Positive scalar curvature and $10\mathrm{/}8$-type inequalities on $4$-manifolds with periodic ends

We show $10\mathrm{/}8$-type inequalities for some end-periodic $4$-manifolds which have positive scalar curvature metrics on the ends. As an application, we construct a new family of closed $4$-manifolds which do not admit positive scalar curvature metrics.