Related papers: Hopf invariants and differential forms

We give a short, mostly elementary and self-contained proof of the classical result that the groups of diffeomorphisms, homeomorphisms, and homotopy equivalences of a surface have the same group of connected components.

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…

By studying periodic points for rational maps on $\bm{C}^d$ with $p$ invariants, we show that they form an invariant variety of dimension $p$ if the periodicity conditions are `fully correlated', and a set of isolated points if the…

We give complete geometric invariants of cobordisms of framed fold maps. These invariants consist of two types. We take the immersion of the fold singular set into the target manifold together with information about non-triviality of the…

For $G$ a commutative group, we give a purely Hopf $G$-coalgebra construction of $G$-colored $3$-manifolds invariants using the notion of modified integral.

Invariant functions under the transformations of a compact linear group $G$ acting in $\real^n$ can be expressed in terms of functions defined in the orbit space of $G$. We develop a method to determine the isotropy classes of the orbit…

Locally stable maps $S^3\to\mathbb{R}^4$ are classified up to homotopy through locally stable maps. The equivalence class of a map $f$ is determined by three invariants: the isotopy class $\sigma(f)$ of its framed singularity link, the…

This paper studies invariants of 3-manifolds derived from certain fin ite dimensional Hopf algebras. The invariants are based on right integrals for these algebras. It is shown that the resulting class of invariants is distinct from the…

For smooth convex disks $A$, i.e., convex compact subsets of the plane with non-empty interior, we classify the classes $G^{\text{hom}}(A)$ and $G^{\text{sim}}(A)$ of intersection graphs that can be obtained from homothets and similarities…

We study the topology of compact manifolds with a Lie group action for which there are only finitely many non-principal orbits, and describe the possible orbit spaces which can occur. If some non-principal orbit is singular, we show that…

This survey/expository article covers a variety of topics related to the "topology at infinity" of noncompact manifolds and complexes. In manifold topology and geometric group theory, the most important noncompact spaces are often…

We study the deformations of a holomorphic symplectic manifold $M$, not necessarily compact, over a formal ring. We show (under some additional, but mild, assumptions on $M$) that the coarse deformation space exists and is smooth,…

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…

An example of a finite dimensional factorizable ribbon Hopf C-algebra is given by a quotient H=u_q(g) of the quantized universal enveloping algebra U_q(g) at a root of unity q of odd degree. The mapping class group M_{g,1} of a surface of…

Let $M$ be a smooth manifold and $\Gamma$ a group acting on $M$ by diffeomorphisms; which means that there is a group morphism $\rho:\Gamma\rightarrow \mathrm{Diff}(M)$ from $\Gamma$ to the group of diffeomorphisms of $M$. For any such…

This work focuses on important step in quantitative topology: given homotopic mappings from $S^m$ to $S^n$ of Lipschitz constant $L$, build the (asymptotically) simplest homotopy between them (meaning having the least Lipschitz constant).…

In this paper, we study a construction of homotopy invariants of open or closed covers, where the homotopy class is defined relative to a pair $(V,r)$, with $V$ a finite set of points in $\mathbb{R}^d$ and $r$ a point in the interior of…

We prove that Hopf manifolds admit holomorphic $(G,X)$-structures, extending to any dimension a result of McKay and Pokrovskiy. For this, we revisit Guysinsky-Katok's group of invertible sub-resonant polynomials, and Bertheloot's approach…

Suppose that the inverse image of the zero vector by a continuous map $f:{\mathbb R}^n\to{\mathbb R}^q$ has an isolated point $P$. There is a local obstruction to removing this isolated zero by a small perturbation, generalizing the notion…

Theorem. Let M be a compact, connected, oriented smooth Riemannian n-manifold with non-empty boundary. Then the cohomology of the complex (Harm*(M),d) of harmonic forms on M is given by the direct sum H^p(Harm*(M),d) = H^p(M;R) +…