The authors consider the problem \[ (-\Delta)^m u=|u|^{2^*-2}u\quad\text{in }\Omega,\qquad u\in D^{m,2}_0(\Omega), \] where \(2^*={2N\over N-2m}\) and either \(\Omega= \mathbb{R}^N\) or \(\Omega\) is a smooth bounded domain in \(\mathbb{R}^N\), \(N> 2m\). Here \(D^{m,2}_0(\Omega)\) is the closure of \(D(\Omega)\) with respect to the norm \[ \| u\|= \begin{cases} |\Delta^{m/2}u|_2 &\text{if }m\text{ is even},\\ |\nabla\Delta^{{m-1\over 2}} u|_2 &\text{if }m\text{ is odd}.\end{cases} \] \(|\cdot|_p\) denotes the \(L^p\)-norm. The authors prove a Sobolev inequality. They also show that the manifold of least energy solutions of the equation \((-\Delta)^m u=|u|^{{4m\over N-2m}} u\) is a nondegenerate critical manifold for the corresponding variational integral. Moreover, the authors generalize the results of J. M. Coron on the existence of solutions of equations with critical exponent on domains with nontrivial topology (reduced homology is not trivial) to the biharmonic operator.