Let . Show that iff extends to the unit disc
Am I right in thinking that the fundamental group of this is trivial, so the identity map is homotopic to any "loop" ? Any hints are greatly appreciated.
Denote an element of by . If then there is a homotopy such that (a fixed point in X), and . Then the map is an extension of f to the unit disc.
That construction is essentially reversible. Given an extension of f to the disc, you can use it to construct a homotopy from f to a constant map.