Volume 14, issue 5 (2014)

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The $(n)$–solvable filtration of link concordance and Milnor's invariants

Carolyn Otto

Algebraic & Geometric Topology 14 (2014) 2627–2654

We establish several new results about both the (n)–solvable filtration of the set of link concordance classes and the (n)–solvable filtration of the string link concordance group, Cm. The set of (n)–solvable m–component string links is denoted by nm. We first establish a relationship between Milnor’s invariants and links, L, with certain restrictions on the 4–manifold bounded by ML, the zero-framed surgery of  S3 on L. Using this relationship, we can relate (n)–solvability of a link (or string link) with its Milnor’s μ̄–invariants. Specifically, we show that if a link is (n)–solvable, then its Milnor’s invariants vanish for lengths up to 2n+2 1. Previously, there were no known results about the “other half” of the filtration, namely n.5mn+1m. We establish the effect of the Bing doubling operator on (n)–solvability and using this, we show that n.5mn+1m is nontrivial for links (and string links) with sufficiently many components. Moreover, we show that these quotients contain an infinite cyclic subgroup. We also show that links and string links modulo (1)–solvability is a nonabelian group. We show that we can relate other filtrations with Milnor’s invariants. We show that if a link is n–positive, then its Milnor’s invariants will also vanish for lengths up to 2n+2 1. Lastly, we prove that the grope filtration of the set of link concordance classes is not the same as the (n)–solvable filtration.

Milnor's invariants, solvable filtration, link concordance
Mathematical Subject Classification 2010
Primary: 57M25
Received: 8 February 2013
Revised: 9 December 2013
Accepted: 12 December 2013
Published: 5 November 2014
Carolyn Otto
Department of Mathematics
University of Wisconsin-Eau Claire
1428 Bell Street
Eau Claire, WI 54703