  We show that for compact Riemannian manifolds of dimension at least $3$ with
nonempty boundary, we can modify the manifold by performing surgeries of
codimension $2$ or higher, while keeping the Steklov spectrum nearly unchanged.
This shows that certain changes in the topology of a domain do not have an
effect when considering shape optimization questions for Steklov eigenvalues in
dimensions $3$ and higher. Our result generalizes the 1-dimensional surgery in
[FS2] to higher dimensional surgeries and to higher eigenvalues. It is proved
in [FS2] that the unit ball does not maximize the first nonzero normalized
Steklov eigenvalue among contractible domains in $\mathbb{R}^n$, for $n \geq
3$. We show that this is also true for higher Steklov eigenvalues. Using
similar ideas we show that in $\mathbb{R}^n$, for $n\geq 3$, the $j$-th
normalized Steklov eigenvalue is not maximized in the limit by a sequence of
contractible domains degenerating to the disjoint union of $j$ unit balls, in
contrast to the case in dimension $2$ [GP1].
