## 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).
[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)

### MSC:

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

Zbl 0722.00043