Abstract
The effective gauge field action due to an odd number of fermion species in three-dimensional $\mathrm{SU}(N)$ gauge theories is shown to change by $\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}|n|$ under a homotopically nontrivial gauge transformation with winding number $n$. Gauge invariance can be restored by use of Pauli-Villars regularization, which, however, introduces parity nonconservation in the form of a parity-nonconserving, topological term in the effective action.
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