Suppose there is a link K consisting of two disjoint embedded 2-spheres S₁², S₂² in R⁵. By the Hurewicz theorem and the Alexander duality, one obtains the isomorphism ₂ ⁵ ₂² ₁² ₂² ₁² ₂ ⁵ ₂² ₁² ₂^{2 5 1}...[ Read more ]

Suppose there is a link K consisting of two disjoint embedded 2-spheres S₁², S₂² in R⁵. By the Hurewicz theorem and the Alexander duality, one obtains the isomorphism

π₂(R⁵ - S₂²)≅Z

By fixing the orientations of S₁² and S₂², it fixs the sign of the above isomorphism and hence defines a fixed integer to S₁² in π₂(R⁵ - S₂²)≅Z. This number is called the linking number of K.

Up to homotopy, this linking number obviously classifies the link. Following from a Zeeman's paper, "Isotopies and knots in manifolds", the linking number also classifies the link up to isotopy. Now adding an extra condition: S₁², S₂² have only one maximum and one mimimum when we project R⁵ → R¹ to a fixed direction (such links is called a good links), and all isotopies keep to have one maximum and one minimum (such isotopies are callded good isotopies). This paper uses a different approach from Zeeman's by the extensive usage of Whitney move to classify the good link, up to good isotopy, by its linking number.