Related papers: Rational homotopy of maps between certain complex …

We show that a rational holonomy representation of any flat manifold except torus must have at least two non-equivalent irreducible subrepresentations. As an application we show that if a K\"ahler flat manifold is not a torus then its…

The main result of this paper is that the k^{\rm th} continuous Hochschild cohomology groups H^k(\cl M,\cl M) and H^k(\cl M,B(H)) of a von Neumann factor ${\cl M}\subseteq B(H) of type {\rm II}_1 with property Gamma are zero for all…

Let $X$ be a compact, Hausdorff topological space. Then $H^M_n(X)=0$ for all $n>0$, where $H_n$ is the multivalued analogue of singular homology. The case $n=1$ is already known [8].

Let F_*(X, Y) be the space of base-point-preserving maps from a connected finite CW complex X to a connected space Y. Consider a CW complex of the form X cup_{alpha}e^{k+1} and a space Y whose connectivity exceeds the dimension of the…

This paper takes its starting point in an idea of Grothendieck on the representation of homotopy types. We show that any locally finite nilpotent homotopy can be represented by a simplicial set which is a finitely generated free group in…

Let $X$ be a nilpotent space such that there exists $p\geq 1$ with $H^p(X,\mathbb Q) \ne 0$ and $H^n(X,\mathbb Q)=0$ if $n>p$. Let $Y$ be a m-connected space with $m\geq p+1$ and $H^*(Y,\mathbb Q)$ is finitely generated as algebra. We…

Let $V_{n,d}$ be the variety of equations for hypersurfaces of degree $d$ in $\mathbb{P}^n(\mathbb{C})$ with singularities not worse than simple nodes. We prove that the orbit map $G'=SL_{n+1}(\mathbb{C}) \to V_{n,d}$, $g\mapsto g\cdot…

We prove that for any open Riemann surface $N,$ natural number $n\geq 3,$ non-constant harmonic map $h:N\to \mathbb{R}^{n-2}$ and holomorphic 2-form $H$ on $N,$ there exists a weakly complete harmonic map $X=(X_j)_{j=1,\ldots,n}:N \to…

We use some detailed knowledge of the cohomology ring of real Grassmann manifolds $G_k(\mathbb{R}^n)$ to compute zero-divisor cup-length and estimate topological complexity of motion planning for $k$-linear subspaces in $\mathbb{R}^n$. In…

Given a simplicial pair $(X,A)$, a simplicial complex $Y$, and a map $f:A \to Y$, does $f$ have an extension to $X$? We show that for a fixed $Y$, this question is algorithmically decidable for all $X$, $A$, and $f$ if $Y$ has the rational…

We prove that the Gauss map of a non-flat complete minimal surface immersed in $\mathbb{R}^n$ can omit a generic hypersurface $D$ of degree at most $ n^{n+2}(n+1)^{n+2}$.

Suppose that $n\neq p^k$ and $n\neq 2p^k$ for all $k$ and all primes $p$. We prove that for any Hausdorff compactum $X$ with a free action of the symmetric group $\mathfrak S_n$ there exists an $\mathfrak S_n$-equivariant map $X \to…

Let $M$ and $N$ be two compact complex manifolds. We show that if the tautological line bundle $\mathscr{O}_{T_M^*}(1)$ is not pseudo-effective and $\mathscr{O}_{T_N^*}(1)$ is nef, then there is no non-constant holomorphic map from $M$ to…

In this note we will provide a gradient estimate for harmonic maps from a complete noncompact Riemannian manifold with compact boundary (which we call "Kasue manifold") into a simply connected complete Riemannian manifold with non-positive…

We propose a generalization of Sullivan's de Rham homotopy theory to non-simply connected spaces. The formulation is such that the real homotopy type of a manifold should be the closed tensor dg-category of flat bundles on it much the same…

In this paper we consider two connected closed Haken manifolds denoted by M^3 and N^3, with the same Gromov simplicial volume. We give a simple homological criterion to decide when a given map f: M^3-->N^3 between M^3 and N^3 can be changed…

Let M be any closed, locally symmetric n-manifold (n>1) of nonpositive curvature. Assume that M has no locally Euclidean factors and no factors locally isometric to SL(3,R). Then for any closed Riemannian manifold N and any continuous map…

Let X be a projective hypersurface in P_k^n of degree d <= n. In this paper we study the relation between the class [X] in K_0(Var_k) and the existence of k-rational points. Using elementary geometric methods we show, for some particular X,…

A Grasstope is the image of the totally nonnegative Grassmannian $\text{Gr}_{\geq 0}(k,n)$ under a linear map $\text{Gr}(k,n)\dashrightarrow \text{Gr}(k,k+m)$. This is a generalization of the amplituhedron, a geometric object of great…

Let $n\geq2$ be a natural number. Let $M_n(\mathbb{K})$ be the ring of all $n \times n$ matrices over a field $\mathbb{K}$. Fix natural number $k$ satisfying $1<k\leq n$. Under a mild technical assumption over $\mathbb{K}$ we will show that…