### Sphere Eversion

#### Posted by David Corfield

*guest post by Scott Carter, a commentary on his and Sarah Gelsinger’s sphere eversion (50MB). See also John Armstrong’s commentary.*

Consider the $2$-category given as follows. The objects correspond to a (possibly empty) set of dots along a vertical axis. Each dot has an associated sign: $-$ will indicate that a $1$-morphism emanating from the dot should be a left pointing strand; $+$ indicates a right pointing strand. The orientation information will be completely suppressed here. The $1$-morphisms are generated (in the sense of composition and tensor products) by $\subset$, $\supset$, and $X$. Of course a predominantly horizontal arc indicates the identity $1$-morphism. There are two flavors of $\subset$ and $\supset$ when orientations are drawn, and 4 flavors of $X$ in that case.

A $1$-morphism from the empty object to the empty object is a collection of generically immersed circles in the plane.

The set of $2$-morphisms are generated by birth, death, saddle, cusp,(the projection of) type II and type III, and the (projection of) Yetter $\psi$ move in which a double point on one arc near an optimum bounces to the other arc near the optimum. Here, of course, height is measured in a left to right direction. Finally, there are tensorators which allow the interchange of critical levels from right to left.

We want a strict symmetric monoidal $2$-category with duals on an object generator $(*)$ whose dual is $(-*)$. So we should define those identities among $2$-morphisms that are allowed. Notice first though that there is no pivotal axiom that would correspond to the type I move. Each of type II, type III and the $\psi$ move are isomorphisms. Loosely speaking their squares are the identity. More precisely, we can assume a naturality axiom for the transposition $X$. And one case of naturality is the $\psi$ move. The naturality axiom, then, is among the $2$-morphisms. There is a direction associated to the move, and the move followed by its opposite is the identity. Similarly there is a direction for type II. Both compositions of type II are equivalent to the identity $2$-morphism. The critical cancelation between a saddle and an optimum (birth or death) is axiomatized as in Baez-Langford. The Zamolochikov relation holds, and a slew of relationships between folds, optima of fold sets, cusps, and double and/or triple points hold. I think that Baez-Langford arguments give that this immersed surface category is the FREE SYMMETRIC MONOIDAL $2$-CATEGORY WITH DUALS ON AN OBJECT GENERATOR.

All of the generating $2$-morphisms (with the exception of the tensorator) are illustrated in the file. Next the $3$-morphisms (or identities among $2$-morphisms) are shown.

A red embedded sphere represents a particular $2$-morphism from the identity $1$-morphism on the empty object to itself. Each step in the eversion represents a relationship among $2$-morphisms. The entire eversion can be thought of as a $3$-morphism from the red $2$-morphism to the blue $2$-morphism.

Smale’s theorem says that there is such a $3$-morphism in this category. It is obvious (or an easy exercise with orange peels) that there is such a $3$-morphism in the pivotal case.

Our eversion differs from all of its predecessors in that every step is a generic codimension $1$ singularity. Consequently, it is not symmetric with respect to the quadruple point.

Scott

## Re: Sphere Eversion

I hadn’t known of the mysteries of sphere eversion before. Watching the video was a surprising amount of fun.

Now I am trying to see what is going on here. On the machine in my current office any attempt to download the 50MB file linked to above fails. I am lacking any information that might be hidden there.

So let me just try to understand what the entry here is saying:

So we have a bunch of arrows drawn on the line, touching each other at their endpoints (but not necessarily target-source but also source-source and target-target), and we want to generate a monoidal category from them, I suppose.

Do I read the above as saying that $\subset$ is to denote concatenation of arrows, while $\supset$ is to denote their monoidal tensor product? Why, then, that unusual notation. And what is $X$? Or is that a typo for $\times$?

I am afraid I didn’t understand the message of this first paragraph. Can anyone help me?