Consider the space \(\text{Riem}(M)\) of Riemannian structures on \(M\) and the function \(f\): \(\text{Riem}(M)\to\mathbb{R}\) given by the injectivity radius. The author shows that the superlevel sets \(f\geq \varepsilon\) are disconnected for \(\varepsilon\) small enough, when \(n=\dim M\geq 5\). Informally speaking, manifolds of dimension \(\geq 5\) admit pairs of Riemannian structures of unit volume and small injectivity radius such that any path joining them in the space of Riemannian structures must “uncontrollably” diminish the injectivity radius of some intermediate structure. The term “uncontrollably” is made precise in terms of Turing computable functions. The proof uses an invariant \(F_M(\varepsilon)\) defined in terms of the filling length of curves, a Riemannian invariant introduced by M. Gromov. The idea is to apply S. Novikov’s theorem on the nonexistence of an algorithm deciding whether or not a given \(n\)-dimensional manifold is homeomorphic to \(S^n\) \((n\geq 5)\). Computing an upper bound for \(F_M(\varepsilon)\) as \(\varepsilon\to 0\) would make possible an explicit construction of an algorithm recognizing whether or not a given homology sphere is simply connected.