  We compute the vacuum fermion current in $(2+1)$ dimensional Jackiw-Rossi
model by using the $1/m$ expansion. The current is expressed through a weighted
$\eta$-function with a matrix weight. In the presence of such a weight, the
usual proof of topological nature of $\eta(0)$ is not longer applicable. Direct
computations confirm the following surprising result: the fermion number
induced by vortices in the Jackiw-Rossi model is \textit{not} topological.
