I am currently working through "Topology of Sobolev Spaces" by Brezis and Li, and they make the conjecture that for $M, N$ compact and smooth manifolds any function $u \in W^{1,p}(M,N)$ is homotopic to a smooth function. So the first theorems, however, analyze the path-connectedness of the Sobolev-Space $W^{1,p}(M,N)$.
My question now is: What do path-connectedness and the conjecture have to do with each other?
I was thinking that maybe it was, because if the Sobolev space was path-connected then every function in the space would be homotopic to the constant function. And the constant function in any function space is smooth, is that correct? Hence the title.
Kind regards.