  We introduce the notion of a \emph{conic sequence} of a convex polytope. It
is a way of building up a polytope starting from a vertex and attaching faces
one by one with certain regulations. We apply this to a toric variety to obtain
an iterated cofibration structure on it. This allows us to prove several
vanishing results in the rational cohomology of a toric variety and to
calculate Poincar\'e polynomials for a large class of singular toric varieties.
