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.

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)

Thus we can ask: \begin{question} Does the exact sequence \begin{equation}\label{eq:Pinskk} 0 \to Z/2 \to SKK{Pin+}_{8k+1} \to \Omega{Pin+}_{8k+1} \to 0 \end{equation}

split for all k \geq 0? \end{question}

eq:Pinskk:

0→Z/2→SKK8k+1Pin+→Ω8k+1Pin+→00 \to Z/2 \to SKK^{Pin^+}_{8k+1} \to \Omega^{Pin^+}_{8k+1} \to 0

— The Euler characteristic of $Pin^{\pm}$ manifolds  (2610.06747 - Hoekzema et al., 5 Oct 2026) in Section 5, “Application to the SKK-sequence,” immediately after Lemma 5.3 (the lemma numbered \label{lem:Pin+counterexamples}); displayed Question and equation (\ref{eq:Pinskk})