TITLE: Fixed-point-free map on a sphere minus a point
QUESTION [7 upvotes]: I want to know of a fixed-point-free map from a 2-sphere minus a point to itself. Please also mention why you thought of your example.
My example: a sphere minus the north pole is the complex plane via stereographic projection, and consider a non-trivial translation of the plane.

REPLY [2 votes]: Take a vector field on a 2-sphere which vanishes only at a single point, and a flow of the identity map will give a diffeomorphism with no fixed points in the complement of that single point.
Your example is the case of taking the constant vector field on the plane.  The inverse of the stereographic projection of this vector field to the sphere gives a vector field with a single index-2 isolated zero.
The Euler characteristic of a sphere is $2$, so I believe your example should be the only example up to homotopy.