Derivatives of the identity and generalizations of Milnor's invariants

TL;DR

This paper generalizes Milnor's link invariants using homotopy-theoretic derivatives of the identity functor, connecting link invariants, bordism, and manifold calculus in a unified framework.

Contribution

It introduces a new class of invariants that extend Milnor's invariants, linking link map invariants with homotopy functor derivatives and manifold calculus.

Findings

Generalization of Koschorke's higher Hopf invariants

Stable range bordism description of derivatives

Integration into multivariable manifold calculus framework

Abstract

We synthesize work of U. Koschorke on link maps and work of B. Johnson on the derivatives of the identity functor in homotopy theory. The result can be viewed in two ways: (1) As a generalization of Koschorke's "higher Hopf invariants", which themselves can be viewed as a generalization of Milnor's invariants of link maps in Euclidean space; and (2) As a stable range description, in terms of bordism, of the cross effects of the identity functor in homotopy theory evaluated at spheres. We also show how our generalized Milnor invariants fit into the framework of a multivariable manifold calculus of functors, as developed by the author and Voli\'{c}, which is itself a generalization of the single variable version due to Weiss and Goodwillie.