Last summer I have had some fun with codimension 2 manifolds M in a purely differential geometric setting: a **positive curvature d-manifold** which admits a circular action of isometries has a fixed point set K which consists of even codimension positive curvature manifold. The **Grove-Searle **situation https://arxiv.org/abs/2006.11973 is when K is a codimension 2 manifold. In that case, the topology of M is severely restricted. The background to that story could not be more interesting because positive curvature manifolds show an affinity with gauge bosons https://arxiv.org/abs/2006.15773 This part of differential geometry is also called **Conner-Kobayashi theory** named after the mathematicians who pioneered that theory.

This summer, I spent a few days with **Fisk theory,** which is also a **codimension-2 story** but it is topologically more interesting as the knots which appear (a codimension 2 sphere in a sphere is called a **knot**) can be more complex than fixed point sets of positive curvature manifolds. The story now is not tied to a **Riemannian metric structure **given on the manifold but given by a** triangulation structure** on the manifold M. The phenomenon of the odd set O(M) immediately disappears after a Barycentric refinement but can be modified with local refinements like edge refinements. If we start with a manifold triangulation for which O(M) is empty, we can make an edge refinement and get a d-2 sphere O(M) which is the intersection of two adjacent unit spheres. We have seen already in 2014 that the ability to make O(M) disappear in the interior of a 3-ball with interior edge refinements is implying the 4-color theorem! This can lead to a **constructive proof **of the **4-color theorem** with a polynomial bound on the number of steps to color it. The only currently available proof of the 4-color theorem is an **uninspiring proof by accountants**: reduce to a finite set of cases, then use a computer to verify these cases.

**Triangulations **of manifolds are difficult as the stories around the **Hauptvermutung **indicate. I prefer therefore to set things up in a purely **combinatorial manner.** In general, for any **discrete d-manifold M **there is a codimension 2 variety O(M) which I call the Fisk variety. Fisk has shown that in the case of a 3-sphere M, the fisk variety can realize any possible knot in M. This makes use of **Seifert surfaces**. In higher dimensions, we don’t yet know what kind of varieties can occur. In the video, I mention a remark of mine which states that if O(S(x)) is empty or a d-3 sphere for all x, then O(M) is a manifold. This is actually necessary and sufficient. The reason is very simple: the Fisk set in S(x) is the intersection of the** Fisk set** O(M) with S(x).