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Generalized lower bound conjecture for triangulated spheres

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Let dd and kk be integers with d≥2k+1d\geq 2k+1, and let XX be a triangulated dd-sphere with face-vector (f0,…,fd)(f_0,\dots,f_d). Generalized lower bound conjecture. For every jj, the face numbers satisfy

fj≥{∑i=−1k−1(−1)k−i+1(j−i−1j−k)(d−i+1j−i)fi,k≤j≤d−k,∑i=−1k−1(−1)k−i+1[(j−i−1j−k)(d−i+1j−i)−(kd−j+1)(d−id−k+1)+∑l=d−jk−1(−1)k−l(ld−j)(d−id−l+1)]fi,d−k+1≤j≤d.f_j \geq \begin{cases} \displaystyle\sum_{i=-1}^{k-1}(-1)^{k-i+1}\binom{j-i-1}{j-k}\binom{d-i+1}{j-i}f_i,& k\leq j\leq d-k,\\ \displaystyle\sum_{i=-1}^{k-1}(-1)^{k-i+1}\left[\binom{j-i-1}{j-k}\binom{d-i+1}{j-i}-\binom{k}{d-j+1}\binom{d-i}{d-k+1}+\sum_{l=d-j}^{k-1}(-1)^{k-l}\binom{l}{d-j}\binom{d-i}{d-l+1}\right]f_i,& d-k+1\leq j\leq d. \end{cases}

Equality holds for some jj if and only if XX is a kk-stacked dd-sphere. The conjecture generalizes the lower bound theorem: its k=1k=1 case is the lower bound theorem for triangulated spheres, while the j=kj=k case was previously stated for polytopal spheres. Stanley proved the inequalities for polytopal spheres, but the equality characterization remains unresolved even there; the conjecture has also been suggested for simply connected triangulated manifolds.

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. Generalized Lower Bound Conjecture for triangulated spheres

    Let d≥2k+1d\geq 2k+1, let SS be a triangulated dd-sphere with face numbers (f0,…,fd)(f_0,\dots,f_d), and let fjf_j denote its number of jj-faces. Generalized Lower Bound Conjecture. For every relevant jj, fjf_j satisfies the piecewise lower bound displayed in the source, with the first formula for k≤j≤d−kk\leq j\leq d-k and the second for d−k+1≤j≤dd-k+1\leq j\leq d. Equality for any jj holds if and only if SS is a kk-stacked dd-sphere. This is presented as the Generalized Lower Bound Conjecture due to McMullen and Walkup; the supplied text gives no resolution status.

    source: Felix Effenberger, “Stacked polytopes and tight triangulations of manifolds”, arXiv:0911.5037 (2011).

References

Primary source

Bhaskar Bagchi and Basudeb Datta, “Lower bound theorem for normal pseudomanifolds”, arXiv:0802.3747 (2012).

face-vectorslower-bound-theoremlower-bound-theoremspolytope-combinatoricstriangulated-spherestriangulations
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