Smith theory (part 2)
In Smith theory (part 1) we discussed the Smith Conjecture which stated that the fixed point set of a smooth cyclic action on the 3-sphere is always unknotted. Today, the goal is to give a hands on disproof of the 4D Smith Conjecture. We'll cook up $\mathbb{Z}/2$-actions on $S^4$ with knotted fixed point set. (The action we build will be smooth and the fixed point set will be knotted in the topological and smooth categories.) To do this, we need a more interesting way to decompose $S^4$ into two pieces. First, an observation. Consider a Mazur manifold $W$ - a four-manifold built by attaching a single 1-handle to $B^4$ and a 2-handle that algebraically links it once. (The framing can be anything, the 2-handle can be knotted in whatever way.) Here's an example: These manifolds will be contractible, but might still not be homeomorphic to $B^4$; in fact, if your 2-handle is attached along anything other than $pt \times S^1$, the resulting boundary will *not*