Smith theory (part 1)
This is a short series on the Smith Conjecture and Smith theory in general
In low-dimensional topology, we're always taking covers (branched or unbranched). For example, many problems in knot theory are approached by studying the topology of the branched double cover. (For example, the branched double cover of an unknotting number one knot is Dehn surgery on a knot by something called the "Montesinos trick", and so unknotting numbers get studied in this way.) So, how can you study group actions on a space:
* What is the fixed point set?
* What is the quotient manifold?
Roughly, if you have a smooth action of a finite group $X^G$ on a smooth manifold $X$, then the fixed point set $X^G$ will be a nice smooth submanifold. If it is a $p$-group action, where $p$ is prime, then we can really understand the fixed point set well by what is referred to as Smith Theory, which for simplicity, I will narrow down to two results:
$$ \dim H_*(X^G; \mathbb{Z}/p) \leq \dim H_*(X;\mathbb{Z}/p), \ \text{ and } \chi(X^G) \equiv \chi(X) \pmod{p}.$$
Here the dimension of the homology is the total dimension, and not in individual gradings. (There are more complicated graded statements, but you can't just compare one grading at a time.) For some proofs, see for example Andy Putman's notes or Bredon's book on transformation groups. (Allan Edmonds used to have nice notes on transformation groups, but I can't find them online anymore.) There is a complementary result for studying the cohomology of the quotient:
$$ H^i(X/G;F) \cong H^i(X;F)^G, $$
whenever $F$ is a field of characteristic 0 or coprime to the order of $G$. (Here, $G$ just needs to be finite and the term on the right-hand side is the fixed points of the induced $G$-action on cohomology.)
These results are actually already awesome. If $X$ has the homology of a ball, then that implies the fixed point set also has the homology of a ball. So, a non-trivial action of $\mathbb{Z}/p^r$ on $D^2$ will have fixed point set either a single point or an arc. (Exercise: Exhibit $\mathbb{Z}/2$-actions that do each.) Similarly, if we have a cyclic group action on the three-sphere, the fixed point set needs to be a sphere. With a little more work, if the action is orientation-preserving, then it can be shown that the fixed point set has to be a circle. In fact, it's not hard to show that for any $n$, there is a $\mathbb{Z}/n$-action on $S^3$ whose fixed point set is even unknotted. Think of $S^3$ as $\mathbb{R}^3$ with a point at infinity, and the unknot as the $z$-axis together with the point at infinity. Now rotate around the $z$-axis by $2\pi/n$ and you see that the fixed point set is the unknot and the quotient is also $S^3$. Note that we can compute from the formula above that the quotient of a cyclic action on $S^3$ always has to be a rational homology sphere.
This now leads us to the Smith Conjecture, which says that a cyclic group action on a sphere can never have a knotted fixed point set. We should really maintain the hypothesis that we want actions to be smooth or PL, or we get into wild knotting, and that's just not en vogue anymore. I do want to mention that there are non-smooth actions on $S^3$ with wild fixed point set so you have to be careful.
It can be helpful to think of this setup slightly differently. Instead of a manifold with an action we take the quotient and the image of the fixed point set downstairs. I.e., we study the action on $S^3$ as coming from taking a branched cover of the quotient, branched along the image of the fixed point set downstairs. (In our examples, we'll just have that $H_1$ of the complement of the fixed point set downstairs is $\mathbb{Z}$, so there will be a unique $p$-sheeted cyclic cover, so the branch set downstairs in the quotient determines the branched cover completely.) In this language, the Smith Conjecture is loosely asking if there's a non-trivial knot in a three-manifold with a cyclic branched cover which is $S^3$. This was proved in the late 70s using an epic combination of techniques coming out of 3-manifold geometry and topology (e.g. hyperbolic structures on 3-manifolds, minimal surface theory, character variety methods, etc) and had contributions from a number of the heavy hitters in the field at the time. See the book by Bass and Morgan for more deets. This result is/was a big deal!!!
Some additional points:
- In the unknot example above, the quotient is also $S^3$, and we know for a counterexample to the Smith conjecture, the quotient would have to be a rational homology sphere. It actually doesn't take too much work to prove that the quotient for a counterexample would have to be a homotopy sphere. This work was all done before Perelman proved the Poincare conjecture, so folks at the time still didn't actually know the quotient had to be a standard sphere for a purported action on $S^3$ with knotted fixed point sets.
- If you don't work with cyclic branched covers, all bets are off. For example, every three-manifold is a branched cover of the figure eight knot!!! See this paper by Hilden, Lozano, and Montesinos.
- If you only care about $\mathbb{Z}/2$-actions this problem is not as difficult.
In higher dimensions, the Smith conjecture is just false in all kinds of ways. For disproofs, see the work of Giffen, Gordon, and Sumners. In the next post, I'm going to give a hands-on construction of a disproof of the four-dimensional Smith conjecture, i.e. give a $\mathbb{Z}/2$-action on $S^4$ whose fixed point set is knotted.
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