  We study the space of "link maps": the space of maps of a disjoint union of
compact, closed manifolds P_1, . . ., P_k into a manifold N whose images 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 connects with
recent work of the author, and is used to show that the Borromean rings are
linked. We also conjecture connectivity estimates for the approximating spaces.
