Cyclic-Homology Chern–Weil Theory for Families of Principal Coactions

Viewing the space of cotraces in the structural coalgebra of a principal coaction as a noncommutative counterpart of the classical Cartan model, we construct the cyclic-homology Chern–Weil homomorphism. To realize the thus constructed Chern–Weil homomorphism as a Cartan model of the homomorphism tautologically induced by the classifying map on cohomology, we replace the unital subalgebra of coaction-invariants by its natural H-unital nilpotent extension (row extension). Although the row-extension algebra provides a drastically different model of the cyclic object, we prove that, for any row extension of any unital algebra over a commutative ring, the row-extension Hochschild complex and the usual Hochschild complex are chain homotopy equivalent. It is the discovery of an explicit homotopy formula that allows us to improve the homological quasi-isomorphism arguments of Loday and Wodzicki. We work with families of principal coactions, and instantiate our noncommutative Chern–Weil theory by computing the cotrace space and analyzing a dimension-drop-like effect in the spirit of Feng and Tsygan for the quantum-deformation family of the standard quantum Hopf fibrations.

Contracting thin disks

We answer a question of Liokumovich–Nabutovsky–Rotman showing that if [Formula: see text] is a Riemannian 2-disk with boundary length [Formula: see text], diameter [Formula: see text] and area [Formula: see text] then [Formula: see text] can be filled by a homotopy [Formula: see text] with [Formula: see text] bounded by [Formula: see text].

Generalized symmetries and higher-order Conservation laws of the Camassa–Holm equation

In the present paper, we derive generalized symmetries of order three of the Camassa–Holm equation by infinite prolongation of a generalized vector field and applying infinitesimal symmetry criterion. In addition, one-dimensional optimal system of Lie subalgebras investigated by applying the adjoint representation. Furthermore, determining equation for multipliers and the 2- dimensional homotopy formula employed to construct higher–order conservation laws for the Camassa–Holm equation.

A homotopy+ solution to the A-B slice problem

The A-B slice problem, a reformulation of the four-dimensional topological surgery conjecture for free groups, is shown to admit a link-homotopy[Formula: see text] solution. The proof relies on geometric applications of the group-theoretic [Formula: see text]-Engel relation. Implications for the surgery conjecture are discussed.

Subdividing three-dimensional Riemannian disks

Papasoglu asked whether for any Riemannian 3-disk [Formula: see text] with diameter [Formula: see text], boundary area [Formula: see text] and volume [Formula: see text], there exists a homotopy [Formula: see text] contracting the boundary to a point so that the area of [Formula: see text] is bounded by [Formula: see text] for some function [Formula: see text]. He further asks whether it is possible to subdivide [Formula: see text] by a disk [Formula: see text] into two regions of volume [Formula: see text] so that the area of [Formula: see text] is bounded by some function [Formula: see text]. In this paper, we answer the questions above in the negative: We prove that given [Formula: see text] and [Formula: see text], one can construct a metric [Formula: see text] so that any 2-disk [Formula: see text] subdividing [Formula: see text] into two regions of volume at least [Formula: see text], the area of [Formula: see text] is greater than [Formula: see text]. We further prove that for any Riemannian 3-sphere [Formula: see text], there is a surface that subdivides the disk into two regions of volume no less than [Formula: see text], and the area of this surface is bounded by [Formula: see text], where [Formula: see text] is the homological filling function of [Formula: see text].

Lagrangian exotic spheres

Let [Formula: see text]. We prove that the cotangent bundles [Formula: see text] and [Formula: see text] of oriented homotopy [Formula: see text]-spheres [Formula: see text] and [Formula: see text] are symplectomorphic only if [Formula: see text], where [Formula: see text] denotes the group of oriented homotopy [Formula: see text]-spheres under connected sum, [Formula: see text] denotes the subgroup of those that bound a parallelizable [Formula: see text]-manifold, and where [Formula: see text] denotes [Formula: see text] with orientation reversed. We further show that if [Formula: see text] and [Formula: see text] admits a Lagrangian embedding in [Formula: see text], then [Formula: see text]. The proofs build on [1] and [18] in combination with a new cut-and-paste argument; that also yields some interesting explicit exact Lagrangian embeddings, for instance of the sphere [Formula: see text] into the plumbing [Formula: see text] of cotangent bundles of certain exotic spheres. As another application, we show that there are re-parametrizations of the zero-section in the cotangent bundle of a sphere that are not Hamiltonian isotopic (as maps rather than as submanifolds) to the original zero-section.

ON FAMILIES IN DIFFERENTIAL GEOMETRY

Families of objects appear in several contexts, like algebraic topology, theory of deformations, theoretical physics, etc. An unified coordinate-free algebraic framework for families of geometrical quantities is presented here, which allows one to work without introducing ad hoc spaces, by using the language of differential calculus over commutative algebras. An advantage of such an approach, based on the notion of sliceable structures on cylinders, is that the fundamental theorems of standard calculus are straightforwardly generalized to the context of families. As an example of that, we prove the universal homotopy formula.