  In this paper the singular hypersurfaces in $\mathbb{C}\mathrm{P}^4$ of
degree $d$ with an isolated singularity are studied. If the singularity is of
type $A_{2k+1}$, under the condition $d<(k+5)/2$, a classification of such
hypersurfaces upto homeomorphism (which is diffeomorphism on the nonsingular
part) is obtained. As an application, cubic hypersurfaces with an
$A_5$-singularity are constructed.
