Related papers: Topological Complexities of Surfaces

The disk complex of a surface in a 3-manifold is used to define its {\it topological index}. Surfaces with well-defined topological index are shown to generalize well-known classes, such as incompressible, strongly irreducible, and critical…

There are many methods for projecting spherical maps onto the plane. Interactive versions of these projections allow the user to centre the region of interest. However, the effects of such interaction have not previously been evaluated. In…

Neutral surfaces, along which most of the mixing in the ocean occurs, are notoriously difficult objects: they do not exist as well-defined surfaces, and as such can only be approximated. In a hypothetical ocean where neutral surfaces are…

Let $S$ be a complete flat surface, such as the Euclidean plane. We determine the homeomorphism class of the space of all curves on $S$ which start and end at given points in given directions and whose curvatures are constrained to lie in a…

We study how to construct explicit deformations of generic smooth maps from closed $n$--dimensional manifolds $M$ with $n \geq 2$ to the $2$--sphere $S^2$ and show that every smooth map $M \to S^2$ is homotopic to a $C^\infty$ stable map…

In this paper, we study Morse functions with regular level sets consisting of spheres, tori, or Klein Bottles on $3$-dimensional closed manifolds. We characterize $3$-dimensional manifolds represented by connected sums each of whose…

We construct examples of nonresolvable generalized $n$-manifolds, $n\geq 6$, with arbitrary resolution obstruction, homotopy equivalent to any simply connected, closed $n$-manifold. We further investigate the structure of generalized…

Charged surfaces in contact with liquids containing ions are accompanied in equilibrium by an electric double layer consisting of a layer of electric charge on the surface that is screened by a diffuse ion cloud in the bulk fluid. This…

The obstruction space T^2 and the cup product T^1 x T^1 -> T^2 are computed for toric singularities.

We classify closed, simply connected $n$-manifolds of non-negative sectional curvature admitting an isometric torus action of maximal symmetry rank in dimensions $2\leq n\leq 6$. In dimensions $3k$, $k=1,2$ there is only one such manifold…

Parametrized topological complexity is a homotopy invariant that represents the degree of instability of motion planning problem that involves external constraints. We consider the parametrized topological complexity in the case of…

We introduce a class of minimal submanfolds $M^n$, $n\geq 3$, in spheres $\mathbb{S}^{n+2}$ that are ruled by totally geodesic spheres of dimension $n-2$. If simply-connected, such a submanifold admits a one-parameter associated family of…

A surface that is the pointwise sum of circles in Euclidean space is either coplanar or contains no more than 2 circles through a general point. A surface that is the pointwise product of circles in the unit-quaternions contains either 2,…

We show that the maximal number of singular moves required to pass between any two regularly homotopic planar or spherical curves with at most n crossings, grows quadratically with respect to n. Furthermore, this can be done with all curves…

We prove that for every $n$-vertex graph $G$, the extension complexity of the correlation polytope of $G$ is $2^{O(\mathrm{tw}(G) + \log n)}$, where $\mathrm{tw}(G)$ is the treewidth of $G$. Our main result is that this bound is tight for…

We consider a surface link in the 4-space which can be presented by a simple branched covering over the standard torus, which we call a torus-covering link. Torus-covering links include spun $T^2$-knots and turned spun $T^2$-knots. In this…

We classify the Lagrangian orientable surfaces in complex space forms with the property that the ellipse of curvature is always a circle. As a consequence, we obtain new characterizations of the Clifford torus of the complex projective…

Various curve complexes with vertices representing multicurves on a surface $S$ have been defined, for example [3], [4] and [8]. The homology curve complex $\mathcal{HC}(S,\alpha)$ defined in [7] is one such complex, with vertices…

In this work, we show that, for any simply-connected elliptic space $S$ admitting a pure minimal Sullivan model with a differential of constant length, we have ${\rm TC}_0(S)\leq 2{\rm cat}_0(S)+\chi_{\pi}(S)$ where $\chi_{\pi}(S)$ is the…

Let S be a finite union of (pairwise disjoint but possibly knotted and linked) closed curves and tubes in the round sphere S^3 or in the flat torus T^3. In the case of the torus, S is further assumed to be contained in a contractible subset…