A New Route to the Interpretation of Hopf Invariant

Abstract

We discuss an object from algebraic topology, Hopf invariant, and reinterpret it in terms of the ϕ-mapping topological current theory. The main purpose of this paper is to present a new theoretical framework, which can directly give the relationship between Hopf invariant and the linking numbers of the higher dimensional submanifolds of Euclidean space R2n-1. For the sake of this purpose we introduce a topological tensor current, which can naturally deduce the (n-1)-dimensional topological defect in R2n-1 space. If these (n-1)-dimensional topological defects are closed oriented submanifolds of R2n-1, they are just the (n - 1)-dimensional knots. The linking number of these knots is well defined. Using the inner structure of the topological tensor current, the relationship between Hopf invariant and the linking numbers of the higher-dimensional knots can be constructed.