Van Kampen theorem? 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.
 A: More in general with Van Kampen theorem one have to deal with notation of the following kind
$$\langle g_1,\dots,g_s \mid m_1(g_1,\dots,g_s) = n_1(g_1,\dots,g_s);\dots; m_k(g_1,\dots,g_s) = n_k(g_1,\dots,g_s) \rangle$$
where $m_i(g_1,\dots,g_s)$ and $n_j(g_1,\dots,g_s)$ are strings containing as symbols $g_i$ or $g_i^{-1}$.
This notation is called a finite presentation of a group:
it simply denote the biggest group having $s$ generators, named $g_1,\dots,g_s$ for which the relations
$$m_i(g_1,\dots,g_s) = n_i(g_1,\dots,g_s)$$
hold.
More formally  the notation above denote the quotient group of the free group with $s$-generators by the smaller normal subgroup containing the elements of the form
$$m_i(g_1,\dots,  g_s) \left(n_i(g_1,\dots,g_s)\right)^{-1}\ \text{.}$$
In this group the equalities $m_i=n_i$ hold and every other group for which $m_i=n_i$ is a quotient of this group, thus the attribute bigger.
Some additional notes: in your examples $\langle a \mid a = 1 \rangle$ means the group with one generator with is equal to the identity, the group with this property is the trivial group $\{1\}$ that contain just the identity (so this proves that you space is indeed simply connected. In the case of the torus $\langle a,b \mid aba^{-1}b^{-1} = 1\rangle$ is also the group $\langle a,b \mid ab=ba\rangle$, this group is the biggest group having two generators which commute, i.e. it is the free abelian group with two generators. 
A: The notation $\langle a|a=1\rangle$ would usually mean the group generated by $a$ subject to the relation $a=1$, so in fact it is the trivial group and your space is null-homotopic as you require. What the notation is really saying is that it's the quotient of the free group on $a$ by itself, which is clearly trivial.
