  Consider the space $W^{2,2}(\Omega;N)$ of second order Sobolev mappings $\ v\
$ from a smooth domain $\Omega\subset\R^m$ to a compact Riemannian manifold $N$
whose Hessian energy $\int_\Omega |\nabla^2 v|^2\, dx$ is finite. Here we are
interested in relations between the topology of $N$ and the $W^{2,2}$ strong or
weak approximability of a $W^{2,2}$ map by a sequence of smooth maps from
$\Omega$ to $N$. We treat in detail $W^{2,2}(\B^5,S^3)$ where we establish the
\underline{sequential weak} $W^{2,2}$ density of
$W^{2,2}(\B^5,S^3)\cap{\mathcal C}^\infty$. The strong $W^{2,2}$
approximability of higher order Sobolev maps has been studied in the recent
preprint \cite{BPV} of P. Bousquet, A. Ponce, and J. Van Schaftigen. For an
individual map $v\in W^{2,2}(\B^5,S^3)$, we define a number $L(v)$ which is
approximately the total length required to connect the isolated singularities
of a strong approximation $u$ of $v$ either to each other or to $\p\B^5$. Then
$L(v)=0$ if and only if $v$ admits $W^{2,2}$ strongly approximable by smooth
maps. Our critical result, obtained by constructing specific curves connecting
the singularities of $u$, is the bound $\ L(u)\leq c\int_{\B^5}|\nabla^2 u|^2\,
dx\ $. This allows us to construct, for the given Sobolev map $v\in
W^{2,2}(\B^5,S^3)$, the desired $W^{2,2}$ weakly approximating sequence of
smooth maps. To find suitable connecting curves for $u$, one uses the twisting
of a $u$ pull-back normal framing of a suitable level surface of $u$
