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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

We study the space of link maps Link(P1,,Pk;N), the space of smooth maps P1 Pk N such that the images of the Pi 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.

calculus of functors, link map, linking number
Mathematical Subject Classification 2000
Primary: 57Q45, 57R99
Secondary: 55P99, 57M25
Received: 16 April 2008
Revised: 30 October 2008
Accepted: 3 November 2008
Published: 20 December 2008
Brian A Munson
Department of Mathematics
Wellesley College
Wellesley, MA 02481