  We examine topological terms of $(2+1)$d sigma models and their consequences
in the light of classifications of invertible quantum field theories utilizing
bordism groups. In particular, we study the possible topological terms for the
$U(N)/U(1)^N$ flag-manifold sigma model in detail. We argue that the Hopf-like
term is absent, contrary to the expectation from a nontrivial homotopy group
$\pi_3(U(N)/U(1)^N)=\mathbb{Z}$, and thus skyrmions cannot become anyons with
arbitrary statistics. Instead, we find that there exist ${N(N-1)\over 2}-1$
types of Chern-Simons terms, some of which can turn skyrmions into fermions,
and we write down explicit forms of effective Lagrangians.
