An approach to higher order linking invariants through holonomy and curvature

Abstract

Abstract. We study the Milnor-Massey linking invariants through the holonomy and curvature of certain nilpotent connections and their flat quotient connections. Versions of the Porter-Turaev Theorem are proved in the context of de Rham cohomology.

A Hydrodynamical Homotopy Co-Momentum Map and a Multisymplectic Interpretation of Higher-Order Linking Numbers

“…Ordinary and higher-order linking numbers provide a quite useful tool for the investigation of Brunnian phenomena in knot theory: recall that a link is almost trivial or Brunnian if upon removing any component therefrom one gets a trivial link. They can be defined recursively in terms of Massey products, or equivalently, Milnor invariants, by the celebrated Turaev-Porter theorem (see [14,23,39,48]). We review, briefly and quite concretely, the basic steps of the Massey procedure, read differential-geometrically as in [23,39,48], presenting at the same time our novel multisymplectic interpretation thereof.…”

“…They can be defined recursively in terms of Massey products, or equivalently, Milnor invariants, by the celebrated Turaev-Porter theorem (see [14,23,39,48]). We review, briefly and quite concretely, the basic steps of the Massey procedure, read differential-geometrically as in [23,39,48], presenting at the same time our novel multisymplectic interpretation thereof.…”

“…Here we wish to apply some recently emerged concepts in multisymplectic geometry (mostly building on [7,45,46]) and construct an explicit homotopy co-momentum map [7] in a hydrodynamical setting, leading to a multisymplectic interpretation of the so-called higher-order linking numbers, viewed à la Massey [23,39,48]. The construction is generalized to cover connected compact oriented Riemannian 2 A. M. Miti and M. Spera [2] manifolds having vanishing intermediate de Rham groups.…”

Homotopy Comomentum Maps in Multisymplectic Geometry

“…Ordinary and higher order linking numbers provide, among others, a quite useful tool for the investigation of Brunnian phenomena in knot theory: recall that a link is almost trivial or Brunnian if upon removing any component therefrom one gets a trivial link (see figure 5.8). They can be defined recursively in terms of Massey products, or equivalently, Milnor invariants, by the celebrated Turaev-Porter theorem (see [FF83,PS02,Spe06,HT12]). We are going to review, briefly and quite concretely, the basic steps of the Massey procedure, read differential geometrically as in [PS02, Spe06, HT12], presenting at the same time our novel multisymplectic interpretation thereof.…”