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I am trying to calculate De Rham cohomology of the following spaces:

  1. $X=\mathbb R^3\setminus r$ where $r$ is a line;
  2. $Y=\mathbb R^3\setminus (r \cup \gamma)$ where $r$ is a line and $\gamma$ is a circumference which concatenates the line;
  3. $Z=\mathbb R^3\setminus (r_1 \cup r_2 \cup \gamma )$ where $r_1,r_2$ are disjoint lines and $\gamma$ is a circumference which concatenates only $r_1$.

Well, about (1) I do not have many doubts: we observe $X$ has $\mathbb R^2 \setminus \{0\}$ as deformation retract hence - by homotopy invariance - we have that $H^{\bullet}(X) \cong H^{\bullet}(\mathbb S^1)$. Is it correct?

Now, what about $Y,Z$? I don't know how to start. Are there any "nice" deformation retracts? Perhaps, $Y$ retracts on $\mathbb R^2$ minus a point and a circumference, but how can I find the cohomology of this space?

The only thing I know are homotopy invariance, Mayer-Vietoris long exact sequence and Kunneth theorem.

Thanks in advance.

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