Related papers: A generalized lower bound theorem for balanced man…

We show that a realization of a closed connected PL-manifold of dimension n-1 in Euclidean n-space (n>2) is the boundary of a convex polyhedron if and only if the interior of each (n-3)-face has a point, which has a neighborhood lying on…

Using an intuition from metric geometry, we prove that any flag and normal simplicial complex satisfies the non-revisiting path conjecture. As a consequence, the diameter of its facet-ridge graph is smaller than the number of vertices minus…

Let (M,g) be a compact Riemannian three-dimensional manifold with boundary. We prove the compactness of the set of scalar-flat metrics which are in the conformal class of g and have the boundary as a constant mean curvature hypersurface.…

The colorful simplicial depth of a collection of d+1 finite sets of points in Euclidean d-space is the number of choices of a point from each set such that the origin is contained in their convex hull. We use methods from combinatorial…

We generalize the notion of graph minors to all (finite) simplicial complexes. For every two simplicial complexes H and K and every nonnegative integer m, we prove that if H is a minor of K then the non vanishing of Van Kampen's obstruction…

We prove that among all flag triangulations of manifolds of odd dimension 2r-1 with sufficiently many vertices the unique maximizer of the entries of the f-, h-, g- and gamma-vector is the balanced join of r cycles. Our proof uses methods…

For positive integers $n,d$, let $\lambda(n,d)$ be the minimal number of vertices of a triangulation of $n$-sphere which admits a degree $d$ simplicial map to the boundary of $(n+1)$-simplex. We show that…

A $k$-edge-coloured graph is colour-balanced if each colour appears equally often. Resolving a conjecture of Pardey and Rautenbach, we show that any colour-balanced $k$-edge-coloured complete graph $K_{2kt}$ contains a perfect matching that…

We introduce "$t$-LC triangulated manifolds" as those triangulations obtainable from a tree of $d$-simplices by recursively identifying two boundary $(d-1)$-faces whose intersection has dimension at least $d-t-1$. The $t$-LC notion…

We introduce the $k$-stellated spheres and consider the class ${\cal W}_k(d)$ of triangulated $d$-manifolds all whose vertex links are $k$-stellated, and its subclass ${\cal W}^{\ast}_k(d)$ consisting of the $(k+1)$-neighbourly members of…

We prove that if an $n$-dimensional space $X$ satisfies certain topological conditions then any triangulation of $X$ as well as any its representation as a simplicial set with contractible faces has at least $2^n$ faces of dimension $n$.…

We prove several vanishing theorems for the cohomology of balanced hyperbolic manifolds that we introduced in our previous work and for the $L^2$ harmonic spaces on the universal cover of these manifolds. Other results include a Hard…

In this article, we study the topological complexity of manifolds with a lower scalar curvature bound. We introduce a small scale index theorem to establish an upper bound for Gromov's simplicial norm of the Poincar\'e dual of the A-hat…

Standard singularity theorems are proven in Lorentzian manifolds of arbitrary dimension n if they contain closed trapped submanifolds of arbitrary co-dimension. By using the mean curvature vector to characterize trapped submanifolds, a…

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…

In 1975 Bollob\'{a}s, Erd\H{o}s, and Szemer\'{e}di asked what minimum degree guarantees an octahedral subgraph $K_3(2)$ in any tripartite graph $G$ with $n$ vertices in each vertex class. We show that $\delta(G)\geq n+2n^{\frac{5}{6}}$…

We prove that if a pure simplicial complex of dimension d with n facets has the least possible number of (d-1)-dimensional faces among all complexes with n faces of dimension d, then it is vertex decomposable. This answers a question of J.…

A $k$-uniform hypergraph $H = (V, E)$ is $k$-partite if $V$ can be partitioned into $k$ sets $V_1, \ldots, V_k$ such that every edge in $E$ contains precisely one vertex from each $V_i$. We call such a graph $n$-balanced if $|V_i| = n$ for…

A conjecture of Kalai asserts that for $d\geq 4$, the affine type of a prime simplicial $d$-polytope $P$ can be reconstructed from the space of affine $2$-stresses of $P$. We prove this conjecture for all $d\geq 5$. We also prove the…

We introduce a version of discrete Morse theory specific for manifolds with boundary. The idea is to consider Morse functions for which all boundary cells are critical. We obtain "Relative Morse Inequalities" relating the homology of the…