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Kalai's manifold g-conjecture

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Let Δ\Delta be an orientable simplicial k\Bbbk-homology dd-manifold without boundary, and let h′′(Δ)=(h0′′(Δ),…,hd+1′′(Δ))h^{\prime\prime}(\Delta)=(h^{\prime\prime}_0(\Delta),\dots,h^{\prime\prime}_{d+1}(\Delta)) be its h′′h^{\prime\prime}-vector. A sequence of integers (h0′′,…,hd+1′′)(h^{\prime\prime}_0,\dots,h^{\prime\prime}_{d+1}) is the h′′h^{\prime\prime}-vector of Δ\Delta if and only if it is symmetric and its first differences form an M-vector. Kalai's manifold gg-conjecture. One has

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

and

(h0′′,h1′′−h0′′,…,h⌊(d+1)/2⌋′′−h⌊(d+1)/2⌋−1′′)(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.

Equivalent formulations 1Other wordings

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

  1. Kalai's manifold gg-conjecture

    Let Δ\Delta be an orientable (d−1)(d-1)-dimensional k\mathbf{k}-homology manifold without boundary. Define the modified hh-numbers hi”h_i” as in the source by

    hi”(Δ)={hi′(Δ)−(di)β~i−1(Δ;k)0≤i<d,hd′(Δ)i=d,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 hi′(Δ)=dim⁡k(k[Δ]/Θ)ih_i'(\Delta)=\dim_{\mathbf{k}}(\mathbf{k}[\Delta]/\Theta)_i for a linear system of parameters Θ\Theta, and let gi”=hi”−hi−1”g_i”=h_i”-h_{i-1}”. Kalai's manifold gg-conjecture. The vector

    (gi”:=hi”−hi−1”)i=0⌊d/2⌋(g_i”:=h_i”-h_{i-1}”)_{i=0}^{\lfloor d/2\rfloor}

    is an MM-vector. Kalai's conjecture is presented as a far-reaching generalization of the sphere gg-conjecture to manifolds; the source does not give a resolution.

    source: Feifei Fan, “Weak Lefschetz property of PL-spheres”, arXiv:2001.06594 (2021).

References

Primary source

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

face-enumerationface-numbersg-conjecturehomology-manifoldssimplicial-manifolds
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