A cohomological criterion for density of smooth maps in Sobolev spaces between two manifolds. (English) Zbl 0735.46017

Nematics. Defects, singularities and patterns in nematic liquid crystals: mathematical and physical aspects, Proc. NATO Adv. Res. Workshop, Orsay/Fr. 1990, NATO ASI Ser., Ser. C 332, 15-23 (1991).

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[For the entire collection see Zbl 0722.00043.]
In [R. Schoen and K. Uhlenbeck, Approximation theorems for Sobolev mappings, preprint (1986)] it was shown that for compact Riemannian manifolds \(M\) and \(N\) the smooth mappings are dense in the Sobolev space \(W^{1,p}(M,N)\) for \(p\geq\dim M\). For the 3-ball \(M\) and the 2-sphere \(N\) this is not true for \(2\leq p<3\). Now in this paper it is shown, that in the case \(1\leq p<\dim M\) this is also true, provided \(N\) is \(([p]-1)\)-connected, the Hurewicz homomorphism from the \([p]\)-th homotopy group \(\Pi_{[p]}(N)\) to the \([p]\)-th homology group with rational coefficients \(H_{[p]}(N,\mathbb{Q})\) is an isomorphism and for all closed \([p]\)-forms \(\omega\) on \(N\) the pullback \(f^*\omega\) along \(f\) is closed. Here \([p]\) denotes the integer part of \(p\).
Reviewer: A.Kriegl (Wien)


46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
57R12 Smooth approximations in differential topology


Zbl 0722.00043