  We consider massless Majorana fermion systems with $G=\mathbb{Z}_N$, $SO(N)$,
and $O(N)$ symmetry in one-dimensional spacetime. In these theories, phase
ambiguities of the partition functions are given as the exponential of the
$\eta$-invariant of the Dirac operators in two dimensions, which is a bordism
invariant. We construct sufficient numbers of bordism invariants to detect all
bordism classes. Then, we classify global anomalies by calculating the
$\eta$-invariant of these bordism classes.
