | . Moreover, | K ( V ) | is a deformation retract of N ( V ) .Proof. The first statement is obvious. In order to obtain a deformation retractionrecall that every point x ∈ | K | can be written uniquely in terms of barycentriccoordinates x = X v ∈ K λ v ( x ) · v. PETAR PAVEˇSI´C
By definition, for every x ∈ N ( V ) there is at least one v ∈ V for which λ v ( x ) > r : N ( V ) → | K ( V ) | as r ( x ) := X v ∈ V λ v ( x ) · v. Clearly, r is homotopic to the identity of N ( V ) through a straight-line homotopy. (cid:3) In particular, if K is partitioned into two disjoint subsets V, V ′ then N ( V ) and N ( V ′ ) form an open cover of | K | and N ( V ) ∩ N ( V ′ ) = N ( V ) − | K ( V ) | = N ( V ′ ) − | K ( V ′ ) | . Lemma 2.2.
Let K be a combinatorial triangulation of a closed d -dimensionalmanifold. If V ⊆ K spans a d -dimensional simplex in K then N ( V ) − | K ( V ) | ishomotopy equivalent to a ( d − -dimensional sphere and for i < dH i ( K ( K − V )) ∼ = H i ( K ) and H i ( K ( K − V )) ∼ = H i ( K ) (integer (co)homology unless | K | is non-orientable and i = d − , in which case oneshould use Z -coefficients).Proof. The first claim follows easily by excision of the interior of the simplex K ( V ).To prove the second statement for homology groups let V ′ = K − V and considerthe following portion of the Mayer-Vietoris sequence H i ( N ( V ) ∩ N ( V ′ )) → H i ( N ( V )) ⊕ H i ( N ( V ′ )) → H i ( K ) → H i − ( N ( V ) ∩ N ( V ′ ))Observe that H i ( N ( V )) = 0, that H i ( N ( V ) ∩ N ( V ′ )) = H i ( S d − ) = 0 for i < d − H d ( | K | ) → H d − ( N ( V ) ∩ N ( V ′ )) is surjective (with Z -coefficients if | K | is non-orientable). By exactness of the above sequence H i ( K ) ∼ = H i ( N ( V ′ )) ∼ = H i ( K ( V ′ ))for i < d . The proof for cohomology groups is similar. (cid:3) Lusternik-Schnirelmann category.
A subset A ⊆ X of a topological space X is said to be categorical if the inclusion map A ֒ → X is nul-homotopic (i.e.,if there exists a homotopy between the inclusion and the constant map). Theminimal cardinality of an open categorical cover of X is denoted cat( X ) and iscalled the Lusternik-Schnirelmann category of X . For example, the category of aspace is 1 if, and only if, it is contractible, and the category of a (non-contractible)suspension is 2, because every suspension has a natural cover by two contractiblecones. See [5] for a comprehensive survey of the results and the vast literatureabout Lusternik-Schnirelmann category and related topics. (Keep in mind whencomparing the results that the survey [5] and the article [6] use the normalizedvalue of cat( X ) which is one less than in our definition so that contractible spaceshave category 0 and non-contractible suspensions have category 1). Lusternik-Schnirelmann category is tightly related to other homotopy invariants, for example,a well-known result states that if cat( X ) ≤ X isfree (see [5, p.44]).We will base our results on a similar but much deeper theorem proved by Dranish-nikov, Katz and Rudyak [6, Corollary 1.2]: if M is a closed d -dimensional manifold( d ≥
3) and if cat( M ) ≤ M is free. Their proof is RIANGULATIONS WITH FEW VERTICES 5 based on the notion of category weight which we briefly recall. Roughly speaking,a non-zero class u ∈ e H ∗ ( M ) (here we omit the coefficients for cohomology fromthe notation) has category weight at least k if the restriction of u to any union of k categorical subsets of M is trivial. Precise definition is slightly more technical -see [5, Section 2.7] or [6, Section 3]. Clearly, if we can find classes u, v ∈ e H ∗ ( M ) ofweight k and l respectively, and such that 0 = u · v ∈ e H ∗ ( M ), then cat( M ) > k + l .We can summarize the main result of [6, Section 4] as follows: Theorem 2.3.
