[Edit: I had a mistake in the numerology (took d=6,5 instead of d=5,4). Edit: I mistakenly identified my mistake, it is 6,5 but I got the indices shifted by one.]

# Background: Polygon spaces

Given a polygon P in space we can consider the space of all its embeddings when the lengths of the edges are fixed and mod out by rigid motion of space. Such spaces are called polygon spaces: see, for example, the paper *Polygon spaces and Grassmannians* by Jean-Claude Hausmann and Allen Knutson. (See also the MO question Space of simple polygons on $n$-vertices as a set of points in $\mathbb{R}^{2n}$.)

The dimension of these spaces can be recovered by a simple numerology: The number of degrees of freedom is $3n$; from this you have to subtract $n$ constraints given by the length of edges, and the dimension 6 of rigid motions of space, and you are left with $2n-6$.

If you consider embeddings into the plane rather then into space then the dimension of your space is $2n-n-3 = n-3$. The idea that such spaces of polygons have symplectic structure is attributed in the Hausmann and Knutson's paper to Cappell. [AK adds: It seems to have first been observed by Klyachko.]

The fact that $n-3$ is half of $2n-6$ is no coincidence and indeed the space of planar embeddings correspond to Lagrangian submanifolds of the polygon spaces.

In between the simple numerology and the complicated spaces we can identify some intermediate objects: the tangent vector spaces. Those are the vector spaces of infinitesimal motions of an embedded polygon in $R^3$ or $R^2$. After modding out with infinitesimal motions arising from rigid motions of the whole space (or plane) we are left with vector spaces of dimensions $2n-6$ and $n-3$ respectively.

## Summary: Talking about polygon spaces we can identify three levels:

**Numerology:** $2n-6$ degrees of freedom to embeddings of polygons into $R^3$, $n-3$ degrees of freedom to embeddings of polygons into $R^2$.

**Linear algebra:** The vector spaces of infinitesimal motions of polygons embedded in the plane and space.

**Symplectic manifolds:** The polygon spaces and their Lagrangian submanifolds.

The polygon spaces have various structures on them and are quite exciting.

# The idea:

Polygons are triangulations of $S^1$. Extend the notion of a "polygon space" to certain embeddings of triangulated $(2k+1)$-dimensional spheres. I will mainly talk about the "next case" which are triangulations of $S^3$.

Problem: Describe an appropriate analog of "polygon spaces" for triangulations of 3-dimensional spheres.

# Details: From numerology to spaces

Numerology --->>> Linear algebra (vector spaces) --->>> Varieties/spaces.

## 1) The Numerology:

The numerology refers to the dimension of our hypothetical analogs for polygon spaces.

Given a sequence of non-negative integers $(f_{-1}=1) ,f_0, f_1, f_2 \dots$ and an integer $d \ge 0$ we define a new sequence $g_0[d]$, $g_1[d]$ , $g_2[d], \dots$ as follows: $$g_0[d] = 1,$$ $$g_1[d] = f_0 - d,$$ $$g_2[d] = f_1 - (d-1) f_0 + {{d} \choose {2}},$$ $$g_3[d] = f_2 - (d-2) f_1 + {{d-1} \choose {2}} f_0 - {{d} \choose {3}},$$ etc. (But below we worry only about g_2 and g_3)

The sequence $f_0, f_1,\dots $ usually comes as the $f$-vector of some simplicial complex and the new sequences $g_i[d]$ are important in studying the combinatorics of these complexes.

### polygons

Now, if we have a polygon with n vertices we have: $$f_0 = f_1 = n ,$$ $$g_2[3] = -(n-3)$$ and $$g_2[4] = -(2n-6),$$ which are the dimensions of the polygon spaces for embeddings in $R^2$ and $R^3$ respectively.

### The next case: triangulations of $S^3$.

The next analogous case is to start with a triangulation $K$ of $S^3$ with $f_0$ vertices $f_1$ edges $f_2$ triangles and $f_3$ 3-simplices.

Here (using Euler's theorem) it is easy to verify that for every $K$ we have $$g_3[6] = 2g_3[5].$$ (Both quantities are negative.)

Thus, the suggestion is that $-g_3[6]$ should be the dimension of a "generalized polygon space" for some sort of embedding of the triangulation $K$ (possibly into $ R^5$), and $-g_3[5]$ the dimension for "embedding" of $K$ (possibly into $R^4$). Hopefully, the latter will correspond to a Lagrangian submanifold of the former.

(**Remark:** in the comments Misha proposed making the embedding into certain CAT (1) spaces.)

Let me just verify the identity I described: $g_3[6] = f_2-4f_1+10f_0-20$ and $g_3[5]=f_2-3f_1+6f_0-10.$ For triangulated 3-spheres, Euler's theorem asserts that $f_0-f_1+f_2-f_3=0$ and also $f_2=2f_3$ and therefore $f_2=2f_1-2f_0$. It follows that $$g_3[6]=-2f_1+8f_0-20,$$ while $$g_3[5]=-f_1+4f_0-10.$$

## 2) The linear algebra

The linear algebra refers to the tangent vector spaces of our hypothetical spaces.

### polygons:

For a graph $G$ consider the quantity

$X$ = (the number of edges - $m$ the number of vertices + ${{m+1} \choose{2}})$.

$X$ ($=g_2$) is a lower bound (interesting of course only when $X$ is non-negative) on the space of infinitesimal stresses when the vertices of the graph are embedded in $R^m$. $-X$ is a lower bound on the number of infinitesimal flexes for such embeddings (this is interesting when $X$ is negative).

For polygons embedded in $R^2$ and $R^3$. $X$ is non-positive and $-X$ is the dimension of the infinitesimal flexes (not just a lower bound). Those are the tangents of the polygon spaces.

### Triangulations of $S^3$.

For a simplicial 2-dimensional complex $K$ we can let $$Y=f_2 - (d-2) f_1 + {{d-1} \choose {2}}f_0 - {{d} \choose {3}}.$$ (We called it $g_3[d]$ before.) We care about $d=6$ so $$Y=f_2-4f_1+10f_0-20.$$

There are some known notions of high dimensional infinitesimal flexes and stresses which are bounded by $Y$ (or $-Y$, respectively).

There are conjectures for simplicial spheres, which are proven for simplicial polytopes, that these bounds are tight.

These spaces are perhaps tangent to some hypothetical polygon-like spaces (but only at very special points).

## 3) How the polygon-like space may look?

This particular suggestion probably does not work, but it allows to describe what I am up to.

We embed the vertices of $k$, the triangulated $S^3$ in some fixed location. We need 4 degrees of freedom for every edge and one constraint for every triangle.

For example, we can let the edges be arcs between the corresponding vertices which are parabolas; the "triangles" will be quadratic (describing a surfaces of minimum area) which extend the arcs corresponding to edges.

The condition is that the areas of the "triangles" are prescribed.