Let M be a triangulation of a connected closed d-manifold. Write g_j(M) for its g-numbers, beta_i(M; F) for Betti numbers over a field F, and let cal K_l(d) denote the class used in the source. Generalized lower bound conjecture for triangulated manifolds. For 1≤ l≤ (d-1)/2, g_l+1(M)≥∑_i=1^l(-1)^l-ibeta_i(M; F). Moreover, equality holds for some l<(d-1)/2 if and only if M∈cal K_l(d). The inequality and the equality implication in one direction were known under additional hypotheses; the l=1 case was proved for F-orientable manifolds, while the full conjecture, which includes the generalized lower bound conjecture for homology spheres, was not expected to be settled soon.

Generalized lower bound conjecture for triangulated manifolds

About 15 years old · traced to 2011

Let $M$ be a triangulation of a connected closed $d$ -manifold. Write $g_j(M)$ for its $g$ -numbers, $\beta_i(M;\mathbb F)$ for Betti numbers over a field $\mathbb F$ , and let ${\cal K}_l(d)$ denote the class used in the source. Generalized lower bound conjecture for triangulated manifolds. For $1\leq l\leq (d-1)/2$ ,

g_{l+1}(M)\geq\sum_{i=1}^l(-1)^{l-i}\beta_i(M;\mathbb F).

Moreover, equality holds for some $l<(d-1)/2$ if and only if $M\in{\cal K}_l(d)$ . The inequality and the equality implication in one direction were known under additional hypotheses; the $l=1$ case was proved for $\mathbb F$ -orientable manifolds, while the full conjecture, which includes the generalized lower bound conjecture for homology spheres, was not expected to be settled soon.

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.

Generalized lower bound conjecture for triangulated manifolds
Let $\Delta$ be a connected triangulated $(d-1)$ -manifold without boundary. Write $h_i(\Delta)$ for its $h$ -numbers and $\beta_i(\Delta)$ for its $i$ th Betti number. A triangulated manifold is locally $r$ -stacked when each vertex link is an $r$ -stacked triangulated sphere. Generalized lower bound conjecture.
$h_{r+1}(\Delta) \geq h_r(\Delta)+ \binom{d+1}{r+1} \sum_{j=1}^{r+1} (-1)^{r+1-j} \beta_{j-1}(\Delta)$
for $r=1,2,\dots,\lfloor d/2\rfloor$ ; moreover, if equality holds for some $r<d/2-1$ , then $\Delta$ is locally $r$ -stacked. This extends the generalized lower bound theorem from simplicial polytopes to triangulated manifolds, with the equality case relating face-number bounds to stackedness.
source: Jonathan Spreer, Uli Wagner, Benjamin A. Burton, Satoshi Murai, Eric Sedgwick and Henry Segerman, “Collection of abstracts of the Workshop on Triangulations in Geometry and Topology at CG Week 2014 in Kyoto”, arXiv:1406.0333 (2014).

References

Primary source

Bhaskar Bagchi and Basudeb Datta, “On stellated spheres, shellable balls, lower bounds and a combinatorial criterion for tightness”, arXiv:1102.0856 (2012).

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