Fundamental group of a difference of topological spaces After days of pondering about this type of problem, I can't came up with nothing.
Let $A=\{(0,1)\}\times S^2$ and $B=S^1\times\{(0,0,1)\}$.
Define $X:=(S^1\times S^2)\setminus(A \cup B)$ and find $\pi_1(X)$.
In general, when dealing with this type of problems, I look for algebraic tricks. So, knowing that $$Y\setminus(Z\cup W)=(Y\setminus Z)\setminus W$$ I can observe that
$1)\,\, (S^1\times S^2)\setminus(A \cup B)=[(S^1\times S^2)\setminus A]\setminus B=[(S^1\setminus \{(0,1)\}) \times S^2]\setminus (S^1\times\{(0,0,1)\}),$
or alternatively,
$2)\,\,(S^1\times S^2)\setminus(A \cup B)=[(S^1\times S^2)\setminus B]\setminus A=[S^1 \times (S^2\setminus \{(0,0,1)\})]\setminus (\{(0,1)\}\times S^2).$
The idea I am following is to write the space as a product $U\times V$, so that knowing the fundamental groups of $U$ and $V$ I can compute $\pi_1(X)$. In this case, even knowing that $S^n\setminus\{N\}\approx \mathbb{R}^n$, I can't conclude.
Please note that I am searching for a general strategy to visualize topological spaces in higher dimensions.
 A: Looking for algebraic tricks without the visualization is unlikely to succeed. In particular, attempting to impose unmotivated algebraic structure on your space such as a product $U \times V$ is more than likely going to send you down a long, pointless wandering path, given that product structures are so special.
Instead, let's look at these two sets, and start simple:
$$A = \{(0,1)\} \times S^2 \qquad B = S^1 \times \{(0,0,1)\}
$$
Each of these is a subspace of $\mathbb R^2 \times \mathbb R^3$.
Let's introduce some coordinates $((x,y),(r,s,t))$ for a point in $\mathbb R^2 \times \mathbb R^3$ (and let's also resist rewriting this as $\mathbb R^5$, for reasons that will be clear).
The first thing I notice is that the set $A$ is a subset of $\{(0,1)\} \times \mathbb R^3$, i.e. it is parallel to $r,s,t$-coordinate 3-space. Also set $B$ is a subset of $\mathbb R^2 \times \{(0,0,1)\}$, i.e. it is parallel to the $x,y$-coordinate 2-space.
What is the intersection of these two coordinate subspaces?
Now I use some high dimensional visualization. In 2-dimensions, any line parallel to the $x$-axis, intersected with any line parallel to the $y$-axis, is a single point. This generalizes very easily to $\mathbb R^m \times \mathbb R^n$: for any $X \in \mathbb R^m$ and any $Y \in \mathbb R^n$ we have
$$(\{X\} \times \mathbb R^n) \cap (\mathbb R^m \times \{Y\}) = \{(X,Y)\} \subset \mathbb R^m \times \mathbb R^n
$$
In particular, in $\mathbb R^2 \times \mathbb R^3$, the subspace $\{(0,1)\} \times \mathbb R^3$ intersected with $\mathbb R^2 \times \{(0,0,1)\}$ is a single point, namely the point
$$P = ((0,1),(0,0,1))
$$
So, we've reached an interesting conclusion: the intersection $A \cap B$ is either the single point subset $\{P\}$ or the emptyset. But if we observe that $(0,1) \in S^1$ and that $(0,0,1) \in S^2$ then we reach the conclusion that
$$A \cap B = \{P\}
$$
From this, we can reach one more conclusion: $A \cup B$ is the wedge sum of $A$ and $B$.
And now we can apply a useful tool: by application of Van Kampen's Theorem,
$$\pi_1(A \cup B,P) = \pi_1(A,P) * \pi_1(B,P)
$$
In words, our group $\pi_1(A \cup B,P)$ is a free product of the two groups $\pi_1(A,P)$ and $\pi_1(B,P)$.
I presume you have the tools to finish this problem off, so I'll leave the pleasure of working out the final answer to you.
