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Pin+ in dimension 8k+2: Wu class behavior versus Euler parity

Conjecture that, in dimension 8k+2 for Pin^+ manifolds, the analogue of the Wu-class vanishing result from Theorem 2.10 does not hold, yet all such Pin^+ manifolds still have even Euler characteristic; establish or refute this by analysing the top Wu class and Euler parity in dimension 8k+2.

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Background

Theorem 2.10 proves vanishing of the top Wu class for Pin- in dimension 8k+4 and for Pin+ in dimension 8k+6. The authors conjecture that the analogous vanishing fails for Pin+ in dimension 8k+2, while the Euler characteristic remains even. This directly impacts applications of the odd-dimensional splitting criterion via Wu classes and the classification of SKK groups in adjacent dimensions.

References

We conjecture that the conclusion of the above Theorem does not hold in the Pin+ case in the dimension 8k+2, but that nonetheless all such manifolds still have even Euler characteristic.

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