Decide the fundamental groups of following topological spaces

Question

Spaces:

• $\{(x, y, z, w): x^2 + y^2 \leq 1 , z^2 + w^2 = 1\}$ - solved,
• $\mathbb{S}^1\times\mathbb{S}^1$ - probably solved,
• $\{(x, y, z): x^2 + y^2 \leq 1, 0 \leq z \leq 1\}$ - maybe solved,
• $\{(x, y, z): x^2 + y^2 \leq 1\}$ - maybe solved,
• $\mathbb{R}^3$ without negative axis $x, y, z$,
• $\{p \in \mathbb{R}^2: ||p|| > 1\}$ - maybe solved,
• $\{p \in \mathbb{R}^2: ||p|| \geq 1\}$ - maybe solved,
• $\{p \in \mathbb{R}^2: ||p|| < 1\}$ - maybe solved,
• $\mathbb{S}^1 \cup (\mathbb{R}_+ \times \{0\})$ - solved,
• $\mathbb{S}^1 \cup (\mathbb{R}_+ \times \mathbb{R})$ - maybe solved,
• $\mathbb{S}^1\times\mathbb{S}^1$ minus one point.

What I know:

• all of them should be trivial, isomorphic to $(\mathbb{Z}, +)$ or isomorphic to fundamental group of figure eight
• $\mathbb{S}^1 \simeq (\mathbb{Z}, +)$
• $\pi_1(X \times Y, (x_0, y_0)) \simeq \pi_1(X, x_0) \times \pi_1(Y, y_0)$

What I think is true:

• homeomorphic spaces have same fundamental groups
• if $X \subseteq \mathbb{R}^n$ is convex then the fundamental group of $X$ is trivial
• fundamental group of deformation retract $A$ of $X$ is isomorphic to the fundamental group of $X$

My tries:

• 2nd one is isomoprhic to the fundamental group of 8, because by statement 2 combined with 3, we have $\mathbb{S}^1 \times \mathbb{S}^1 \simeq (\mathbb{Z}, +) \times (\mathbb{Z}, +)$, which is neither trivial nor isomoprhic to $(\mathbb{Z}, +)$
• 7th one's f. group is isomoprhic to $\mathbb{S}^1$'s, because of 3rd of my conjectures
• 8th one is convex, hence its group is trivial
• 6th one's f. group is isomoprhic to $\mathbb{S}^1$'s, because it contains space homeomorphic to $\mathbb{S}^1$ and I think we can use my third conjecture again
• 3rd + 4th: they both seem convex to me, hence their groups are trivial
• 10th: we could retract it to $\mathbb{S}^1$

edit1: I will check for duplicates after I return, really didn't have time now, just wanted to post. edit2: I found this Determing if the fundamental group of the following is isomorphic to either the trivial, infinite cyclic, figure eight fundamental groups, I guess I can understand his 3rd one (hence my 9th one), but not his 1st and 2nd (my 11th and 5th).

Answer 1

For the first space you can apply the product formula. The fundamental group of the fifth space is isomorphic to that of $\mathbb{S}^2$ with three points removed, which is then homeomorphic to the disc with two points removed, which is retracted to ficure eight. For the last one, $\mathbb{S} \times \mathbb{S}$ is actually torus. If you know that a torus can be realized from a square, then the sides are retract of $\mathbb{S} \times \mathbb{S}$ with one point removed. So the fundamendal group of the last space is isomorphic to that of figure eight.