Let M be a closed d -dimensional ( d ≥ ) manifold M whose fun-damental group is not free. Then there exist suitable systems of coefficients on M and cohomology classes u ∈ H ( M ) of weight 2 and v ∈ H d − ( M ) of weight 1, suchthat = u · v ∈ H d ( M ) . As a consequence, cat( M ) ≥ . Homotopy triangulations and covering type.
Let us denote by ∆( X ) theminimal number of vertices in a triangulation of a compact polyhedron. Clearly,∆( X ) is a topological invariant of compact polyhedra but it is in general very farfrom being a homotopy invariant. As an easy example let X := S ∨ S ∨ S ,the one-point union of three circles, let X be the graph with two vertices and fourparallel edges between them and let X := ∆ (1)3 , the 1-skeleton of the tetrahedron.All three spaces have the same homotopy type and yet easy geometric reasoningshows that ∆( X ) = 7, ∆( X ) = 5, ∆( X ) = 4. To obtain a homotopy invariantnotion recall that a homotopy triangulation of X is a simplicial complex K togetherwith a homotopy equivalence X ≃ | K | . Then the minimal number of vertices amongall possible homotopy triangulations of X is not only a homotopy invariant of X butit also provides a link to the concept of covering type that was recently introducedby M. Karoubi and C. Weibel [9].Recall that a cover U of a space X is said to be good if all finite non-empty in-tersections of elements of U are contractible. Standard examples are covers byconvex sets, covers of polyhedra by open stars of vertices and covers of Riemannianmanifolds by geodesic balls. One of the main facts about good covers is the NerveTheorem (see [8, Corollary 4.G3]): if U is a good open cover of a paracompact space X , then X ≃ | N ( U ) | , where | N ( U ) | is the geometric realization of the nerve of U .Karoubi and Weibel defined the covering type of X as the minimum cardinality ofa good open cover of a space that is homotopy equivalent to X .If X admits a homotopy triangulation X ≃ | K | , where the simplicial complex K has n vertices, then the open stars of the vertices form a good cover for | K | , thereforect( X ) ≤ n . Conversely, if there exists a homotopy equivalence X ≃ Y where Y has a good open cover U with n elements, then X ≃ Y ≃ | N ( U ) | is a homotopytriangulation of X with n vertices. Thus we have proved the following result (cf.[7, Theorem 1.2]): Proposition 2.4. If X has the homotopy type of a compact polyhedron, then ct( X ) equals the minimal number of vertices in a homotopy triangulation of X . For every compact polyhedron X there is the obvious relation ∆( X ) ≥ ct( X ) andwe have seen previously that ∆( X ) can be in fact much bigger that ct( X ). However,if M is a closed triangulable manifold then there is some evidence that ∆( M ) arect( M ) close and often equal. Notably, Borghini and Minian [2] showed that for PETAR PAVEˇSI´C closed surfaces ∆( M ) and ct( M ) coincide, with the sole exception of the orientablesurface of genus 2, where the two quantities differ by one.There are several useful estimates of ct( X ) based on other homotopy invariantsof X . For example, let hdim( X ) denote the homotopy dimension of X , i.e. theminimal dimension of a homotopy triangulation of X . Then we have the followingestimate (cf. [9, Proposition 3.1]): Proposition 2.5.
Let k = hdim( X ) . If ct( X ) = k + 2 , then X ≃ S k , otherwise ct( X ) ≥ k + 3 .Proof. If ct( X ) ≤ n , then by Nerve theorem X has a homotopy triangulation bya subcomplex of ∆ n − . However, | ∆ n − | is contractible, the only subcomplex of∆ n − whose homotopy dimension is its ( n − | ∆ ( n − n − | ≈ S n − , while allthe other subcomplexes have the homotopy dimension at most n − (cid:3) Govc, Marzantowicz and Paveˇsi´c [7] applied techniques from Lusternik-Schnirelmanncategory to obtain further estimates of the covering type of a space and proved thefollowing results:
Theorem 2.6. ( [7, Theorem 4.1] ) The covering type of a r -fold wedge of sphere ofdimension i equals the minimal integer n for which (cid:0) n − i +1 (cid:1) ≥ r . and Theorem 2.7. ( [7, Corollary 2.4] ) Let M be a d -dimensional closed manifold.Then every triangulation of M has at least d + 12 cat( M )(cat( M ) − vertices. Proofs
In this section we provide the proofs for the results states in Section 1.
