An \(\alpha\)-approximation theorem for \(R^{\infty}\)-manifolds (Q1100773)

The main result of this paper is the \(\alpha\)-approximation theorem for \(R^{\infty}\)-manifolds formulated in the classical form. Main theorem. Let N be an \(R^{\infty}\)-manifold and \(\alpha\) an open cover of N. There is an open cover \(\beta\) of N such that if M is an \(R^{\infty}\)- manifold and \(f: M\to N\) is a \(\beta\)-equivalence, then f is \(\alpha\)- close to a homeomorphism. In the process of proving the main theorem results analogous to those in Q and \(\ell_ 2\)-manifold theory are formulated and proved. Among them there are analogues of the unknotting theorem and Collar theorem. The analogy for these theorems is not direct and moreover it is known that it couldn't be direct.