Related papers: Topological Complexity of the Klein bottle

We prove that a 2-convex closed surface $S\subset E^4$ in the four-dimensional Euclidean space $E^4$, which is either $C^2$-smooth or polyhedral, provided that each vertex is incident to at most five edges, admits a mapping of degree one to…

We study the complexity of the isomorphism relation for various classes of closed subgroups of the group of permutations of the natural numbers. We use the setting of Borel reducibility between equivalence relations on Polish spaces. For…

Let X be a (not-necessarily homotopy-associative) H-space. We show that TC_{n+1}(X) = cat(X^n), for n >= 1, where TC_{n+1}(-) denotes the so-called higher topological complexity introduced by Rudyak, and cat(-) denotes the…

We prove the formula \begin{equation*} TC(X\vee Y)=\max\{TC(X),TC(Y),cat(X\times Y)\} \end{equation*} for the topological complexity of the wedge $X\vee Y$.

How complex must two finite 2-complexes be to admit a common, but not finite common, covering? We obtain an almost answer: the minimum possible number of triangles in a pseudo-simplicial triangulation of each complex is 3, 4, or 5.

We show that a topologically minimal disk in a tetrahedron with index $n$ is either a normal triangle, a normal quadrilateral, or a normal helicoid with boundary length 4(n+1). This mirrors geometric results of Colding and Minicozzi.

We exhibit an infinite family of indecomposable Klein bottles in the 4-sphere with order-4 meridians.

We study the topological index of some irregular surfaces that we call generalized Lagrangian. We show that under certain hypotheses on the base locus of the Lagrangian system the topological index is non-negative. For the minimal surfaces…

Let the complexity of a closed manifold M be the minimal number of simplices in a triangulation of M. Such a quantity is clearly submultiplicative with respect to finite coverings, and by taking the infimum on all finite coverings of M…

We study the topological complexity, in the sense of Smale, of three enumerative problems in algebraic geometry: finding the 27 lines on cubic surfaces, the 28 bitangents and the 24 inflection points on quartic curves. In particular, we…

As the sequel to [3], we construct a minimal complex surface of general type with p_g=0, K^2=2 and H_1=Z/2Z using a rational blow-down surgery and Q-Gorenstein smoothing theory. We also present an example of p_g = 0,K^2 = 2 and H_1 = Z/3Z.

We compute the higher topological complexity of ordered configuration spaces of orientable surfaces, thus extending Cohen-Farber's description of the ordinary topological complexity of those spaces.

We introduce \emph{local Urysohn width}, a complexity measure for classification problems on metric spaces. Unlike VC dimension, fat-shattering dimension, and Rademacher complexity, which characterize the richness of hypothesis…

We survey results on the topological complexity of classical configuration spaces of distinct ordered points in orientable surfaces and related spaces, including certain orbit configuration spaces and Eilenberg-Mac Lane spaces associated to…

The topological (resp. geodesic) complexity of a topological (resp. metric) space is roughly the smallest number of continuous rules required to choose paths (resp. shortest paths) between any points of the space. We prove that the geodesic…

We prove that for any compact orientable connected 3-manifold with torus boundary, a concatenation of it and the direct product of the circle and the Klein bottle with an open 2-disk removed admits a Lagrangian embedding into the standard…

We show that the topological complexity of an aspherical space $X$ is bounded below by the cohomological dimension of the direct product $A\times B$, whenever $A$ and $B$ are subgroups of $\pi_1(X)$ whose conjugates intersect trivially. For…

The simplicial complexity is an invariant for finitely presentable groups that was recently introduced by Babenko, Balacheff and Bulteau to study systolic area. The simplicial complexity $\kappa(G)$ was proved to be a good approximation of…

Let M be a compact, connected, orientable, irreducible 3-manifold and T' an incompressible torus boundary component of M such that the pair (M,T') is not cabled. By a result of C. Gordon, if S and T are incompressible punctured tori in M…

We compute the low dimensional cohomologies $\tilde H^q(gc_N,C)$, $H^q(gc_N,\C)$ of the infinite rank general Lie conformal algebras $gc_N$ with trivial coefficients for $q\le3, N=1$ or $q\le2, N\ge2$. We also prove that the cohomology of…