Related papers: Topological complexity of oriented Grassmann manif…
We prove exact complexity dichotomies for two quantum invariants of closed oriented three-manifolds, with the categorical data fixed. For a modular category $\mathcal{C}$, computing the Reshetikhin--Turaev invariant $Z_{\mathcal{C}}(M)$…
We study the variety $\rm{VAPS}_G(q_2^3, 10)$ (resp. $\rm{VAPS}_G(q_3^2, 10)$), a Grassmannian compactification of the variety of finite schemes of length $10$ apolar to $q_{2}^3$ (resp. $q_3^2$), where $\{q_{n} = 0\}\subset \mathbb P^{n} $…
Lower and upper bounds on the size of a covering of subspaces in the Grassmann graph $\cG_q(n,r)$ by subspaces from the Grassmann graph $\cG_q(n,k)$, $k \geq r$, are discussed. The problem is of interest from four points of view: coding…
We introduce and study oscillator topologies on paratopological groups and define certain related number invariants. As an application we prove that a Hausdorff paratopological group $G$ admits a weaker Hausdorff group topology provided $G$…
We give new lower bounds for the (higher) topological complexity of a space, in terms of the Lusternik-Schnirelmann category of a certain auxiliary space. We also give new lower bounds for the rational topological complexity of a space, and…
We propose a way to define and compute invariants of general smooth 4-manifolds based on topological twists of non-Lagrangian 4d N=2 and N=3 theories in which the problem is reduced to a fairly standard computation in topological A-model,…
We introduce certain canonical blow-ups $\mathcal T_{s,p,n}$, as well as their distinct submanifolds $\mathcal M_{s,p,n}$, of Grassmann manifolds $G(p,n)$ by partitioning the Pl\"ucker coordinates with respect to a parameter $s$. Various…
In this paper, we prove a version of the classical Cartan-Hadamard theorem for negatively curved manifolds, of dimension $n\neq 5$, with non-empty totally geodesic boundary. More precisely, if $M_1^n,M_2^n$ are any two such manifolds, we…
We study the family of closed Riemannian n-manifolds with holonomy group isomorphic to $Z_2^{n-1}$, which we call generalized Hantzsche-Wendt manifolds. We prove results on their structures, compute some invariants, and find relations…
We show that the Lusternik-Schnirelmann category of the homotopy cofiber of the diagonal map for non-orientable surfaces equals three. Also, we prove that the topological complexity of non-orientable surfaces of genus $>3$ is four.
In this paper, we investigate homogeneous Riemannian geometry on real flag manifolds of the split real form of $\mathfrak{g}_2$. We characterize the metrics that are invariant under the action of a maximal compact subgroup of $G_2.$ Our…
We introduce the effectual topological complexity (ETC) of a $G$-space $X$. This is a $G$-equivariant homotopy invariant sitting in between the effective topological complexity of the pair $(X,G)$ and the (regular) topological complexity of…
A Theorem due to Guillemin and Sternberg about geometric quantization of Hamiltonian actions of compact Lie groups $G$ on compact Kaehler manifolds says that the dimension of the $G$-invariant subspace is equal to the Riemann-Roch number of…
In this paper we describe the algebra of differential invariants for GL(n,C)-structures. This leads to classification of almost complex structures of general positions. The invariants are applied to the existence problem of…
The smooth (resp. metric and complex) Nielsen Realization Problem for K3 surfaces $M$ asks: when can a finite group $G$ of mapping classes of $M$ be realized by a finite group of diffeomorphisms (resp. isometries of a Ricci-flat metric, or…
We construct new compact manifolds endowed with closed $\mathrm{G}_2$ structures that satisfy the topological properties found by Joyce and Baraglia for the existence of a torsion-free $\mathrm{G}_2$ structure in the same cohomology class.…
The object of this paper is to study GL(2,R) orbit closures in hyperelliptic components of strata of abelian differentials. The main result is that all higher rank affine invariant submanifolds in hyperelliptic components are branched…
Periodic Hamiltonians on a three-dimensional (3-D) lattice with a spectral gap not only on the bulk but also on two edges at the common Fermi level are considered. By using K-theory applied for the quarter-plane Toeplitz extension, two…
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…
In this manuscript, we study the interplay between symmetry and topology with a focus on the $Z_2$ topological index of 2D/3D topological insulators and high-order topological insulators. We show that in the presence of either a…