Combinatorial manifolds with few vertices

Brehm, Ulrich; Kühnel, Wolfgang

doi:10.1016/0040-9383(87)90042-5

Cited by 74 publications

(107 citation statements)

References 16 publications

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“…[3, 4,7], [3,4,5], [4,5,6], [5,6,8], [6,8,9] For (iii)⇒(i), the previous proposition shows that it is sufficient to find a triangulation of CP 2 with g-vector (1, 3, 6) and simple 4-tree which satisfies the conditions of the previous proposition. Table 1 contains such a triangulation, originally due to Kühnel [22], and Table 2 shows an appropriate simple 2-tree in the link of an edge.…”

Section: Constructionsmentioning

confidence: 98%

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Face enumeration—from spheres to manifolds

Swartz¹

2009

J. Eur. Math. Soc.

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Abstract. We prove a number of new restrictions on the enumerative properties of homology manifolds and semi-Eulerian complexes and posets. These include a determination of the affine span of the fine h-vector of balanced semi-Eulerian complexes and the toric h-vector of semi-Eulerian posets.The lower bounds on simplicial homology manifolds, when combined with higher dimensional analogues of Walkup's 3-dimensional constructions [47], allow us to give a complete characterization of the f -vectors of arbitrary simplicial triangulations of S 1 × S 3 , CP 2 , K3 surfaces, and. We also establish a principle which leads to a conjecture for homology manifolds which is almost logically equivalent to the g-conjecture for homology spheres. Lastly, we show that with sufficiently many vertices, every triangulable homology manifold without boundary of dimension three or greater can be triangulated in a 2-neighborly fashion.

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Section: Constructionsmentioning

confidence: 98%

“…Theorem 4.18 implies 6 ≤ g 2 , and hence (ii)⇒(iii). [1,3,4,7,8], [1,3,4,8,9] , [1,3,5,6,8], [1,3,5,6,9], [1,3,6,8,9], [1,4,5,6,7] , [1,4,5,7,9], [1,4,7,8,9], [1,5,6,7,9], [2,3,4,5,9] , [2,3,4,6,…”

Section: Constructionsmentioning

confidence: 99%

Face enumeration—from spheres to manifolds

Swartz¹

2009

J. Eur. Math. Soc.

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“…First, each component of the boundary of ∆ is a closed surface, so the Dehn-Sommerville relations tell us that the g 2 of each component is 3β 1 . Second, to show that Σ is k-rigid the induction must begin with any closed surface instead of just S 2 .…”

Section: It Follows From Proposition 55 Below That If ∆ Ismentioning

confidence: 99%

Applications of Klee’s Dehn–Sommerville Relations

Novik

Swartz

2009

Discrete Comput Geom

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We use Klee's Dehn-Sommerville relations and other results on face numbers of homology manifolds without boundary to (i) prove Kalai's conjecture providing lower bounds on the f -vectors of an even-dimensional manifold with all but the middle Betti number vanishing, (ii) verify Kühnel's conjecture that gives an upper bound on the middle Betti number of a 2k-dimensional manifold in terms of k and the number of vertices, and (iii) partially prove Kühnel's conjecture providing upper bounds on other Betti numbers of odd-and even-dimensional manifolds. For manifolds with boundary, we derive an extension of Klee's Dehn-Sommerville relations and strengthen Kalai's result on the number of their edges.

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“…Example 7 corresponds to the 6-vertex triangulation of projective 2-plane, and can be generalized as vertex-minimal triangulation of projective space only in dimensions 4, 8, 16, see [2,3].…”

Section: 1mentioning

confidence: 99%

“…It appeared as "simple game with constant sum" in game theory, see [12,4] and also as "strongly complementary simplicial complex", see [2,3].…”

Section: Introductionmentioning

confidence: 99%

Manifolds associated to simple games

Galashin

Panina

2016

J. Knot Theory Ramifications

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Abstract. Starting by a simple game Q as a combinatorial data, we build up a cell complex M (Q), whose construction resembles combinatorics of the permutohedron. The cell complex proves to be a combinatorial manifold; we call it the simple game induced manifold. By some motivations coming from polygonal linkages, we think of Q and of M (Q) as of a quasilinkage and the moduli space of the quasilinkage respectively. We present some examples of quasilinkages and show that the moduli space retains many properties of moduli space of polygonal linkages. In particular, we show that the moduli space M (Q) is homeomorphic to the space of stable point configurations on S 1 , for an associated with a quasilinkage notion of stability.

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Combinatorial manifolds with few vertices

Cited by 74 publications

References 16 publications

Face enumeration—from spheres to manifolds

Face enumeration—from spheres to manifolds

Applications of Klee’s Dehn–Sommerville Relations

Manifolds associated to simple games

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