For $C^r$ differentiable manifolds $E$ and $B$ of the same dimension, and $\pi \colon E \to B$ a local $C^r$ diffeomorphism and onto, we examine, as with the topological case, the extent to which some basic properties of $\pi$ are inherited by the induced map $p$ on spaces of fiber transferring maps with respect to $\pi$.
"Induced Fibrations on Spaces of Fiber Transferring Maps: The $C^r$ Case." Missouri J. Math. Sci. 13 (1) 24 - 28, Winter 2001. https://doi.org/10.35834/2001/1301024