TITLE: Examples of $\pi_1 (X) = \mathbb{Z}$
QUESTION [7 upvotes]: I want to know some examples of topological spaces whose fundamental group is isomorphic to set of integers. 
First, of course i know $\mathbb{S}^1$, and its deformation retract, $\mathbb{R}^2 - \{ 0, 0\}$, $\mathbb{C}^1 - \{0 \}$, etc, are isomorphic to set of integers. 
And of course from Van kampen theorem, attaching simply connected spaces, or by wedge sum of simpily connected space for  $\mathbb{S}^1$, etc, we can produces $\mathbb{Z}$.   
Can you give me some other examples than that?

REPLY [3 votes]: Two examples of particular importance in symplectic topology are the Lie groups of unitary matrices $U(n)$ and symplectic matrices $Sp(2n, \mathbb{R})$. The symplectic matrices are the real $2n \times 2n$ matrices that satisfy $A^T J A = J$, where 
$$
J = \begin{pmatrix}
0 & -I_n \\
I_n & 0 
\end{pmatrix}
$$
is the standard complex structure on $\mathbb{R}^{2n}$.
In the case of $U(n)$, an isomorphism of $\pi_1 (U(n))$ with $\mathbb{Z}$ is induced by $\det : U(n) \to S^1$.
Another Lie group example is $GL(n,\mathbb{C})$. One can show via polar decomposition that it actually deformation retracts onto $U(n)$.