A manifold calculus approach to link maps and the linking number

Munson, Brian A

doi:10.2140/agt.2008.8.2323

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A manifold calculus approach to link maps and the linking number

Brian A Munson

Algebraic & Geometric Topology 8 (2008) 2323–2353

DOI: 10.2140/agt.2008.8.2323

Abstract

We study the space of link maps $Link (P_{1}, \dots, P_{k}; N)$ , the space of smooth maps $P_{1} ⊔ \dots ⊔ P_{k} \to N$ 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

References

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

Volume 8, issue 4 (2008)