Positive weights orthogonal to integer subgroups

A subgroup of ι → ℤ contains no nonzero nonnegative vector exactly when it is orthogonal to a vector of strictly positive integer weights. This is the integer-lattice form of Stiemke's theorem.

For a pointed Heegaard diagram, ι indexes its regions, and P is the subgroup of periodic domains. Weak admissibility, for all Spin^c structures at once, says every nonzero periodic domain has positive and negative multiplicities. Since P is closed under negation, this is the condition below. The weights give positive target areas for the regions, with zero signed area for every periodic domain. Applying the result to diagrams requires a separate construction of their periodic-domain subgroup.

Main result

• AddSubgroup.exists_pos_dotProduct_eq_zero_iff.

Reference

• P. Ozsváth, Z. Szabó, Holomorphic disks and topological invariants for closed three-manifolds, Ann. of Math. 159 (2004), Lemma 4.12.

theorem AddSubgroup.exists_pos_dotProduct_eq_zero_iff {ι : Type u_1} [Fintype ι] (P : AddSubgroup (ι → ℤ)) : (∃ (c : ι → ℤ), (∀ (i : ι), 0 < c i) ∧ ∀ p ∈ P, c ⬝ᵥ p = 0) ↔ ∀ p ∈ P, 0 ≤ p → p = 0

A subgroup P of ι → ℤ contains no nonzero nonnegative vector exactly when some vector of strictly positive integer weights is orthogonal to all of P.

For the group of periodic domains of a pointed Heegaard diagram, the weights are the areas of the regions for an area form in which every periodic domain has signed area zero; compare Ozsváth–Szabó, Holomorphic disks and topological invariants for closed three-manifolds, Lemma 4.12.