  For every $g\ge 2$ and $n\ge4$, we provide an $n-$manifold $M$ and a
continuous $2-$sided map $f\colon S\longrightarrow M$, where $S$ is a closed
genus $g$ surface, such that no simple loop is contained in
$\text{ker}(\,f_*\,)$. This provides a counterexample to the the classical
simple loop conjecture for surfaces to manifolds of dimensions at least four.
