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We establish several new results about both the (n)-solvable filtration,F_n^m, of the set of link concordance classes and the (n)-solvable filtrationof the string link concordance group. We first establish a relationship betweenMilnor's invariants and links, L, with certain restrictions on the 4-manifoldbounded by M_L. Using this relationship, we can relate (n)-solvability of alink (or string link) with its Milnor's invariants. Specifically, we show thatif a link is (n)-solvable, then its Milnor's invariants vanish for lengths upto 2^{n+2}-1. Previously, there were no known results about the "other half" ofthe filtration, namely F_{n.5}^m/F_{n+1}^m. We establish the effect of the Bingdoubling operator on (n)-solvability and using this, we show thatF_{n.5}^m/F_{n+1}^m is nontrivial for links (and string links) withsufficiently many components. Moreover, we show that these quotients contain aninfinite cyclic subgroup. We also show that links and string links modulo(1)-solvability is a nonabelian group. We show that we can relate otherfiltrations 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 2^{n+2}-1.Lastly, we prove that the Grope filtration, G_n^m, of the set of linkconcordance classes is not the same as the (n)-solvable filtration.

The (n)-solvable filtration of the link concordance group and Milnor's invariants