*This is the fourth 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 known to hold for simplicial spheres and the Lefshetz property which is known for simplicial polytopes and is wide open for simplicial spheres. Eran’s third post was about the connection to algebraic shifting. The fifth and last post is planned to deal with connections to rigidity, a fascinating topic on its own which is related also like yesterday’s post to architecture and art.*

(From right): Udo Pachner, Peter Kleinschmidt, and Günter Ewald. (Oberwolfach photo collection)

## All kinds of spheres

It is possible that the link of a vertex in a simplicial sphere is no longer a sphere. This is not good if we look for a proof by induction. A subfamily which is closed under links is the piecewise linear spheres. Another advantage of this family for inductive proof is given by Pachner: any PL--sphere can be obtained from the boundary of the -simplex by a finite sequence of bistellar moves aka Pachner moves. This is a family of possible local moves, defined combinatorially.

However, so far no proof of the -conjecture for PL-spheres is known.

A superfamily of simplicial spheres, closed under links, is that of *homology spheres*, where we just require that the *homology* of all face links are as of the sphere of the appropriate dimension. The hard Lefschetz property is conjectured for this family too.

What we can show is that the hard Lefschetz property is preserved under some combinatorial constructions on spheres, namely: connected sum, join, stellar subdivision – or more generally the inverse moves of topology preserving edge contractions, which are exactly the admissible contractions defined above for *minors*. Let us just mention two remarks:

1. These last moves played a role in Murai’s refinement of Billera-Lee theorem. He showed that any squeezed sphere can be obtained from the boundary of the simplex in this way, hence squeezed spheres have the hard Lefschetz property.

2. Any PL-sphere can be obtained from the boundary of a simplex by a finite sequence of stellar and inverse-stellar moves, so it remains to show that inverse-stellar moves on PL-spheres preserve the hard Lefschetz property.

## 2-CM

is *doubly Cohen-Macaulay* (2-CM) if is CM and deleting any vertex of , the induced complex on the rest of the vertices is also CM of the same dimension as . Examples include all homology spheres, as well as other examples where the vector is known to be an -vector. This led Björner and Swartz to conjecture that the -vector of any 2-CM complex is an -vector. Note that the -vector need not be symmetric anymore. Algebraically, the *weak Lefschetz* property may hold, i.e. injections

for .

What we can show is only that is an -vector for 2-CM complexes. This is done using *rigidity theory for graphs*, to be discussed next time.

Another case were is an -vector is when is the barycentric subdivision of a simplicial sphere, or more generally of any CM complex!

This was shown recently in a work with Martina Kubitzke, by proving a hard Lefschetz type result.

## Manifolds

Let a finite triangulation of a connected orientalble -manifold without boundary. We have seen that and depends on the Euler characteristic of , so may not be symmetric. Neither it is an -vector.

Can we fix that?

Kalai defined a new vector which is a function of and of the Betti numbers of

where is the dimension of the reduced j-th homology of with coefficients.

The vector is symmetric and an -vector, and recently Isabella Novik and Ed Swartz figured out its algebraic meaning:

The *socle* of a graded standard ring is the ideal of elements in annihilated by the maximal ideal (generated by the variables). Let be generic elements on , so they make as small as possible (it is a finite vector. Such , that makes this quotient ring of zero Krull-dimension is called a linear system of parameters).

Let be the part of the degree at most in the socle of this ring. Then

Now define and for as for the -vector.

Kalai conjectured that is an -vector, and it is possible that has the hard Lefschetz property. Novik and Swartz showed further that the hard Lefschetz property for the vertex links of implies that is an -vector.

So we are back to the algebraic -conjecture.

To end, even is not known for say PL-5-spheres.

Let me just mention that all 5 parts were written at the same time, ~8 years ago. There were some developments and new approaches to the g-conjecture since then. Notably Bagchi’s mu-vector, and a very recent claim for a proof of the conjecture, posted on the arXiv by Chong and Tay that await fixing (the algebra structure as *currently* appears there is not well defined).

Many thanks Eran and sorry for the 8-years delay for part 4. I am looking forward for publishing part 5 and perhaps a new part 6 with updates.