Calculating Fundamental Groups

**shmounal** · March 18th 2012, 01:05 PM

Calculate the fundamental groups of the following spaces:

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

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

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

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

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

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

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

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

**xxp9** · March 19th 2012, 07:41 AM

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

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