For every finite bounded lattice L and integers a,b, if widetildeH_a+b-2!(Delta(L setminushat0,hat1);Bbbk)≠ 0, then there exist complementary elements x,y∈ L, meaning x wedge y=hat0 and x vee y=hat1, such that widetildeH_a!(Delta((hat0,x));Bbbk)≠ 0 and widetildeH_b!(Delta((hat0,y));Bbbk)≠ 0, where Bbbk is a coefficient field.

Global-to-local lattice-complement homology conjecture

For every finite bounded lattice $L$ and integers $a,b$ , if $\widetilde{H}_{a+b-2}\!\left(\Delta\left(L\setminus\{\hat{0},\hat{1}\}\right);\Bbbk\right)\neq 0$ , then there exist complementary elements $x,y\in L$ , meaning $x\wedge y=\hat{0}$ and $x\vee y=\hat{1}$ , such that $\widetilde{H}_{a}\!\left(\Delta((\hat{0},x));\Bbbk\right)\neq 0$ and $\widetilde{H}_{b}\!\left(\Delta((\hat{0},y));\Bbbk\right)\neq 0$ , where $\Bbbk$ is a coefficient field.

References

Primary source

arXiv

Additional references

Pairs of lattice complements with complementary homology — arXiv — Sara Faridi, Dharm Veer, Volkmar Welker

combinatorial-topology commutative-algebra

Posted 2 weeks ago · edited 2 weeks ago

Progress summary

Refreshed 2 weeks ago

Claimed progress

A new paper finds supporting cases and counterexamples to stronger versions, but does not settle the main conjecture.

The conjecture asks whether global homology forces complementary lattice pairs with prescribed local homology. Faridi, Veer, and Welker report positive evidence while ruling out stronger formulations, without claiming a complete resolution.

September 2026 paper

A September 22, 2026 report on Faridi, Veer, and Welker's paper describes positive evidence for the proposed principle and counterexamples to strengthened versions. This is progress on the conjectural picture, not a proof of the central statement.

Current status (as of September 2026): Positive evidence and counterexamples to stronger variants are reported, but the central global-to-local conjecture remains unresolved.

Sources

No solutions have been posted yet.