Hello there i am having trouble to determine isomorphisms of the following fundamental groups:
1) the torus $T$ with a removed point.
2) $\mathbb{R}^3$ with nonnegative axes
3) $S^1 \cup (\mathbb{R} \times 0 )$
i am using deformation retractions most of the time to find a solution for such problems. But for these 3 particular problems i can't seem to figure out the required retraction :(.
Are there any other theorems/definitions which can also determine isomorphisms of particular fundamental groups?
Kees Til
1) the Torus with a point removed deformation retracts onto the figure 8. One way to see this is to use the square with edges identified. Remove a point from the middle and then imagine stretched the point out to the edges. The edges are a wedge of two circles.
2) $\mathbb{R}^3$ with the nonnegative axis removed deformation retracts onto a "3-pipe space"
This space deformation retracts to a figure 8 by pushing the pipes in to the center and kind of twisting the top pipe onto the others.
3) $S^1 \cup (\mathbb{R}\times 0)$ deformation retracts to a circle with a line segment connecting two antipodal points. This is homotopic to the figure 8 by contracting the line segment to a point (or you can use van kampens to compute the fundamental group).