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So I'm trying to use Van Kampen theorem to prove that a space is null-homotopic.

The thing is I got it down to this $\langle a\mid a=1\rangle$, however I'm confused what does this mean.

For calculating the the torus you get it down to this $\langle a,b\mid a^{-1}b^{-1}ab=1\rangle \cong \mathbb{Z} \times \mathbb{Z}$.

But, was thinking does the brackets mean $\langle 1\rangle \cong \mathbb{Z}$ . I'm confused as normally the $\langle$ , $\rangle$ brackets means the generating set.

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    $\langle a\;|\;a=1 \rangle$ is the group with one generator $a$, which satisfies $a = 1$, hence the trivial group. BTW, what does it mean for a space to be null-homotopic? Did you mean simply connected?2012-03-14
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    In the brackets we have generators G and relations R: . If there is no relation and one generator, then this group is isomorphic to $\mathbf{Z}$.2012-03-14

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