Proof. ( of Theorem 1.1 ) Observe that M is by assumption simply-connectedand hence orientable, which implies that Poincar´e duality holds with arbitrarycoefficients.Let K be a combinatorial triangulation of M . Since M is d -dimensional, thereexists a ( d + 1)-element subset V ⊂ K spanning a simplex. Lemma 2.2, togetherwith Seifert-van Kampen theorem imply that K ( K − V ) is simply connected andthat H i ( K ( K − V )) = H i ( K ) for i < d .Under the assumption (a), if d = 2 i and if H i ( M ) is the first non-trivial homologygroup of M , then the homology of K ( K − V ) is free and concentrated in dimension i . It follows that K ( K − V ) is homotopy equivalent to a wedge of i -dimensionalspheres. By Theorem 2.6 the covering type of a wedge of r spheres of dimension i isequal to i + k + 1 where k is the minimal integer satisfying (cid:0) i + ki +1 (cid:1) ≥ r . We concludethat K has at least ( d + 1) + ( d + k + 1) = 3 d + k + 2 elements.Moreover, if k = 1 then clearly rank H i ( M ) = 1, therefore M admits a CW-decomposition with three cells in dimensions 0 , i and d , respectively. Then the RIANGULATIONS WITH FEW VERTICES 7 i -dimensional skeleton is the sphere S i and the d -dimensional cell is attached to S i by a map with Hopf invariant 1. By the celebrated theorem of Adams, this ispossible only if i ∈ { , , , } .Under the assumption (b) H i ( M ) = 0. If H i ( M ) ∼ = Z , then by the UniversalCoefficients Theorem H i ( M ) ∼ = Z and by Poincar´e duality H d − i ( M ) ∼ = Z . On theother hand if H i ( M ) = Z , then by Poincar´e duality H d − i ( M ) = 0 or Z . Lemma2.2 yields H k ( K ( K − V )) ∼ = H k ( M ) for k < d , which in both cases implies thathdim( K ( K − V )) ≥ d − i , and that the cohomology of K ( K − V ) is not that ofa sphere. By Proposition 2.5 the covering type of K ( K − V ) is at least d − i + 3.We conclude that K has at least ( d + 1) + ( d − i + 3) = 2 d − i + 4 elements. (cid:3) Proof. ( of Theorem 1.2 ) Let K be a combinatorial triangulation of M . Since M is d -dimensional, its triangulation must contain at least one d -simplex, and so thereexist vertices v , . . . , v d +1 ∈ K that span a d -dimensional simplex in K . Let usenumerate the remaining vertices so that K = { v , . . . , v d +1 , . . . , v n } .By adding one vertex at a time we obtain a sequence of subcomplexes∆ d = K d +1 < . . . < K k < K n = K, where K k = K ( v , . . . , v k ) ≤ K . Since π ( M ) is non-trivial, there exists a minimal l , such that π ( | K ( v , . . . , v l ) | ) is non-trivial. By expressing K l as in formula (1) K l = K l − ∪ v l ∗ (lk( v l ) ∩ K l − ) , we see that K l is a union of two simply-connected subcomplexes. By Seifert-vanKampen theorem its fundamental group can be non-trivial only if (the geomet-ric realization of) the intersection L := lk( v l ) ∩ K ( v , . . . , v l − ) has at least twocomponents. Let us denote L ′ := lk( v l ) ∩ K ( v l +1 , . . . , v n ). Then L and L ′ arefull subcomplexes of lk( v l ) and their vertices determine a partition of the ver-tices of lk( v l ). By lemma 2.1 | L ′ | is a deformation retract of | lk( v l ) | − | L | . Since | lk( v l ) | ≈ S d − , we can apply Alexander duality [8, Theorem 3.44] and obtain that H d − ( | L ′ | ) ∼ = e H ( | L | ) = 0 . By Proposition 2.5 there exist d − v l +1 , . . . , v l + d − , that span a simplex in L ′ . Since these vertices arecontained