  we show that the space of metrics of positive scalar curvature on a manifold
is, when nonempty, homotopy equivalent to a space of metrics of positive scalar
curvature that restrict to a fixed metric near a given submanifold of
codimension greater or equal than 3. Our main tool is a parameterized version
of the Gromov-Lawson construction, which was used to show that the existence of
a metric of positive scalar curvature on a manifold was invariant under
surgeries in codimension greater or equal than 3.
