# Determing if the fundamental group of the following is isomorphic to either the trivial, infinite cyclic, figure eight fundamental groups

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

• What do you mean "with nonnegative axes"? Commented May 10, 2014 at 17:20
• @ThomasAndrews he means remove the sets $\{(x,0,0)|x\geq 0\}$, $\{(0,y,0)|y\geq 0\}$ and $\{(0,0,z)|z\geq 0\}$
– Seth
Commented May 10, 2014 at 17:46

2) $\mathbb{R}^3$ with the nonnegative axis removed deformation retracts onto a "3-pipe space"
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).