Strong density for higher order Sobolev spaces into compact manifolds

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Given a compact manifold Nⁿ, an integer k ∈ N* and an exponent 1 ≤ p < ∞, we prove that the class C∞(Qᵐ; Nⁿ) of smooth maps on the cube with values into Nⁿ is dense with respect to the strong topology in the Sobolev space Wᵏ, ᵖ(Qᵐ; Nⁿ) when the homotopy group π kp (Nⁿ) of order kp is trivial. We also prove the density of maps that are smooth except for a set of dimension m - kp - 1, without any restriction on the homotopy group of Nⁿ