Parity constraints beyond the two-carrier decomposition

Determine whether the odd-parity conclusion for the soul persists for self-complementary periodic surfaces of higher genus when the two-carrier, two-cylinder decomposition (S) is not assumed, and characterize which even values of the number of carriers can occur.

Background

The main theorem proves that, for a Morse function satisfying f(x+v)=-f(x), every soul L in a stability zone whose foliation has exactly two genus-one carriers and two cylinders satisfies ⟨L,2v⟩ odd. The paper observes that with ν carriers and ν cylinders, the same argument imposes an arithmetic condition involving ν and the shift s, but only establishes that ν must be even. The unresolved issue is whether the parity conclusion extends to higher-genus settings and which even carrier numbers are geometrically realizable.

References

Does the conclusion persist beyond (S), for surfaces of higher genus? The remark above reduces this to the arithmetic of $\nu$ and $s$ but, beyond the parity of $\nu$, we do not know which $\nu$ occur.

— A parity constraint on the topological numbers of self-complementary triply-periodic surfaces  (2610.18416 - Leo, 13 Oct 2026) in Section “Complements and questions”

Is there a converse: for which $L$ with $\langle L,w\rangle$ odd is $D_L\neq\emptyset$ for some self-complementary $f$?

— A parity constraint on the topological numbers of self-complementary triply-periodic surfaces  (2610.18416 - Leo, 13 Oct 2026) in Section “Complements and questions”

Can the parity constraint be strengthened in that direction?

— A parity constraint on the topological numbers of self-complementary triply-periodic surfaces  (2610.18416 - Leo, 13 Oct 2026) in Section “Complements and questions”