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Odd-dimensional Pin+ case at n ≡ 1 (mod 8)

Determine the SKK group SKK_n^{Pin^+} for n ≡ 1 (mod 8) by resolving whether there exist 8k+2-dimensional Pin^+ manifolds with even Euler characteristic and, if so, whether the short exact sequence 0 → Z/2 → SKK_{8k+1}^{Pin^+} → Ω_{8k+1}^{Pin^+} → 0 splits.

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Background

For Pin+ structures in odd dimensions, the authors determine SKK groups for three congruence classes, but the n ≡ 1 (mod 8) case remains unresolved. This hinges on whether 8k+2-dimensional Pin+ manifolds necessarily have even Euler characteristic (known in low dimensions but open in general). If all such Euler characteristics are even, a nontrivial extension by Z/2 arises whose splitting status is also unknown.

References

To our knowledge SKK_n{Pin+} is unknown in general for n\equiv 1\pmod{8}, both because it remains unresolved whether 8k+2-dimensional Pin+ manifolds have even Euler characteristic for k\geq 2, and because, if they do, it remains unclear whether the sequence is split for k \geq 1.

— SKK groups of manifolds and non-unitary invertible TQFTs (2504.07917 - Hoekzema et al., 10 Apr 2025) in Subsection “Pin±-manifolds” (after Proposition 3.11)