Four-way complementary homology in lattices

Determine whether every finite lattice with nonvanishing reduced homology in degree k admits an element x and complements y and y' of x whose lower and upper intervals have nonvanishing reduced homology in the four complementary degrees specified by a+b=k-2.

Background

The paper’s main results establish complementary lower-interval homology for pairs of lattice complements in several settings, including arbitrary lattices under a weaker conclusion, geometric lattices, and face lattices of simplicial complexes. Question 1.1 strengthens this by simultaneously requiring nonvanishing homology below and above one element x, together with corresponding homology below and above complementary elements y and y'.

The authors prove the strengthened assertion for geometric lattices and discuss positive cases arising from ortho-complemented lattices, but do not resolve it for arbitrary finite lattices. The question is therefore an explicitly stated unresolved extension of the main complementation problem.

References

Does there exist $x \in \overline{L}$ and complements $y$ and $y'$ of $x$ such that \begin{align*} &_a\big(\,(\hat{0},x);K\,\big) \neq 0, \qquad _b\big(\,(x,\hat{1});K\,\big) \neq 0, \ &_b\big(\,(\hat{0},y);K \,\big) \neq 0, \qquad _a\big(\,(y',\hat{1});K \,\big) \neq 0\? \end{align*}

— Pairs of lattice complements with complementary homology  (2609.26397 - Faridi et al., 22 Sep 2026) in Question 1.1 (que:extmain-new), Section 8, “Open questions”

\cref{lem:updown} leads to the following question, to which we have no counterexample.

Is the answer to \cref{que:extmain-new} positive if $k$ coincides with the dimension of the order complex of $\overline{L}$?

— Pairs of lattice complements with complementary homology  (2609.26397 - Faridi et al., 22 Sep 2026) in Question following Proposition 8.5 (lem:updown), Section 8, “Open questions”