k-orientable case for k ≥ 4: existence and splitting in odd dimensions

Ascertain whether odd Euler characteristic k-orientable manifolds exist for k ≥ 4, and, in the event that none exist in the relevant dimensions, determine whether the short exact sequence 0 → Z/2 → SKK_n^{Or_k} → Ω_n^{Or_k} → 0 splits for odd n.

Background

The authors generalise splitting results across many tangential structures using Kervaire-type invariants. For k-orientable manifolds, the parity of Euler characteristics is strongly constrained; however, for k ≥ 4 it is unsettled whether odd Euler characteristic examples exist. If such examples do not exist in the relevant odd-dimensional contexts, the splitting of the Z/2-extension remains an open question.

References

For k-orientable manifolds with k\geq 4, in particular 8-parallelisable manifolds, it is unknown whether any odd Euler characteristic manifolds exist (if they do they would live in dimensions multiples of 2{k+1}, see \cref{section:k-orientability}) or whether the sequence splits if this is not the case.

— SKK groups of manifolds and non-unitary invertible TQFTs (2504.07917 - Hoekzema et al., 10 Apr 2025) in Section “Splitting results for odd-dimensional SKK groups,” following Table 1