Group actions on spheres with one fixed point

The question of what groups act on spheres in certain ways has an impressive history. Milnor, pioneering the methods of group cohomology, computed the exact set of finite groups which act on a sphere.

Atiyah and Bott, in their groundbreaking work on Lefschetz fixed point theorems, derive as a corollary that a diffeomorphism $f$ of a compact manifold whose order is a power of an odd prime cannot have precisely one fixed point (a conjecture of Conner and Floyd). They also prove an old conjecture that if a compact Lie group $G$ acts differentiably on a homology sphere with precisely two fixed points, the two linear representations on the tangent spaces of the fixed points are equivalent. Specializing to the case $G=\mathbb{Z}/n\mathbb{Z}$, it follows that two cobordant lens spaces are isometric.

It would take me a lot of effort to learn and give an account of these results although this is certainly something I would like to do at some point. Instead, I would like to give a brief survey of some interesting results on groups acting on manifolds with exactly one fixed point that I found by accident while looking for something completely different. I leave the proofs to the interested reader.

The first example of a smooth action of a group with one fixed point was constructed by Stein: an action of $\mathrm{SL}_2(5)$, also known as the double cover of $A_5$ on $S^7$.

Petrie found the first infinite class of groups for which such an action is known. Using equivariant surgery techniques he proves

$\textbf{Theorem 1:}$  The following groups act smoothly on a homotopy sphere with one fixed point: (i) $S^3$, $\mathrm{SO(3)}$; (ii)$\mathrm{SL(2,F), PSL(2,F)}$ with characteristic $F$ odd; and (iii) any odd order abelian group having three noncyclic Sylow subgroups.

One can also consider the induced representation on the tangent space at the fixed point and Petrie shows that many representations do occur.

It is then natural to ask in what dimensions can one find manifolds with such actions. Using gauge-theoretic techniques based on the work of Taubes Buchdal, Kwasik, and Schultz show the following:
$\textbf{Theorem 2:}$ Let $M$ be a compact oriented 3-manifold such that $\pi_1(M)$ has no nontrivial representations into $\mathrm{SU}(2)$. If G is a finite group acting smoothly (or locally linearly) on $M$ then the number of isolated fixed points of the action is 0 or 2.

$\textbf{Corollary 3:}$ if $G$ acts smoothly on the 3-disc $D^3$ [resp. $\mathbb{R}^3$] then the fixed point set $(D^{3})^{(G)}$ is a disc of dimension 0,1, or 2. [resp. $\mathbb{R}^0$, $\mathbb{R}^1$, $\mathbb{R}^2$]

Using more group theoretic methods, they also prove
$\textbf{Theorem 4:}$ For each $n\geq6$ there is a locally linear action of $A^5$ on $S^n$ with exactly one fixed point.

On the other hand, if $M$ is a closed integral homology sphere of dimension 4 or 5, then they show that there is no locally linear action of a finite group $G$ on $M$ with exactly one fixed point.

One can then ask for this action to be smooth, which culminates in the following due to a great many people's hard work, especially in surgery theory:

$\textbf{Theorem 5:}$ $S^n$ admits an action of a finite group with one fixed point if and only if $n \geq 6$.

Finally, coming full circle back to where we started this survey, since $\mathrm{SL}_2(5)$ has a smooth one-fixed-point action on $S^7$, by taking a reduced suspension S8 has an exotic topological $\mathrm{SL}_2(5)$-action. Borowiecka shows that there is no effective smooth $\mathrm{SL}_2(5)$-action on any 8-dimensional Z-homology sphere with exactly one fixed point. The proof is based on a case-by-case analysis of possible tangential $\mathrm{SL}_2(5)$-modules, obtained from the character table, at the fixed point of 8-dimensional homology spheres, and in each case a contradiction is obtained by using the intersection form.