Non-slice links in 4-manifolds

Clayton McDonald, University of Glasgow
Fine Hall 314

For $X = S^2 \times S^2$ and $\CP2 \# \bCP2$, we show that there exists a link with 2 components which is not smoothly slice in $X$. By contrast, it is well-known that every knot (i.e., link with 1 component) is smoothly slice in both $S^2 \times S^2$ and $\CP2 \# \bCP2$. Our proof uses classical topological and smooth obstructions, as well as constructive arguments to exploit the symmetries of the problem. As a side note, we also show that for every compact 4-manifold there exists a link that is not slice in it (either smoothly or topologically).