A minimum dimensional counterexample to Ganea's conjecture

A 7-dimensional CW-complex having Lusternik-Schnirelmann category equal to 2 is constructed. Using a divisibility phenomenon for Hopf invariants, it is proved that the Cartesian product of the constructed complex with a sphere of sufficiently large dimension also has category 2. This space hence constitutes the minimum dimensional known counterexample to Ganea's conjecture on the Lusternik-Schnirelmann category of spaces.     

Two special cases of the assignment problem

The assignment problem may be stated as follows: Given finite sets of points S and T, with|S| ? |T|, and given a “metric” which assigns a distance d(x, y) to each pair (x, y) such that x ∈ T and y ∈ S find a 1?1 function Q: T→ S which minimizes Σ_x∈Td(x, Q(x)) We consider the two special cases in which the points lie (1) on a line segment and (2) on a circle, and the metric is the distance along the line segment or circle, respectively. In each case, we show that the optimal assignment Q can be computed in a number of steps (additions and comparisons) proportional to the number of points. The problem arose in connection with the efficient rearrangement of desks located in offices along a corridor which encircles one floor of a building.     

A special case of the Stahl conjecture

Let _Gn,k$G_{n,k}$ denote the Kneser graph whose vertices are the n$n$-element subsets of a (2n+k)$(2n+k)$-element set and whose edges are the disjoint pairs. In this paper we prove that for any non-negative integer s$s$ there is no graph homomorphism from _G4,2$G_{4,2}$ to _G4s+1,2s+1$G_{4s+1,2s+1}$. This confirms a conjecture of Stahl in a special case.     

Generalized Mukai conjecture for special Fano varieties

Let X be a Fano variety of dimension n, pseudoindex i _X and Picard number ρ_X. A generalization of a conjecture of Mukai says that ρ_X(i _X −1)≤n. We prove that the conjecture holds for a variety X of pseudoindex i _X ≥n+3/3 if X admits an unsplit covering family of rational curves; we also prove that this condition is satisfied if ρ_X> and either X has a fiber type extremal contraction or has not small extremal contractions. Finally we prove that the conjecture holds if X has dimension five.     

A special case of mahler’s conjecture

A special case of Mahler's conjecture on the volume-product of symmetric convex bodies in n -dimensional Euclidean space is treated here. This is the case of polytopes with at most 2n+2 vertices (or facets). Mahler's conjecture is proved in this case for n≤ 8 and the minimal bodies are characterized. <lsiheader> <onlinepub>7 August, 1998 <editor>Editors-in-Chief: &lsilt;a href=../edboard.html#chiefs&lsigt;Jacob E. Goodman, Richard Pollack&lsilt;/a&lsigt; <pdfname>20n2p163.pdf <pdfexist>yes <htmlexist>no <htmlfexist>no <texexist>no <sectionname> </lsiheader> Received May 28, 1996, and in revised form November 7, 1996.     

Ganea's Conjecture on Lusternik-Schnirelmann Category

A series of complexes Q_p indexed by all primes p is constructedwith catQ_p=2 and catQ_pxSⁿ=2 for either n2 or n=1 and p=2. Thisdisproves Ganea's conjecture on Lusternik–Schnirelmann(LS) category. 1991 Mathematics Subject Classification 55M30.     

The strong no loop conjecture for special biserial algebras

We establish the strong no loop conjecture for some special cases, in particular, for special biserial algebras.

     

The strong Atiyah conjecture for virtually cocompact special groups

We provide new conditions for the strong Atiyah conjecture to lift to finite group extensions. In particular, we show that fundamental groups of compact special cube complexes satisfy these conditions, so the conjecture holds for finite extensions of these groups.     