in lk( v l ), they can be joined to v l in K , therefore vertices v l , . . . , v l + d − span a simplex in K .Let us denote A := { v , . . . , v d +1 } and B := { v l , . . . , v l + d − } . A and B are disjointand together contain 2 d + 1 vertices of K . To conclude the proof, we must showthat K − A − B contains at least d vertices.Since π ( M ) is not free, Theorem 2.3 states that there exist cohomology classes u ∈ H ( M ) of weight 2 and v ∈ H d − ( M ) of weight 1, such that u · v = 0. Both K ( A ) and K ( B ) are contractible, therefore N ( A ∪ B ) = N ( A ) ∪ N ( B ) is a unionof two categorical sets. It follows that u | N ( A ∪ B ) = 0, and so the restriction of v to N ( K − A − B ) cannot be trivial, as it would contradict u · v = 0. Therefore H d − ( N ( K − A − B )) = 0 and the Proposition 2.5 implies that K − A − B mustcontain at least d vertices, as claimed. (cid:3) Proof. ( of Corollaries 1.3 and 1.4 ) Every 1- or 2-dimensional homology sphereis a PL-sphere so we may assume d ≥
3. If there is a PL-triangulation of M with less than 3 d + 1 vertices, then π ( M ) is free by by Theorem 1.2. Therefore, PETAR PAVEˇSI´C assumptions H ( M ) = 0 or H ( M ; Z p ) = 0 imply that M is simply-connected, andso it is homeomorphic to S d by the positive answer to the Poincar´e conjecture. (cid:3) Proof. ( of Theorem 1.5 ) Let σ be a simplex in K of codimension k + 1 andlet x ∈ | K | be a point lying in the interior of σ . Then we may use excision andhomology sequence of the pair to relate the homology of lk( σ ) to the local homologyof | K | at x : Z ∼ = H d ( | K | , | K | − x ) ∼ = H d ( σ ∗ lk( σ ) , ∂σ ∗ lk( σ )) ∼ = ∼ = H d − ( ∂σ ∗ lk( σ )) = H d − (Σ d − k − lk( σ )) ∼ = e H k (lk( σ )) . It follows that lk( σ ) is a k -dimensional homology sphere.If codimension of σ is at most 3, then dim lk( σ ) ≤
2, therefore lk( σ ) is a combina-torial triangulation of a sphere. We will use this as a base for induction.Let σ be a simplex of codimension k + 1, and assume that for each v ∈ lk( σ ) thelink lk( v, lk( σ )) = lk( { v } ∪ σ ) is combinatorially equivalent to S k − . It followsthat lk( σ ) is a combinatorial triangulation of a k -dimensional homology sphere. Byassumption lk( σ ) has at most 3 k vertices, so Corollary 1.3 implies that lk( σ ) is acombinatorial triangulation of S k . We conclude that links of all simplices in K are homeomorphic to spheres of suitable dimensions, hence the triangulation K iscombinatorial. (cid:3) References [1] B. Bagchi and B. Datta, Combinatorial triangulations of homology spheres,
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Geometry & Topology (2008) 1711-1727.[7] D. Govc, W. Marzantowicz and P. Paveˇsi´c, Estimates of covering type and the number ofvertices of minimal triangulations, arXiv:1710.03333, 2017, 19 pages.[8] A. Hatcher, Algebraic Topology , (Cambridge University Press, 2002).[9] M. Karoubi, C. Weibel, On the covering type of a space, arXiv:1612.00532, 2016, 16 pages.[10] S. Klee, I. Novik, Face enumeration on simplicial complexes, in
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Result. Math. (1986) 95-106.[12] F.H. Lutz, Triangulated Manifolds with Few Vertices: Combinatorial Manifolds,arXiv:math.CO/0506372v1, 2005, 37 pages.(Petar Paveˇsi´c) Faculty of Mathematics and Physics, University of Ljubljana, Slovenija
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