View as the unit sphere in . Now,
1. What are the path connected components of the subset of described by the equation , where the x and y refer to the coordinates (in )?
2. Is it true that the similar subset is homeomorphic to the circle?
3. what is the fundamental group of , where K is the subset in the 2nd part of the problem?
thnx
Let's start with one that seems easiest, whether the curve defined by is homeomorphic to . I have literally put no real work into this, and so please do not think I am dropping hints because I know something, but my intuition (God knows how useful that is) says they are not. is a -manifold, is that subspace?