#### Volume 8, issue 4 (2008)

 Recent Issues
 The Journal About the Journal Subscriptions Editorial Board Editorial Interests Editorial Procedure Submission Guidelines Submission Page Author Index To Appear ISSN (electronic): 1472-2739 ISSN (print): 1472-2747
A manifold calculus approach to link maps and the linking number

### Brian A Munson

Algebraic & Geometric Topology 8 (2008) 2323–2353
##### 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.

##### Keywords
calculus of functors, link map, linking number
##### Mathematical Subject Classification 2000
Primary: 57Q45, 57R99
Secondary: 55P99, 57M25
##### Publication
Received: 16 April 2008
Revised: 30 October 2008
Accepted: 3 November 2008
Published: 20 December 2008
##### Authors
 Brian A Munson Department of Mathematics Wellesley College Wellesley, MA 02481 USA http://palmer.wellesley.edu/~munson