Kalai's manifold g-conjecture

From papers

Let $\Delta$ be an orientable simplicial $\Bbbk$ -homology $d$ -manifold without boundary, and let $h^{\prime\prime}(\Delta)=(h^{\prime\prime}_0(\Delta),\dots,h^{\prime\prime}_{d+1}(\Delta))$ be its $h^{\prime\prime}$ -vector. A sequence of integers $(h^{\prime\prime}_0,\dots,h^{\prime\prime}_{d+1})$ is the $h^{\prime\prime}$ -vector of $\Delta$ if and only if it is symmetric and its first differences form an M-vector. Kalai's manifold $g$ -conjecture. One has

h^{\prime\prime}_i=h^{\prime\prime}_{d+1-i}\quad\text{for all }0\leq i\leq d+1,

and

(h^{\prime\prime}_0,h^{\prime\prime}_1-h^{\prime\prime}_0,\dots,h^{\prime\prime}_{\lfloor(d+1)/2\rfloor}-h^{\prime\prime}_{\lfloor(d+1)/2\rfloor-1})

is an M-vector. The symmetry condition is known, and the M-vector condition is known in several special cases, but the full conjecture remains open.

Progress summary

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Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

Kalai's manifold $g$ -conjecture
Let $\Delta$ be an orientable $(d-1)$ -dimensional $\mathbf{k}$ -homology manifold without boundary. Define the modified $h$ -numbers $h_i”$ as in the source by
$h_i”(\Delta)=\begin{cases}h_i'(\Delta)-\binom{d}{i}\widetilde\beta_{i-1}(\Delta;\mathbf{k})&0\leq i<d,\h_d'(\Delta)&i=d,\end{cases}$
where $h_i'(\Delta)=\dim_{\mathbf{k}}(\mathbf{k}[\Delta]/\Theta)_i$ for a linear system of parameters $\Theta$ , and let $g_i”=h_i”-h_{i-1}”$ . Kalai's manifold $g$ -conjecture. The vector
$(g_i”:=h_i”-h_{i-1}”)_{i=0}^{\lfloor d/2\rfloor}$
is an $M$ -vector. Kalai's conjecture is presented as a far-reaching generalization of the sphere $g$ -conjecture to manifolds; the source does not give a resolution.
source: Feifei Fan, “Weak Lefschetz property of PL-spheres”, arXiv:2001.06594 (2021).

Sources & referencesView supporting material

Primary source

Kai Fong Ernest Chong and Tiong Seng Tay, “The face numbers of homology spheres”, arXiv:1706.03322 (2024).

1 week ago

Progress summary

Equivalent formulations 1

Kalai's manifold ggg-conjecture

Primary source

Kalai's manifold $g$ -conjecture