Let \(M\) be a smooth compact, Riemannian \(n\)-dimensional manifold. One considers the following Sobolev spaces: \(w^{1,p}(M)\), the completion of \(C^ \infty(M)\) in the norm \(\| f\|_{1,p}= \Bigl(\int_ n (| f|^ b+ |\nabla f|^ p)dx\Bigr)^{1/p}\), and \(w^{1,p}(M, M)= \{f\in w^{1,b}(M, \mathbb{R}^ k)\mid f(x)\in M\) a.e. \(x\in M\}\), where \(w^{1,p}(M, \mathbb{R}^ k)= \{(f_ 1, \dots, f_ k)\mid f_ i\in w^{1,p}(M)\), \(i= 1,\dots, k\}\) and \(\mathbb{R}^ k\) is a Euclidean space that contains \(M\) as a submanifold. The space \(w^{1,p}(M, M)\) is independent of metrics on \(M\) and of embeddings of \(M\) in Euclidean spaces. The author proves that the Poincaré conjecture is equivalent to the following conjecture: ``A smooth compact, connected \(n\)-dimensional manifold \(M\) without boundary is homeomorphic with the sphere \(S^ n\) if and only if \(C^ \infty(M, M)\) is dense in \(w^{1,p}(M, M)\) for all \(1\leq p< \infty\).'' Moreover, the author notes that the above result follows from a very difficult theorem of \textit{F. Bethuel} [Acta Math. 167, No. 3/4, 153-206 (1991; Zbl 0756.46017)].