## (Eran Nevo) The g-Conjecture III: Algebraic Shifting

This is the third in a series of posts by Eran Nevo on the g-conjecture. Eran’s first post was devoted to the combinatorics of the g-conjecture and was followed by a further post by me on the origin of the g-conjecture. Eran’s second post was about the commutative-algebra content of the conjecture. It described the Cohen-Macaulay property (which is largely understood and known to hold for simplicial spheres) and the Lefshetz property which is known for simplicial polytopes and is wide open for simplicial spheres.

## The g-conjecture and algebraic shifting

### Squeezed spheres

Back to the question from last time, Steinitz showed that

any simplicial 2-sphere is the boundary of a convex 3-polytope.

However, in higher dimension

there are many more simplicial spheres than simplicial polytopes,

on a fixed large number of vertices. We will need Kalai’s squeezed spheres (of dimension $\geq 4$) which demonstrate this.

It is not known whether the hard Lefschetz property holds for all simplicial spheres. Recently Satoshi Murai showed that that the hard Lefschetz property holds for squeezed spheres. This gives a refinement of Billera-Lee part of the $g$-theorem in terms of generic initial ideals. We will phrase it more combinatorially, via algebraic shifting.

### Symmetric algebraic shifting

Symmetric algebraic shifting is an operator on simplicial complexes, $K\rightarrow \Delta(K)$, defined by Kalai. $\Delta(K)$ carries the same information as the generic initial ideal of $I_K$.

$\Delta(K)$ has the same $f$-vector as $K$, and it is shifted (see our earlier post on shifting). What properties of $K$ can be read off $\Delta(K)$? Well, the hard Lefschetz property can be read!

Let $\Delta(d,n)$ be the maximal simplicial complex on the vertex set $[n]$ with all maximal faces of the same dimension $d-1$ (such complex is called pure such that it doesn’t contain any of the sets  $T^d_d, T^d_{d-1}...,T^d_{\lceil d/2\rceil}$, where

$T^d_{d-k}=\{k+2,k+3,...,d-k,d-k+2,d-k+3,...,d+2\}$, $0\leq k\leq \lfloor d/2\rfloor.$

For example, $T^3_2=\{2,3,5\}$ and $T^3_1=\{4,5\}$ so the maximal faces in $\Delta(3,n)$ are the ones of the form $\{1,2,m\}$ or $\{1,3,m\}$ or $\{2,3,4\}$. In particular, $\Delta(d,n)$ is shifted, and actually it equals $\Delta(C(d,n))$, the symmetric shifting of the boundary of a cyclic polytope.

Now, our simplicial $(d-1)$-sphere on $n$ vertices $K$ has the hard Lefschetz property iff

$\Delta(K)\subseteq \Delta(d,n)$.

Murai showed the following: suppose that a simplicial complex $L$ is pure $(d-1)$-dimensional on $n$ vertices, with $h(L)$ symmetric and $L\subseteq \Delta(d,n)$. Then there exists a (squeezed) sphere $K$ such that $\Delta(K)=L$. $K$ is the squeezed sphere which Kalai constructed from the half-dimensional skeleton of $L$. Given an $M$-vector $g$, the Billera-Lee polytope corresponds to this construction, where the half-dimensional skeleton of $L$ is the compressed complex (w.r.t. the rev-lex order) with $f$-vector equals $g$.

### Van Kampen-Flores complexes

Kalai and Sarkaria (independently) conjectured that if a simplicial complex $L$ on $n$ vertices can be embedded in the $(d-1)$-sphere, then

$\Delta(L)\subseteq \Delta(d,n)$.

In particular, for $L$ a triangulation of the $(d-1)$-sphere, the $g$-conjecture would follow.

Note that $T^{2d+1}_d=\{d+3,d+4,...,2d+3\}$. $T^{2d+1}_d\in \Delta(L)$ iff the $d$-skeleton of the $(2d+2)$-simplex, $\sigma^{2d+2}_{\leq d}$, a.k.a the van Kampen-Flores complex, is contained in $\Delta(L)$, because $\Delta(L)$ is shifted. It is known that $\sigma^{2d+2}_{\leq d}$ does not embed in $S^{2d}$, and we would like to conclude that if $\sigma^{2d+2}_{\leq d}\subseteq \Delta(L)$ then $L$ does not embed in $S^{2d}$. Building on a result of Ed Swartz, if we could prove it, then it would follow that the $g$-vector of any piecewise linear sphere is an $M$-vector!

Say that a simplicial complex $H$ is a minor of a simplicial complex $L$ if you can obtain $H$ from $L$ by successive deletions and (admissible) contractions. Here deletion means taking a subcomplex, and contraction means identifying two vertices $u,v$ which satisfy the link condition, i.e. $lk(v)\cap lk(u) = lk(\{v,u\})$. If $H,L$ are one dimensional, this does recover the usual definition of minors for graphs.

We can show that if $\sigma^{2d+2}_{\leq d}$ is a minor of $L$ then $L$ does not embed in $S^{2d}$. Is it true that $\sigma^{2d+2}_{\leq d}\subseteq \Delta(L)$ implies that $\sigma^{2d+2}_{\leq d}$ is a minor of $L$? The answer is Yes for $d=0,1$, and we don’t know the situation for $d>1$. If it is true, then the $g$-conjecture for PL-spheres would follow.

We just mentioned PL-spheres. Can we solve the $g$-conjecture for special families of spheres? And what about other manifolds?? Next time…