The order-two bordism conjecture for M-theory on unoriented manifolds

Let $\Sigma$ be the Klein bottle with one of its two nonbounding $\operatorname{pin}^{+}$ structures, and let $S^1$ have its nonbounding string structure. Let $B$ be the Bott manifold, and define

$$N=S^1\times\Sigma\times B.$$

The group of bordism classes of the relevant M-theory manifolds is denoted by $\pi_{11}M\mathfrak m_c$.

Order-two bordism conjecture. The group $\pi_{11}M\mathfrak m_c$ is cyclic of order $2$, the bordism class of $(N,0)$ represents its generator, and the mod-$2$ index of the $\operatorname{pin}^{+}$ Dirac operator is an isomorphism

$$\pi_{11}M\mathfrak m_c\longrightarrow\mathbb Z/2\mathbb Z.$$

This identifies the possible reflection-positive invertible topological eleven-dimensional theories relevant to the ambiguity in the M-theory action. The statement is based on computations announced elsewhere and is presented as a conjecture in the source.