The Hopf invariant is linked to null-homotopy properties of maps.
problem Understanding the relationship between the Hopf invariant and null-homotopy of maps.
method Using the generalized Hopf invariant and constructing smooth null-homotopies, the paper explores the relationship between the Hopf invariant and null-homotopy properties of maps.
result Sharp results on the relationship between the Hopf invariant and null-homotopy properties of maps, showing the necessity and sufficiency of certain conditions.
We provide an efficient algorithm to compute the minimum area of a homotopy between two closed plane curves, given that they divide the plane into finite number of regions. For any positive real number ε>0, we construct a closed plane curve γ such that the minimum area of a null homotopy of 2⋅γ is l…
Enhanced bounds on rho-invariants for 3-manifolds.
problem Establishing bounds on Cheeger-Gromov rho-invariants for 3-manifolds.
method Constructing chain null-homotopies with linear complexity.
result Linearly bounded complexity of constructed null-homotopies.
This work focuses on important step in quantitative topology: given homotopic mappings from Sm to Sn of Lipschitz constant L, build the (asymptotically) simplest homotopy between them (meaning having the least Lipschitz constant). The present paper resolves this problem for the first case where Hopf invariant p…
Filling length measures the length of the contracting closed loops in a null-homotopy. The filling length function of Gromov for a finitely presented group measures the filling length as a function of length of edge-loops in the Cayley 2-complex. We give a bound on the filling length function in terms of the log of an …
The study proves a gap theorem for CAT(0) spaces with a constant below 1/(6√π).
problem Proving isoperimetric inequalities in non-positive curvature spaces.
method Introduced minimal tetrahedra to prove a linear inequality.
result Established a gap theorem for CAT(0) spaces with a constant below 1/(6√π).
A new method calculates the minimum volume swept by a sphere's homotopy in 3D space.
problem Finding the minimum volume swept by a sphere's homotopy in 3D space.
method Cable system approach to define and compute cable indices.
result A linear-time algorithm computes all cable indices and achieves the lower bound for the swept volume.
We propose a general strategy to derive null-homotopy operators for differential complexes based on the Bernstein-Gelfand-Gelfand (BGG) construction and properties of the de Rham complex. Focusing on the elasticity complex, we derive path integral operators P for elasticity satisfying $\mathscr{D}\mathscr{P…
The quaternions are non-commutative. The deviation from commutativity is encapsulated in the commutator of unit quaternions. It is known that the k-th power of the commutator is null-homotopic if and only if k is divisible by 12. The main purpose of this paper is to construct a concrete null-homotopy of the 12-th power…
The paper proves existence of minimal homotopies for immersed planar curves.
problem Existence of area-minimizing homotopies between homotopic curves in the plane.
method Geometric and variational approach, lifting curves into higher co-dimension, applying Douglas's solution of the Plateau problem.
result Uniform convergence of Douglas minimizers and minimal homotopy area minimization.