Related papers: Diagrams and the second homotopy group

Given an $n$-gon, the poset of all collections of pairwise non-crossing diagonals is isomorphic to the face poset of some convex polytope called \textit{associahedron}. We replace in this setting the $n$-gon (viewed as a disc with $n$…

A discrete group is matricially stable if every function from the group to a complex unitary group that is "almost multiplicative" in the point-operator norm topology is "close" to a genuine unitary representation. It follows from a recent…

The two pillars of Algebraic topology - Homology and homotopy theory rely on the availability of basic building blocks called cells. Cells take the form of simplexes, and have properties such as faces, sub-cells, convexity and…

In this paper we generalize some of these results for loop algebras and groups as well as for the Virasoro algebra to the two-dimensional case. We define and study a class of infinite dimensional complex Lie groups which are central…

We build on the results of [6] to show that the homology groups $\mathrm{H}_{r_1+r_2}(Y_0(\mathcal{N}_\Sigma),\mathcal{O})_{\mathfrak{m}_\Sigma}$ of arithmetic manifolds are free over certain deformation rings $R_\Sigma$, when there are…

Deformation K-theory associates to each discrete group G a spectrum built from spaces of finite dimensional unitary representations of G. In all known examples, this spectrum is 2-periodic above the rational cohomological dimension of G…

A general expansion scheme based on the concept of linked cluster expansion from the theory of classical spin systems is constructed for models of interacting electrons. It is shown that with a suitable variational formulation of mean-field…

Main Theorem: Spaces of r-branch point 3-cycle covers, degree n or Galois of degree n!/2 have one (resp. two) component(s) if r=n-1 (resp. r\ge n). Improves Fried-Serre on deciding when sphere covers with odd-order branching lift to…

This paper studies the free group of rank two from the point of view of Stallings core graphs. The first half of the paper examines primitive elements in this group, giving new and self-contained proofs for various known results about them.…

Let $k$ be a finite field and let $G$ be a reductive group over $k[[\pi]]$. Suppose $\mathrm{char}(k)$ is larger than twice the Coxeter number of $G$, we prove the standard endoscopic fundamental lemma for the spherical Hecke algebra of $G$…

An integral homology theory on the category of undirected reflexive graphs was constructed in [2]. A geometrical method to understand behaviors of $1$- and $2$-simplices under differential maps of the theory was developed in [3] and led us…

First we confirm a conjecture asserting that any compact K\"ahler manifold $N$ with $\Ric^\perp>0$ must be simply-connected by applying a new viscosity consideration to Whitney's comass of $(p, 0)$-forms. Secondly we prove the projectivity…

A framework for higher gauge theory based on a 2-group is presented, by constructing a groupoid of connections on a manifold acted on by a 2-group of gauge transformations, following previous work by the authors where the general notion of…

Let $G=*_\lambda G_\lambda$ be a free product of torsion-free groups, and let $g\in[G,G]$ be any element not conjugate into a $G_\lambda$. Then scl$_G(g)\ge1/2$. This generalizes, and gives a new proof of a theorem of Duncan-Howie.

The fundamental group of the complement of a hyperplane arrangement plays an important role in studying the corresponding arrangements. In particular, for large families of hyperplane arrangements, this fundamental group, being isomorphic…

There is a lattice of torsion theories in simplicial groups such that the torsion/torsion-free categories are given by simplicial groups with truncated Moore complex below/above a certain degree. We study the restriction of these torsion…

For a group $G,$ let $\Gamma(G)$ denote the graph defined on the elements of $G$ in such a way that two distinct vertices are connected by an edge if and only if they generate $G$. Moreover let $\Gamma^*(G)$ be the subgraph of $\Gamma(G)$…

There is a striking similarity between Macdonald's reduced word formula and the image of the Schubert class in the cohomology ring of the permutahedral variety $\mathrm{Perm}_n$ as computed by Klyachko. Toward understanding this better, we…

We prove a rank 1 version of the Hanna Neumann Theorem. This shows that every one-relator 2-complex without torsion has the nonpositive immersion property. The proof generalizes to staggered and reducible 2-complexes.

Given a smooth compact complex surface together with a holomorphic line bundle on it, using the theory of Hodge modules, we compute the twisted Hodge groups/numbers of Hilbert schemes (or Douady spaces) of points on the surface with values…