### A combinatorial theorem for a symmetric triangulation of the sphere ${S}^{2}$

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We introduce and study a rough (approximate) curvature-dimension condition for metric measure spaces, applicable especially in the framework of discrete spaces and graphs. This condition extends the one introduced by Karl-Theodor Sturm, in his 2006 article On the geometry of metric measure spaces II, to a larger class of (possibly non-geodesic) metric measure spaces. The rough curvature-dimension condition is stable under an appropriate notion of convergence, and stable under discretizations as...

We give a self-contained introduction to the theory of shadows as a tool to study smooth 3-manifolds and 4-manifolds. The goal of the present paper is twofold: on the one hand, it is intended to be a shortcut to a basic use of the theory of shadows, on the other hand it gives a sketchy overview of some of the most recent results on shadows. No original result is proved here and most of the details of the proofs are left out.

The paper gives an illustrated introduction to the theory of hyperbolic virtual polytopes and related counterexamples to A.D. Alexandrov’s conjecture.

We consider locally standard 2-torus manifolds, which are a generalization of small covers of Davis and Januszkiewicz and study their equivariant classification. We formulate a necessary and sufficient condition for two locally standard 2-torus manifolds over the same orbit space to be equivariantly homeomorphic. This leads us to count the equivariant homeomorphism classes of locally standard 2-torus manifolds with the same orbit space.

Let ${P}_{k}$ be a path on $k$ vertices. In an earlier paper we have proved that each polyhedral map $G$ on any compact $2$-manifold $\mathbb{M}$ with Euler characteristic $\chi \left(\mathbb{M}\right)\le 0$ contains a path ${P}_{k}$ such that each vertex of this path has, in $G$, degree $\le k\u230a\frac{5+\sqrt{49-24\chi \left(\mathbb{M}\right)}}{2}\u230b$. Moreover, this bound is attained for $k=1$ or $k\ge 2$, $k$ even. In this paper we prove that for each odd $k\ge \frac{4}{3}\u230a\frac{5+\sqrt{49-24\chi \left(\mathbb{M}\right)}}{2}\u230b+1$, this bound is the best possible on infinitely many compact $2$-manifolds, but on infinitely many other compact $2$-manifolds the upper bound can be lowered to $\u230a(k-\frac{1}{3})\frac{5+\sqrt{49-24\chi \left(\mathbb{M}\right)}}{2}\u230b$.