1. ## Calculating Fundamental Groups

Calculate the fundamental groups of the following spaces:

i) $\displaystyle X_{1} = \{ (x, y, z) \in \mathbb{R}^3 | x>0 \}$

ii) $\displaystyle X_{2} = \{ (x, y, z) \in \mathbb{R}^3 | x \neq 0 \}$

iii) $\displaystyle X_{3} = \{ (x, y, z) \in \mathbb{R}^3 | (x, y, z) \neq (0, 0, 0) \}$

iv) $\displaystyle X_{4} = \mathbb{R}^3$\$\displaystyle \{ (x, y, z) | x=0, y=0, 0\leq z\leq1 \}$

v) $\displaystyle X_{5} = \mathbb{R}^3$\$\displaystyle \{ (x, y, z) | x=0, 0\leq y\leq 1 \}$

For i) $\displaystyle \pi_{1}(X_{1}) = \{1\}$ as $\displaystyle X_{1}$ is obviously simply connected so has trivial fundamental group.

ii) As $\displaystyle (X_{2})$ is the the disjoint union of two contractible spaces I think this makes $\displaystyle \pi_{1}(X_{2}) = \{1\}$ too?

iii) and iv) are both homotopically equivalent to $\displaystyle \mathbb{S}^2$ and therefore simply connected so does that make them $\displaystyle \{1\}$ also?

for v) I have no idea?

I'm just doubting myself as I find that they are just the trivial fundamental groups for the first four and would like some reassurance?

any help on v) much appreciated too.

2. ## Re: Calculating Fundamental Groups

You're right for 1-4. For #5, it is equivalent to a cylinder. So the fundamental group is Z.