A manifold calculus approach to link maps and the linking number

Abstract

We study the space of link maps $Link\left(P_1,\dots ,P_k;N\right)$, the space of smooth maps $P_1\bigsqcup \cdots \bigsqcup P_k\to N$ such that the images of the $P_i$ are pairwise disjoint. We apply the manifold calculus of functors developed by Goodwillie and Weiss to study the difference between it and its linear and quadratic approximations. We identify an appropriate generalization of the linking number as the geometric object which measures the difference between the space of link maps and its linear approximation. Our analysis of the difference between link maps and its quadratic approximation resembles recent work of the author on embeddings, and is used to show that the Borromean rings are linked.