Clasper concordance, Whitney towers and repeating Milnor invariants

We show that for each [Formula: see text], a link [Formula: see text] bounds a degree [Formula: see text] Whitney tower in the 4-ball if and only if it is [Formula: see text]-concordant to the unlink. This means that [Formula: see text] is obtained from the unlink by a finite sequence of concordances and degree [Formula: see text] clasper surgeries. In our construction, the trees associated to the Whitney towers coincide with the trees associated to the claspers. As a corollary to our previous obstruction theory for Whitney towers in the 4-ball, it follows that the [Formula: see text]-concordance filtration of links is classified in terms of Milnor invariants, higher-order Sato–Levine and Arf invariants. Using a new notion of [Formula: see text]-repeating twisted Whitney towers, we also classify a natural generalization of the notion of link homotopy, called twisted self [Formula: see text]-concordance, in terms of [Formula: see text]-repeating Milnor invariants and [Formula: see text]-repeating Arf invariants.