The moduli space of smooth fiberings is shown to be a Fréchet manifold, and its homotopy type is computed for circle and torus fiberings on manifolds of dimension at most three.
Kihara, Smooth homotopy of infinite-dimensional 𝐶∞-manifolds, Mem
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On Topology of the Infinite-Dimensional Space of Fibrations
The moduli space of smooth fiberings is shown to be a Fréchet manifold, and its homotopy type is computed for circle and torus fiberings on manifolds of dimension at most three.