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Existence of odd-Euler-characteristic k-orientable manifolds in high connectivity

Determine whether there exist k-orientable manifolds of dimension 2^{k+1} m with odd Euler characteristic for k ≥ 4; in particular, ascertain whether there exists a 4-orientable manifold of dimension 32 m with odd Euler characteristic.

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Background

The authors show that k-orientability forces many Wu classes to vanish, greatly restricting the parity of the Euler characteristic. For k ≤ 3, examples with odd Euler characteristic are known in the minimal allowed dimensions (e.g., complex, quaternionic, and octonionic projective spaces). For k ≥ 4, the existence of such manifolds remains unsettled, and this question directly impacts the structure of SKK groups and splitting problems in odd dimensions.

References

Open Question Does there exist a 4-orientable manifold \mathcal{X}{32m} with odd Euler characteristic? More generally, does there exist a k-orientable 2{k+1}m-dimensional manifold for k\geq 4 with odd Euler characteristic?

— SKK groups of manifolds and non-unitary invertible TQFTs (2504.07917 - Hoekzema et al., 10 Apr 2025) in Open Question, Section 2.6 (k-orientability)