Homology of $\mathbb{R}^3 - S^1$ I've been looking for a space on the internet for which I cannot write down the homology groups off the top of my head so I came across this:
Compute the homology of $X : = \mathbb{R}^3 - S^1$.
I thought that if I stretch $S^1$ to make it very large then it looks like a line, so $\mathbb{R}^3 - S^1 \simeq (\mathbb{R}^2 - (0,0)) \times \mathbb{R}$. Then squishing down this space and retracting it a bit will make it look like a circle, so $(\mathbb{R}^2 - (0,0)) \times \mathbb{R} \simeq S^1$. Then I compute 
$ H_0(X) = \mathbb{Z}$
$ H_1( X) = \mathbb{Z}$
$ H_n(X) = 0 (n > 1)$
Now I suspect something is wrong here because if you follow the link you will see that the OP computes $H_2(X,A) = \mathbb{Z}$. I'm not sure why he computes the relative homologies but if the space is "nice" then the relative homologies should be the same as the absolute ones, so I guess my reasoning above is flawed.
Maybe someone can point out to me what and then also explain to me when $H(X,A) = H(X)$. Thanks for your help!
Edit $\simeq$ here means homotopy equivalent.

Consider $X = \mathbb{R}^3 \setminus (S^1 \times \{ 0 \}) \subseteq \mathbb{R}^3$, $U = X \setminus z \text{-axis}$ and $V = B(0,1/2) \times \mathbb{R}$, where $B(0, 1/2)$ is the open ball with radius $1/2$ and center in the origin of $\mathbb{R}^2$. It is clear that $\{ U, V \}$ is an open cover of $X$.
Now let's compute the homotopy type of $U$. Consider a deformation $f$ of $((0, +\infty) \times \mathbb{R}) \setminus \{(1,0) \}$ onto the circle of center $(1,0)$ and radius $1/2$. Let revolve $f$ around the $z$-axis: obtain a deformation of $U$ onto the doughnut ($2$-torus) of radii $R=1$ and $r = 1/2$.
Since $V$ is contractible and $U \cap V$ is homotopically equivalent to $S^1$, Mayer-Vietoris sequence gives the solution: $H_0(X) = H_1(X) = H_2(X) = \mathbb{Z}$ and $H_i(X) = 0$ for $i \geq 3$.
