Related papers: The Topological Complexity of a Surface

A projective surface S is said to be isogenous to a product if there exist two smooth curves C, F and a finite group G acting freely on C \times F so that S=(C \times F)/G. In this paper we classify all surfaces with p_g=q=1 which are…

We show that a real rational (over $\C$) surfaces are quasi-simple, i.e., that such a surface is determined up to deformation in the class of real surfaces by the topological type of its real structure.

Given a polynomial map $\psi:S^m\to \mathbb{R}^k$ with components of degree $d$, we investigate the structure of the semialgebraic set $Z\subseteq S^m$ consisting of those points where $\psi$ and its derivatives satisfy a given list of…

Let $S$ be a closed, orientable surface of genus at least 2. The cotangent bundle of the "hyperbolic'' Teichm\"uller space of $S$ can be identified with the space $\CP$ of complex projective structures on $S$ through measured laminations,…

We determine the symmetrized topological complexity of the circle, using primarily just general topology.

Given a finite subgroup G of the mapping class group of a surface S, the Nielsen realization problem asks whether G can be realized as a finite group of homeomorphisms of S. In 1983, Kerckhoff showed that for S a finite-type surface, any…

A {\em $1-$vertex triangulation} of an oriented compact surface $S$ of genus $g$ is an embedded graph $T\subset S$ with a unique vertex such that all connected components of $S\setminus T$ are triangles (adjacent to exactly 3 edges of $T$).…

We use an alternative definition of topological complexity to show that the topological complexity of the mapping telescope of a sequence $X_1\rightarrow X_2\rightarrow X_3\rightarrow...$ is bounded above by $2max{TC(X_i); i=1,2,...}$.

In this article we determine, for an infinite family of maps on the plane, the topology of the surface on which the minimal regular covering occurs. This infinite family includes all Archimedean maps.

Let $p:\Sigma'\to\Sigma$ be a finite Galois cover, possibly branched, with Galois group $G$. We are interested in the structure of the cohomology of $\Sigma'$ as a module over $G$. We treat the cases of branched and unbranched covers…

Stable compact minimal submanifolds of the product of a sphere and any Riemannian manifold are classified whenever the dimension of the sphere is at least three. The complete classification of the stable compact minimal submanifolds of the…

We consider a finite analytic morphism $\varphi =(f,g)$ defined from a complex analytic normal surface $(Z,z)$ to ${\mathbb C}^2$. We describe the topology of the image by $\varphi$ of a reduced curve on $(Z,z)$ by means of iterated pencils…

We show that if S is a finite type orientable surface of genus g and p punctures where 3g+p > 4, then EL(S) is (n-1)-connected and (n-1)-locally connected where dim(PML(S))=2n+1=6g+2p-7. Furthermore, if g=0, then EL(S) is homeomorphic to…

In this article we describe relations of the topology of closed 1-forms to the group theoretic invariants of Bieri-Neumann-Strebel-Renz. Starting with a survey, we extend these Sigma invariants to finite CW- complexes and show that many…

We present upper and lower bounds for symmetrized topological complexity $TC^\Sigma(X)$ in the sense of Basabe-Gonz\'alez-Rudyak-Tamaki. The upper bound comes from equivariant obstruction theory, and the lower bounds from the cohomology of…

In this paper, we introduce relative LS category of a map and study some of its properties. Then we introduce `higher topological complexity' of a map, a homotopy invariant. We give a cohomological lower bound and compare it with previously…

Using quasiconformal mappings, we prove that any Riemann surface of finite connectivity and finite genus is conformally equivalent to an intrinsic circle domain U in a compact Riemann surface S. This means that each connected component B of…

Given $I,B\in\mathbb{N}\cup \{0\}$, we investigate the existence and geometry of complete finitely branched minimal surfaces $M$ in $\mathbb{R}^3$ with Morse index at most $I$ and total branching order at most $B$. Previous works of…

We determine, for all genus $g\geq2$ the Riemann surfaces of genus $g$ with $4g$ automorphisms. For $g\neq$ $3,6,12,15$ or $30$, this surfaces form a real Riemann surface $\mathcal{F}_{g}$ in the moduli space $\mathcal{M}_{g}$: the Riemann…

Let ${\mathcal M}_g$ be the moduli space of compact connected Riemann surfaces of genus $g\geq 2$ and let $\widehat{{\mathcal M}_g}$ be its Deligne-Mumford compactification, which is stratified by the topological type of the stable Riemann…