Topology and Sobolev Spaces
نویسندگان
چکیده
with 1 ≤ p <∞. W (M,N) is equipped with the standard metric d(u, v) = ‖u− v‖W1,p . Our main concern is to determine whether or not W (M,N) is path-connected and if not what can be said about its path-connected components, i.e. its W -homotopy classes. We say that u and v are W -homotopic if there is a path u ∈ C([0, 1],W (M,N)) such that u = u and u = v. We denote by ∼p the corresponding equivalence relation. Let ∼ denote the equivalence relation on C(M,N), i.e. u ∼ v if there is a path u ∈ C([0, 1], C(M,N)) such that u = u and u = v. First an easy result
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