Homotopy classes of differential maps VS those of continuous maps - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T05:51:59Z http://mathoverflow.net/feeds/question/19571 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/19571/homotopy-classes-of-differential-maps-vs-those-of-continuous-maps Homotopy classes of differential maps VS those of continuous maps Fei YE 2010-03-28T02:26:25Z 2010-03-28T17:12:08Z <p>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.</p> <p><strong>Edit:</strong> Thanks to the comments below, I should ask whether it is true when $M$ and $N$ are differentiable manifolds. </p> http://mathoverflow.net/questions/19571/homotopy-classes-of-differential-maps-vs-those-of-continuous-maps/19592#19592 Answer by Ryan Budney for Homotopy classes of differential maps VS those of continuous maps Ryan Budney 2010-03-28T05:53:15Z 2010-03-28T05:53:15Z <p>There's no way this can be literally true:</p> <p>$$[M,N]^{diff} = [M,N]^{cont}$$</p> <p>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:</p> <p>$$[M,N]^{diff} \to [M,N]^{cont}$$</p> <p>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"). </p>