Shake slice and shake concordant links

Abstract

We can construct a [Formula: see text]-manifold by attaching [Formula: see text]-handles to a [Formula: see text]-ball with framing [Formula: see text] along the components of a link in the boundary of the [Formula: see text]-ball. We define a link as [Formula: see text]-shake slice if there exists embedded spheres that represent the generators of the second homology of the [Formula: see text]-manifold. This naturally extends [Formula: see text]-shake slice, a generalization of slice that has previously only been studied for knots, to links of more than one component. We also define a relative notion of shake[Formula: see text]-concordance for links and versions with stricter conditions on the embedded spheres that we call strongly[Formula: see text]-shake slice and strongly[Formula: see text]-shake concordance. We provide infinite families of links that distinguish concordance, shake concordance, and strong shake concordance. Moreover, for [Formula: see text] we completely characterize shake slice and shake concordant links in terms of concordance and string link infection. This characterization allows us to prove that the first non-vanishing Milnor [Formula: see text] invariants are invariants of shake concordance. We also argue that shake concordance does not imply link homotopy.