Embedding spheres in knot traces

The trace of the -framed surgery on a knot in is a 4-manifold homotopy equivalent to the 2-sphere. We characterise when a generator of the second homotopy group of such a manifold can be realised by a locally flat embedded -sphere whose complement has abelian fundamental group. Our characterisation is in terms of classical and computable -dimensional knot invariants. For each, this provides conditions that imply a knot is topologically -shake slice, directly analogous to the result of Freedman and Quinn that a knot with trivial Alexander polynomial is topologically slice.
Compositio Mathematica, 157 (10)
ISSN:0010-437X
ISSN:1570-5846