Related papers: Higher dimensional flexible polyhedra
Paper withdrawn by the author.
This article has been withdrawn due to an error in a proof of the main result.
This paper has been withdrawn as the statements in Proposition 4.4 and Theorem 1.4(i) are not correct.
This paper has been withdrawn by the author due to serious flaws in certain proofs. For instance, the method used to construct certain automorphic representations is flawed.
We demonstrate the existence of four types of flexible prismatic polyhedra that can be derived or inferred from a consideration of Bricard octahedra and generalizations of Bricard octahedra. These flexible polyhedra are of genus 0 and 1,…
A soft presentation of hyperbolic spaces, free of differential apparatus, is offered. Fifth Euclid's postulate in such spaces is overthrown and, among other things, it is proved that spheres (equipped with great-circle distances) and…
For each $d\geq 3$ we construct cube complexes homeomorphic to the $d$-sphere with $n$ vertices in which the number of facets (assuming $d$ constant) is $\Omega(n^{5/4})$. This disproves a conjecture of Kalai's stating that the number of…
This paper has been withdrawn by the author due to a critical error in the proof of Theorem 5.4 on which the proof of the main theorem on the non-simplenss was based.
We present a self-contained proof that the number of diameter pairs among n points in Euclidean 3-space is at most 2n-2. The proof avoids the ball polytopes used in the original proofs by Grunbaum, Heppes and Straszewicz. As a corollary we…
A classical Theorem of Alexandrov states that the map associating its boundary to a convex polyhdedron of the 3-dimensional Euclidean space is a bijection from the set of convex polyhdedron up to congruence to the set of isometry classes of…
This paper has been withdrawn by the author due to mathematical errors.
This paper proves the existence of homeomorphic (diffeomorphic) complex 6-dimensional (7-dim) complete intersections that belong to components of the moduli space of different dimensions. These results are given as a supplement to earlier…
We construct 2^{\Omega(n^{5/4})} combinatorial types of triangulated 3-spheres on n vertices. Since by a result of Goodman and Pollack (1986) there are no more than 2^{O(n log n)} combinatorial types of simplicial 4-polytopes, this proves…
This paper has been withdrawn by the author. The most updated version can be accessed by arXiv:1806.07290.
This paper has been withdrawn by the author due to a crucial error in the proofs. The error has been corrected and the paper has been expanded in arXiv:0910.5327
We explain in some detail the geometric structure of spheres in any dimension. Our approach may be helpful for other homogeneous spaces (with other signatures) such as the de Sitter and anti-de Sitter spaces. We apply the procedure to the…
We introduce topological prismatoids, a combinatorial abstraction of the (geometric) prismatoids recently introduced by the second author to construct counter-examples to the Hirsch conjecture. We show that the `strong $d$-step Theorem'…
This paper has been withdrawn by the author due to a critical error in the proof of Theorem A pointed out by Burkhard Wilking.
This paper has been withdrawn by the author, due a crucial mistake in proof of lemma 4.2.
We say that a topologically embedded 3-sphere in a smoothing of Euclidean 4-space is a barrier provided, roughly, no diffeomorphism of the 4-manifold moves the 3-sphere off itself. In this paper we construct infinitely many one parameter…