Here's a puzzle I think I know the answer to, but I don't have a proof: Let K denote the Klein bottle, the ideal surface you get when you identify the top and bottom edges of the unit square [0,1] x [0,1] normally, by (x,0) ~ (x,1), but identify the left and right edges by a flip: (0,y) ~ (1,1-y). Puzzle: ----------------------------------------------------------------------- Consider the space Homeo(K) of self-homeomorphisms of the Klein bottle. I.e., Homeo(K) consists of all continuous bijections h : K —> K having a continuous inverse. Two self-homeomorphisms h_0, h_1 of K are *in the same path component* of Homeo(K) if there is a continuous *family* {h(t) | 0 <= t <= 1} of homeomorphisms h(t) in Homeo(K) such that h(e) = h_e for e = 0, 1. The continuity of this family just amounts to there being a continuous map H : K x [0,1] —> K such that the restriction of H to any time-slice K x {t}: H | K x {t} —> K is the homeomorphism h(t) : K —> K. QUESTION: ————————— How many path components does Homeo(K) have? ----------------------------------------------------------------------- —Dan