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).