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# Homotopy classes of differentaildifferential maps VS those of continuous maps

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Let $M$ and $N$ be two topological manifolds. Denote by $[M,N]^{\text{diff}}$ and $[M,N]^{\text{cont}}$ the set of homotopy classes of differential and continuous maps respectively. Is it true that $[M,N]^{\text{diff}}=[M,N]^{\text{cont}}$ ? Any reference? Thank you.

Edit: The reason that I am interested in this result is Thanks to understand the classification of differential fiber bundle. Consider the differential fiber bundles with fiber $F$, base $B$ and structure group $\text{Diff}^+(F)$. Is comments below, I should ask whether it true that the isomorphism classes of such differential fiber bundles form the set $[B, \text{BDiff}^+(F)]^{cont}$ ? Let me make the notations clear. $\text{Diff}^+(F)$ is the orientation preserved diffeomorphism group of true when $F$. M$and$\text{BDiff}^+(F)$is the classifying space. Isomorphism of two differential bundle means there is a diffeomorphism between the total spaces which commutes with the bundle mapsN$ are differentiable manifolds.

2 Question explained

Let $M$ and $N$ be two topological manifolds. Denote by $[M,N]^{\text{diff}}$ and $[M,N]^{\text{cont}}$ the set of homotopy classes of differential and continuous maps respectively. Is it true that $[M,N]^{\text{diff}}=[M,N]^{\text{cont}}$ ? Any reference? Thank you.

Edit: The reason that I am interested in this result is to understand the classification of differential fiber bundle. Consider the differential fiber bundles with fiber $F$, base $B$ and structure group $\text{Diff}^+(F)$. Is it true that the isomorphism classes of such differential fiber bundles form the set $[B, \text{BDiff}^+(F)]^{cont}$ ? Let me make the notations clear. $\text{Diff}^+(F)$ is the orientation preserved diffeomorphism group of $F$. $\text{BDiff}^+(F)$ is the classifying space. Isomorphism of two differential bundle means there is a diffeomorphism between the total spaces which commutes with the bundle maps.

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