  We study diffeomorphisms $f$ with heterodimensional cycles, that is,
heteroclinic cycles associated to saddles $p$ and $q$ with different indices.
Such a cycle is called fragile if there is no diffeomorphism close to $f$ with
a robust cycle associated to hyperbolic sets containing the continuations of
$p$ and $q$. We construct a codimension one submanifold of $\Difr(\SS^2\times
\SS^1)$, $r\ge 1$, that consists of diffeomorphisms with fragile
heterodimensional cycles. Our construction holds for any manifold of dimension
$\ge 4$.
