• Corpus ID: 235490328

Systolic inequalities for the number of vertices

@inproceedings{Avvakumov2021SystolicIF,
  title={Systolic inequalities for the number of vertices},
  author={Sergey Avvakumov and Alexey Balitskiy and Alfredo Hubard and Roman N. Karasev},
  year={2021}
}
. Inspired by the classical Riemannian systolic inequality of Gromov we present a combinatorial analogue providing a lower bound on the number of vertices of a simplicial complex in terms of its edge-path systole. Similarly to the Riemannian case, where the inequality holds under a topological assumption of “essentiality”, our proofs rely on a combinatorial analogue of that assumption. Under a stronger assumption, expressed in terms of cohomology cup-length, we improve our results… 
1 Citations
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References

SHOWING 1-10 OF 28 REFERENCES
Combinatorial systolic inequalities
We establish combinatorial versions of various classical systolic inequalities. For a smooth triangulation of a closed smooth manifold, the minimal number of edges in a homotopically non-trivial loop
Discrete Systolic Inequalities and Decompositions of Triangulated Surfaces
TLDR
This work builds upon Riemannian systolic inequalities, which bound the minimum length of non-trivial closed curves in terms of the genus and the area of the surface, and proves the existence of pants decompositions of length O(g3/2n1/2) for any triangulated combinatorial surface of genus g with n triangles.
The minimal length product over homology bases of manifolds
Minkowski’s second theorem can be stated as an inequality for n -dimensional flat Finsler tori relating the volume and the minimal product of the lengths of closed geodesics which form a homology
Estimates of Covering Type and the Number of Vertices of Minimal Triangulations
TLDR
This work relates the covering type of a triangulable space to the number of vertices in its minimal triangulations and derives within a unified framework several estimates of vertex-minimalTriangulations which are either new or extensions of results that have been previously obtained by ad hoc combinatorial arguments.
Filling minimality of finslerian 2-discs
We prove that every Riemannian metric on the 2-disc such that all its geodesics are minimal is a minimal filling of its boundary (within the class of fillings homeomorphic to the disc). This improves
How many simplices are needed to triangulate a Grassmannian?
We compute a lower bound for the number of simplices that are needed to triangulate the Grassmann manifold $G_k(\mathbb{R}^n)$. In particular, we show that the number of top-dimensional simplices
Rigidity and the lower bound theorem 1
SummaryFor an arbitrary triangulated (d-1)-manifold without boundaryC withf0 vertices andf1 edges, define $$\gamma (C) = f_1 - df_0 + \left( {\begin{array}{*{20}c} {d + 1} \\ 2 \\ \end{array} }
On Isosystolic Inequalities for T^n, RP^n, and M^3
If a closed smooth n-manifold M admits a finite cover whose Z/2Z-cohomology has the maximal cup-length, then for any riemannian metric g on M, we show that the systole Sys(M,g) and the volume
Filling metric spaces
We prove an inequality conjectured by Larry Guth that relates the $m$-dimensional Hausdorff content of a compact metric space with its $(m-1)$-dimensional Urysohn width. As a corollary, we obtain
Linear bounds for constants in Gromov's systolic inequality and related results
Let $M^n$ be a closed Riemannian manifold. Larry Guth proved that there exists $c(n)$ with the following property: if for some $r>0$ the volume of each metric ball of radius $r$ is less than
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