Related papers: Hopf invariants and differential forms

We show that the set of harmonic maps from the 2-dimensional stratified spheres with uniformly bounded energies contains only finitely many homotopy classes. We apply this result to construct infinitely many harmonic map flows and mean…

We provide a simplified approach to the the stable Hopf invariant. We provide short elementary proofs of the Cartan Formula, the Composition Formula, and the Transfer formula. In addition, when $\pi$ is a discrete group, we show how to…

In this note we study logarithmic transformations in the sense of differential topology on two fibers of the Hopf surface. It is known that such transformations are susceptible to yield exotic smooth structures on four-manifolds. We will…

We show that for any smooth Hausdorff manifolds M and N, which are not necessarily second countable, paracompact or connected, any isomorphism from the algebra of smooth (real or complex) functions on N to the algebra of smooth functions on…

In this paper, we discuss the associated family of harmonic maps $\mathcal{F}: M \rightarrow G/K$ from a Riemann surface $M$ into inner symmetric spaces of compact or non-compact type which are either algebraic or totally symmetric. These…

Given a homotopy equivalence f between two topological spaces we assemble well known pieces and unfold them into an explicit formula for a strong deformation retraction of the mapping cylinder of f onto its top.

We prove that for any two closed Riemannian manifolds $M^{2m}$ ($m\geq 1$) and $N$, there exists a minimizing (extrinsic) $m$-polyharmonic map for every free homotopy class in $[M^{2m}, N]$, provided that the homotopy group $\pi_{2m}(N)$ is…

We describe the behaviour of the homotopy similarity relations and finite-order invariants under the function $[X,Y]\to[X,Z]$ induced by a map $Y\to Z$ strongly $r$-similar to the constant map.

Both bi-harmonic map and $f$-harmonic map have nice physical motivation and applications. In this paper, by combination of these two harmonic maps, we introduce and study $f$-bi-harmonic maps as the critical points of the $f$-bi-energy…

This paper presents two algorithms. In their simplest form, the first algorithm decides the existence of a pointed homotopy between given simplicial maps f, g from X to Y and the second computes the group $[\Sigma X,Y]^*$ of pointed…

Let X be a smooth complex algebraic variety. Morgan [Mor78] showed that the rational homotopy type of X is a formal consequence of the differential graded algebra defined by the first term of its weight spectral sequence. In the present…

We show that if $M$ and $N$ have the same homotopy type of simply connected closed smooth $m$-manifolds such that the integral and mod-$2$ cohomologies of $M$ vanish in odd degrees, then their homotopy inertia groups are equal. Let $M^{2n}$…

Let $G$ be a simply-connected simple compact Lie group and let $M$ be an orientable smooth closed 4-manifold. In this paper we calculate the homotopy type of the suspension of $M$ and the homotopy types of the gauge groups of principal…

In this paper we show that a (non necessarily integrable) holomorphic plane field on a compact complex manfold $M$ having an infinite number of invariant hypersurfaces must admit a meromorphic first integral $F:M\longrightarrow…

We study notions of homotopy in the Newtonian space $N^{1,p}(X;Y)$ of Sobolev type maps between metric spaces. After studying the properties and relations of two different notions we prove a compactness result for sequences in homotopy…

Let $B$ be a M\"obius band and $f:B \to \mathbb{R}$ be a Morse map taking a constant value on $\partial B$, and $\mathcal{S}(f,\partial B)$ be the group of diffeomorphisms $h$ of $B$ fixed on $\partial B$ and preserving $f$ in the sense…

In this paper we compute the mapping class group of simply-connected closed smooth manifolds $M$ with integral homology $H_{*}(M) \cong \mathbb Z \oplus \mathbb Z \oplus \mathbb Z$ provided that $\dim M \ne 4$.

Subtle issues arise when extending homotopy invariants to spaces of functions having little regularity, e.g., Sobolev spaces containing discontinuous functions. Sometimes it is not possible to extend the invariant at all, and sometimes,…

We compare Hofer's geometries on two spaces associated with a closed symplectic manifold M. The first space is the group of Hamiltonian diffeomorphisms. The second space L consists of all Lagrangian submanifolds of $M \times M$ which are…

The Hopf index, a topological invariant that quantifies the linking of preimage fibers, is fundamental to the structure and stability of hopfions. In this work, we propose a new mathematical framework for modeling hopfions with high Hopf…