Chen–Nagano parity conjecture for compact connected Riemannian symmetric spaces

Prove that every compact connected Riemannian symmetric space has a two-number congruent modulo 2 to its topological Euler characteristic.

Background

The paper recalls the Chen–Nagano conjecture for compact connected Riemannian symmetric spaces. For such a space M, the conjecture asserts that the two-number and the topological Euler characteristic have the same parity.

The authors note that compact connected irreducible Riemannian symmetric spaces have been classified and that computations based on this classification suggest the conjecture is valid. However, they state that a conceptual proof is still desirable, so the conjecture is presented as unresolved.

References

Let $M$ be a compact connected Riemannian symmetric space. Then it satisfies #_2(M) \equiv \chi{\mathrm{top}(M) \pmod{2}. We refer to this conjecture as the Chen--Nagano conjecture. Compact connected irreducible Riemannian symmetric spaces have been classified, and computations based on this classification suggest that the conjecture is valid. Nevertheless, a conceptual proof is desirable, as it may clarify the underlying mechanism responsible for the parity agreement.

— Two-numbers and Euler characteristics for quandles  (2609.05187 - Kai et al., 4 Sep 2026) in Conjecture in Section 2, following the Chen–Nagano theorem

Find a condition for a finite quandle $(X,s)$ to satisfy the Chen-Nagano inequality $#_2 (X,s) \geq \chi{\mathrm{qdl} (X,s)$.

— Two-numbers and Euler characteristics for quandles  (2609.05187 - Kai et al., 4 Sep 2026) in Problem 2, final Problem environment