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Generalized lower bound conjecture for triangulated spheres

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 1

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).

Sources & referencesView supporting material

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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