Wednesday, 14 October 2009

dg.differential geometry - Homotopy classes of differential maps VS those of continuous maps

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]diffto[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 Cinfty-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 Cinfty 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").

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