Related papers: Topological complexity of real Grassmannians

Invariants for Riemann surfaces covered by the disc and for hyperbolic manifolds in general involving minimizing the measure of the image over the homotopy and homology classes of closed curves and maps of the $k$-sphere into the manifold…

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 compute the small cohomology ring of the Cayley Grassmannian, that parametrizes four-dimensional subalgebras of the complexified octonions. We show that all the Gromov-Witten invariants in the multiplication table of the Schubert classes…

Nielsen's geometric approach offers a powerful framework for quantifying the complexity of unitary transformations. In this formulation, complexity is defined as the length of the minimal geodesic in a suitably constructed geometric space…

In this paper we use toric geometry to investigate the topology of the totally non-negative part of the Grassmannian (Gr_{kn})_{\geq 0}. This is a cell complex whose cells Delta_G can be parameterized in terms of the combinatorics of…

We study the geometry of fanning curves in the Grassmann manifold of n-dimensional subspaces of $\mathbb{R}^{kn}$; we construct a complete system of invariants which solve the congruence problem. The geometry of the invariants themselves…

We complement our previous computation of the Chow-Witt rings of classifying spaces of special linear groups by an analogous computation for the general linear groups. This case involves discussion of non-trivial dualities. The computation…

In this article, we computed the homology groups of real Grassmann manifold $G_{n,m}(\mathbb{R})$ by Witten complex.

Contents 1 Mappings and distortion 2 The mathematics of good behavior much of the time, and the BMO frame of mind 3 Finite polyhedra and combinatorial parameterization problems 4 Quantitative topology, and calculus on singular spaces 5…

We prove the Riemann-Roch theorem for homotopy invariant $K$-theory and projective local complete intersection morphisms between finite dimensional noetherian schemes, without smoothness assumptions. We also prove a new Riemann-Roch theorem…

The geodesic complexity of a metric space X is the smallest k for which there is a partition of X x X into ENRs E_0,...,E_k on each of which there is a continuous choice of minimal geodesic sigma(x_0,x_1) from x_0 to x_1. We prove that the…

Our main result offers a new (quite systematic) way of deriving bounds for the cup-length of Poincare spaces over fields; we outline a general research program based on this result. For the oriented Grassmann manifolds, already a limited…

A real toric space is a topological space which admits a well-behaved $\mathbb{Z}_2^k$-action. Real moment-angle complexes and real toric varieties are typical examples of real toric spaces. A real toric space is determined by a pair of a…

In this survey article, given a smooth closed manifold M we study the space of Riemannian metrics of positive scalar curvature on M. A long-standing question is: when is this space non-empty (i.e. when does M admit a metric of positive…

We introduce a bivariate version of topological complexity, $\mathrm{TC}(f,g)$, associated with two continuous maps $f\colon X\to Z$ and $g\colon Y\to Z$. This invariant measures the minimal number of continuous motion planning rules…

Grassmann manifolds $G_{k,n}$ are among the central objects in geometry and topology. The Borel picture of the mod 2 cohomology of $G_{k,n}$ is given as a polynomial algebra modulo a certain ideal $I_{k,n}$. The purpose of this paper is to…

We give an upper bound on the topological complexity of varieties $\mathcal{V}$ obtained as complements in $\mathbb{C}^m$ of the zero locus of a polynomial. As an application, we determine the topological complexity of unordered…

We introduce a notion of retraction between continuous maps of topological spaces and study the behavior of several numerical invariants under such retractions. These include (co)homological dimensions, the Lusternik-Schnirelmann category,…

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…

We introduce a version of Farber's topological complexity suitable for investigating mechanical systems whose configuration spaces exhibit symmetries. Our invariant has vastly different properties to the previous approaches of Colman-Grant,…