Families over special base manifolds and a conjecture of Campana

Consider a smooth, projective family of canonically polarized varieties over a smooth, quasi-projective base manifold Y, all defined over the complex numbers. It has been conjectured that the family is necessarily isotrivial if Y is special in the sense of Campana. We prove the conjecture when Y is a surface or threefold. The proof uses sheaves of symmetric differentials associated to fractional boundary divisors on log canonical spaces, as introduced by Campana in his theory of Orbifoldes Géométriques. We discuss a weak variant of the Harder?CNarasimhan Filtration and prove a version of the Bogomolov?CSommese Vanishing Theorem that take the additional fractional positivity along the boundary into account. A brief, but self-contained introduction to Campana??s theory is included for the reader??s convenience.     

Toward the abelian root groups conjecture for special Moufang sets

We prove that if a root group of a special Moufang set contains an element of order then it is abelian.     

The Rubin-Stark conjecture for a special class of function field extensions

We prove a strong form of the Brumer-Stark Conjecture and, as a consequence, a strong form of Rubin's integral refinement of the abelian Stark Conjecture, for a large class of abelian extensions of an arbitrary characteristic p global field k. This class includes all the abelian extensions K/k contained in the compositum k_p∞?k_p·k_∞ of the maximal pro-p abelian extension k_p/k and the maximal constant field extension k_∞/k of k, which happens to sit inside the maximal abelian extension k^ab of k with a quasi-finite index. This way, we extend the results obtained by the present author in (Comp. Math. 116 (1999) 321-367).     

Some cases of a conjecture on L-functions of twisted Carlitz modules

We prove two polynomial identies which are particular cases of a conjecture arising in the theory of L-functions of twisted Carlitz modules. This conjecture is stated in [6 Grishkov, A., Logachev, D. (2016). Resultantal varieties related to zeroes of L-functions of Carlitz modules. Finite Fields Appl. 38:116–176. Available at: arxiv.org/pdf/1205.2900.pdf.[Crossref], [Web of Science ®] , [Google Scholar], p. 153, 9.3, 9.4] and [8 Logachev, D., Zobnin, A. (2017). L-functions of Carlitz Modules, Resultantal Varieties and Rooted Binary Trees, arXiv:1607.06147v3 [math.AG]. [Google Scholar], p. 5, 0.2.4].     

A unified approach to known and unknown cases of Berge's conjecture

Berge's elegant dipath partition conjecture from 1982 states that in a dipath partition P of the vertex set of a digraph minimizing , there exists a collection C^k of k disjoint independent sets, where each dipath P∈P meets exactly min{|P|, k} of the independent sets in C. This conjecture extends Linial's conjecture, the Greene–Kleitman Theorem and Dilworth's Theorem for all digraphs. The conjecture is known to be true for acyclic digraphs. For general digraphs, it is known for k=1 by the Gallai–Milgram Theorem, for k?λ (where λis the number of vertices in the longest dipath in the graph), by the Gallai–Roy Theorem, and when the optimal path partition P contains only dipaths P with |P|?k. Recently, it was proved (Eur J Combin (2007)) for k=2. There was no proof that covers all the known cases of Berge's conjecture. In this article, we give an algorithmic proof of a stronger version of the conjecture for acyclic digraphs, using network flows, which covers all the known cases, except the case k=2, and the new, unknown case, of k=λ?1 for all digraphs. So far, there has been no proof that unified all these cases. This proof gives hope for finding a proof for all k.     

Two proofs of the Bermond-Thomassen conjecture for tournaments with bounded minimum in-degree

The Bermond-Thomassen conjecture states that, for any positive integer r, a digraph of minimum out-degree at least 2r−1 contains at least r vertex-disjoint directed cycles. Thomassen proved that it is true when r=2, and very recently the conjecture was proved for the case where r=3. It is still open for larger values of r, even when restricted to (regular) tournaments. In this paper, we present two proofs of this conjecture for tournaments with minimum in-degree at least 2r−1. In particular, this shows that the conjecture is true for (almost) regular tournaments. In the first proof, we prove auxiliary results about union of sets contained in another union of sets, that might be of independent interest. The second one uses a more graph-theoretical approach, by studying the properties of a maximum set of vertex-disjoint directed triangles.