Homotopy classes of differential maps VS those of continuous maps

2010-03-28T02:26:25Z

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: Thanks to the comments below, I should ask whether it is true when $M$ and $N$ are differentiable manifolds.

Answer by Ryan Budney for Homotopy classes of differential maps VS those of continuous maps

2010-03-28T05:53:15Z

There's no way this can be literally true:

$$[M,N]^{diff} = [M,N]^{cont}$$

Most of the continuous functions from $M$ to $N$ are not differentiable. So there's no way the above equality can be an equality of sets. I think what you want to ask is if the inclusion:

$$[M,N]^{diff} \to [M,N]^{cont}$$

a bijection? This is answered affimatively in Hirsch's "Differential Topology" textbook. It boils down to a smoothing argument, that every continuous function can be uniformly approximated by a $C^\infty$-smooth function and the smoothing is unique up to a small homotopy. The argument goes further, to state the the space of continuous functions has the same homotopy-type as the space of $C^\infty$ functions. The smoothing argument can be done with bump functions and partitions of unity, and also via a standard convolution with a bump function argument ("smoothing operators").

Homotopy classes of differential maps VS those of continuous maps - MathOverflow

Homotopy classes of differential maps VS those of continuous maps

Answer by Ryan Budney for Homotopy classes of differential maps VS those of continuous maps