Related papers: A note on the connectivity of certain complexes as…

In this paper, second installment in a series of three, we give a correspondence theorem to relate the count of genus $g$ curves in a fixed linear system in an abelian surface to a tropical count. To do this, we relate the linear system…

A viable and still unproved conjecture states that, if $X$ is a smooth algebraic surface and $C$ is a smooth algebraic curve in $X$, then $C$ realizes the smallest possible genus amongst all smoothly embedded $2$-manifolds in its homology…

We give a short, mostly elementary and self-contained proof of the classical result that the groups of diffeomorphisms, homeomorphisms, and homotopy equivalences of a surface have the same group of connected components.

One of the ways that connectedness has been studied through the history of topology is by using chains, the so called chain connectedness. Here we combine this notion together with continuity up to a covering to provide the inheritance of…

The aim of this small note is to prove an elementary yet useful properties of finitely presented groups. Let G be a finitely generated group with one end. Fix a (finite) generating set and let $B_n$ be the ball of radius $n$ around $e$. Let…

We formulate a tropical analogue of Grothendieck's section conjecture: that for every stable graph G of genus g>2, and every field k, the generic curve with reduction type G over k satisfies the section conjecture. We prove many cases of…

Progress on the problem whether the Hilbert schemes of locally Cohen-Macaulay curves in projective 3 space are connected has been hampered by the lack of an answer to a question that was raised by Robin Hartshorne in his paper "On the…

Given two finite covers $p: X \to S$ and $q: Y \to S$ of a connected, oriented, closed surface $S$ of genus at least $2$, we attempt to characterize the equivalence of $p$ and $q$ in terms of which curves lift to simple curves. Using…

An interesting problem in classical differential geometry is to find methods to prove that two surfaces defined by different charts actually coincide up to position in space. In a previous paper we proposed a method in this direction for…

We establish a criterion that ensures a bounded almost complex curve in a bounded almost complex 4-manifold minimizes genus amongst all smooth surfaces that share its homology class and the transverse link on its boundary. An immediate…

We define the $k$-cut complex of a graph $G$ with vertex set $V(G)$ to be the simplicial complex whose facets are the complements of sets of size $k$ in $V(G)$ inducing disconnected subgraphs of $G$. This generalizes the Alexander dual of a…

A translation surface is a surface formed by identifying edges of a collection of polygons in the complex plane that are parallel and of equal length using only translations. We determined that the same circle packing can be realized on…

For a genus g handlebody H a simplicial complex, with vertices being isotopy classes of certain incompressible surfaces in H, is constructed and several properties are established. In particular, this complex naturally contains, as a…

Any smooth projective curve embeds into $\mathbb{P}^3$. More generally, any curve embeds into a rationally connected variety of dimension at least three. We prove conversely that if every curve embeds in a threefold $X$, then $X$ is…

The paper is devoted to metric properties of singularities. We investigate the relations among topology, metric properties and smoothness. In particular, we present some higher dimensional analogous of Mumford's theorem on smoothness of…

If $X$ is a compact set, a {\it topological contraction} is a self-embedding $f$ such that the intersection of the successive images $f^k(X)$, $k>0$, consists of one point. In dimension 3, we prove that there are smooth topological…

We prove that the bounded derived category of coherent sheaves on a smooth projective complex variety reconstructs the isomorphism classes of fibrations onto smooth projective curves of genus $g\geq 2$. Moreover, in dimension at most four,…

We extend the notion of graph homomorphism to cellularly embedded graphs (maps) by designing operations on vertices and edges that respect the surface topology; we thus obtain the first definition of map homomorphism that preserves both the…

We give necessary and sufficient topological conditions for a simple closed curve on a real rational surface to be approximable by smooth rational curves. We also study approximation by smooth rational curves with given complex…

In the present note we describe geometrically the homology classes in the total space of a surface bundle over a surface in terms of the holonomy map. We treat the cases where the base surface is closed or has one boundary component. We…