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We determine those smooth $n$--dimensional closed manifolds with $n \geq 4$which admit round fold maps into ${\mathbb{R}}^{n-1}$, i.e.\ fold maps whosecritical value sets consist of disjoint spheres of dimension $n-2$ isotopic toconcentric spheres. We also classify such round fold maps up to $C^{\infty}$$\mathcal{A}$--equivalence.

Round fold maps of n--dimensional manifolds into ${\mathbb{R}}^{n-1}$