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Path and quasi-homotopy for Sobolev maps between manifolds. (English) Zbl 1462.58005

Let \(p>1\) and \(W^{1,p}(M^n,N^n)\) be the Sobolev space between \(M^n\) and \(N^n\), two \(n\)-dimensional compact Riemannian manifolds. We say that \((u,v)\in W^{1,p}(M^n,N^n)\) are path homotopic (resp. \(p\)-quasihomotopic) if there exists a continuous path \(h\in C([0,1];W^{1,p}(M^n,N^n))\) joining \(u\) and \(v\) (resp. if they are homotopic outside sets of arbitrarily small \(p\)-capacity). Therefore the author states that if \((u,v)\in W^{1,n}(M^n,N^n)\) \(\big(\)resp. \(W^{1,p}(M^n,N^n)\) \(\big)\) are path homotopic (resp. \(p\)-quasihomotopic), then they are \(n\)-quasihomotopic (resp. path homotopic) (Theorems 1.1 and 1.2). Also the author states that \((u,v)\in W^{1,p}(M^n,N^n)\) are path homotopic if and only if they are \(p\)-quasihomotopic whenever \(N^n\) is furthermore an aspherical manifold, i.e., their higher homotopy groups vanish (Theorem 1.3).

MSC:

58B05 Homotopy and topological questions for infinite-dimensional manifolds
55P99 Homotopy theory
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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