Approximation by homeomorphisms and solution of P. Blass problem on pseudo-isotopy

Abstract:

For every map of $f:{S^1} \to {S^1} = \{ z \in C:|z| = 1\}$ of degree 1, existence of apseudo-isotopy $h:{S^1} \times I \to R = \{ z \in C:|z| \geqq 1\}$ such that $h(z,0) = z$ and $h(z,1) = f(z)$ is established. On the other hand (i) maps of ${I^n}$ into ${I^n} \times 0 \subset {E^{n + 1}}$ cannot be, in general, uniformly approximated by homeomorphic embeddings of ${I^n}$ in ${E^{n + 1}}$ for $n > 1$, and (ii) maps of ${S^n}$ into ${S^n} \subset {E^n}$ of degree 1 cannot be, in general, extended to a pseudo-isotopy of ${S^n}$ into ${E^{n + 1}}$.