  We prove that for cobordant closed spin manifolds of dimension $n\geq 3$ the
associated spaces of metrics with invertible Dirac operator are homotopy
equivalent. This is the spinorial counterpart of a similar result on positive
scalar curvature of Chernysh/Walsh and generalizes the surgery result of
Ammann-Dahl-Humbert on the existence of metrics with invertible Dirac operator
under surgery. We also give a relative statement of this homotopy equivalence.
