# Fundamental group of annulus with relation

I'm trying to find the fundamental group of space $X$ obtained from an annulus $\{p \in \mathbb R^2 : 1 \leq |p| \leq2 \}$ by identifying the point $(x,y)$ on the inner circle of radius $1$ with the point $(-2x,-2y)$ on the outer circle of radius $2$.

I think that we can use CW-complexes do that, but I'm not sure how to do it in this case. Moreover, I think I can apply van Kampen here too by taking the sets $A= X \smallsetminus \{\text{inner circle}\}$ and $B= X \smallsetminus \{\text{outer circle}\}$. Is that correct?

Thanks...

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Careful: in $X$ the inner circle is identified with the outer circle, so you'll want to rethink your choice of $A$ and $B$. – Adina G May 6 '12 at 16:29

## 1 Answer

The use of Van Kampen isn't quite correct - you can't remove the inner circle from $X$ without also removing the outer circle. Try Van Kampening with two half-annuli intersecting at the $y$-axis (or a thickened neighborhood of it). Note that the intersection is actually connected, due to the unusual gluing map.

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Can you please illustrate how to use cellular homology to find the homology of this space? I couldn`t do it because I have not seen any example how to use it. I found it by simplicial homology only. Please. – Danny May 7 '12 at 15:52