Thread: Fundamental Group of Plane Minus Two Points

Need help with the following:

Calculate the fundamental group of the plane minus the points (0,1) and (0,-1).

Thanks in advance!

Originally Posted by DJDorianGray

Need help with the following:

Calculate the fundamental group of the plane minus the points (0,1) and (0,-1).

Thanks in advance!

An idea: draw two circles with radius 1 and center $\displaystyle (0,-1),(0,1)$ resp. around the two given points, and now "shrink" the exterior of both circles to the point (0,0) and ALSO the interior of both circles shrink to the same point (0,0) ==> you get the plane minus those two points is homotopically equivalent to a bouquet of two circles (i.e., the figure $\displaystyle \infty$), and it is well known, and as far as I remember not hard at all, to prove that this bouquet's fund. group is the free group in 2 generators.

Tonio