  We show that the periodic $\eta$-invariants introduced by
Mrowka--Ruberman--Saveliev~\cite{MRS3} provide obstructions to the existence of
cobordisms with positive scalar curvature metrics between manifolds of
dimensions $4$ and $6$. The proof combines a relative version of the
Schoen--Yau minimal surface technique with an end-periodic index theorem for
the Dirac operator. As a result, we show that the bordism groups
$\Omega^{spin,+}_{n+1}(S^1 \times BG)$ are infinite for any non-trivial group
$G$ which is the fundamental group of a spin spherical space form of dimension
$n=3$ or $5$.
