We show that the fundamental groups of any two closed irreducible nongeometric graph manifolds are quasi-isometric. We also classify the quasi-isometry types of fundamental groups of graph manifolds with boundary in terms of certain finite two-colored graphs. A corollary is the quasi-isometric classification of Artin groups whose presentation graphs are trees. In particular, any two right-angled Artin groups whose presentation graphs are trees of diameter greater than $2$ are quasi-isometric; further, this quasi-isometry class does not include any other right-angled Artin groups

This article contains a description of one connected component of the space of stability conditions on the bounded derived category of coherent sheaves on a complex algebraic K$3$ surface

We define slow quasiregular mappings and study cohomology and universal coverings of closed manifolds receiving slow quasiregular mappings. We show that closed manifolds receiving a slow quasiregular mapping from a punctured ball have the de Rham cohomology type of either ${\mathbb{S}}^n$ or ${\mathbb{S}}^{n-1}\times {\mathbb{S}}^1$. We also show that in the case of manifolds of the cohomology type of ${\mathbb{S}}^{n-1}\times {\mathbb{S}}^1$, the universal covering of the manifold has exactly two ends, and the lift of the slow mapping into the universal covering has a removable singularity at the point of punctuation. We also obtain exact growth bounds and a global homeomorphism–type theorem for slow quasiregular mappings into the manifolds of the cohomology type ${\mathbb{S}}^{n-1}\times {\mathbb{S}}^1$

Let $k$ be a field of characteristic zero, and let $\stackrel{̲}{k}$ be an algebraic closure of $k$. For a geometrically integral variety $X$ over $k$, we write $\stackrel{̲}{k}(X)$ for the function field of $\stackrel{̲}{X}=X{\times }_k\stackrel{̲}{k}$. If $X$ has a smooth $k$-point, the natural embedding of multiplicative groups ${\stackrel{̲}{k}}^{*}\hookrightarrow \stackrel{̲}{k}(X{)}^{*}$ admits a Galois-equivariant retraction.

In the first part of this article, equivalent conditions to the existence of such a retraction are given over local and then over global fields. Those conditions are expressed in terms of the Brauer group of $X$.

In the second part of the article, we restrict attention to varieties that are homogeneous spaces of connected but otherwise arbitrary algebraic groups, with connected geometric stabilizers. For $k$ local or global, and for such a variety $X$, in many situations but not all, the existence of a Galois-equivariant retraction to ${\stackrel{̲}{k}}^{*}\hookrightarrow \stackrel{̲}{k}(X{)}^{*}$ ensures the existence of a $k$-rational point on $X$. For homogeneous spaces of linear algebraic groups, the technique also handles the case where $k$ is the function field of a complex surface.

Soient $k$ un corps de caractéristique nulle et $\stackrel{̲}{k}$ une clôture algébrique de $k$. Pour une $k$-variété $X$ géométriquement intègre, on note $\stackrel{̲}{k}(X)$ le corps des fonctions de $\stackrel{̲}{X}=X{\times }_k\stackrel{̲}{k}$. Si $X$ possède un $k$-point lisse, le plongement naturel de groupes multiplicatifs ${\stackrel{̲}{k}}^{*}\hookrightarrow \stackrel{̲}{k}(X{)}^{*}$ admet une rétraction équivariante pour l'action du groupe de Galois de $\stackrel{̲}{k}$ sur $k$.

Dans la première partie de l'article, sur les corps locaux puis sur les corps globaux, on donne des conditions équivalentes à l'existence d'une telle rétraction équivariante. Ces conditions s'expriment en terme du groupe de Brauer de la variété $X$.

Dans la seconde partie de l'article, on considère le cas des espaces homogènes de groupes algébriques connexes, non nécessairement linéaires, avec groupes d'isotropie géométriques connexes. Pour $k$ local ou global, pour un tel espace homogène $X$, dans beaucoup de cas mais pas dans tous, l'existence d'une rétraction équivariante à ${\stackrel{̲}{k}}^{*}\hookrightarrow \stackrel{̲}{k}(X{)}^{*}$ implique l'existence d'un point $k$-rationnel sur $X$. Pour les espaces homogènes de groupes linéaires, la technique permet aussi de traiter le cas où $k$ est un corps de fonctions de deux variables sur les complexes

We extend knot contact homology to a theory over the ring $\mathbb{Z}[{\lambda }^{\pm 1},{\mu }^{\pm 1}]$ with the invariant given topologically and combinatorially. The improved invariant, which is defined for framed knots in $S^3$ and can be generalized to knots in arbitrary manifolds, distinguishes the unknot and can distinguish mutants. It contains the Alexander polynomial and naturally produces a two-variable polynomial knot invariant that is related to the $A$-polynomial