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I would like to know if it is possible to use van Kampen's theorem without knowing exactly what the intersecting space looks like.

So given $U$ and $V$, and an attaching map, is it possible to work out the fundamental group of $X=U\cup_{f}V$ without knowing how $X$ looks like, i.e. how do you get the relations only with the attaching map?

Consider the example: Given 2 solid tori ($S^1\times D^2$) $U$ and $V$, and attaching map $f: S^1\times S^1\rightarrow S^1\times S^1$, $f(z,w)=(z,zw)$, such that $X=U\cup_{f}V$, find the fundamental group of $X$. Can I proceed without knowing what $X$ looks like? (Do let me know what it looks like anyway, thanks!)

BlackAdder
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  • I don't understand your example: in that case, you know that $U \cap V$ looks like the common boundary of $U$ and $V$, so a torus $S^1 \times S^1$, don't you? – PseudoNeo Apr 25 '13 at 16:57
  • Yes indeed it is, I think I asked the wrong question there. I would like to know what the whole space $X$ looks like. – BlackAdder Apr 25 '13 at 17:00
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    https://en.wikipedia.org/wiki/Lens_space – PseudoNeo Apr 25 '13 at 17:11

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The answer is Yes. Using your example where both $U$ and $V$ are both solid tori. van Kampen theorem tells you that $\pi_1(X)=\mathbb{Z}/<f(\alpha)>$ where $\alpha$ the meridian curve of the attached solid torus. In this case $X$ is a lens space.This example you can think of as performing Dehn filling on knot exterior of the unknot.

Hesky Cee
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