TITLE: Fundamental group of the product of 3-tori minus the diagonal
QUESTION [14 upvotes]: I have a past qual question here: let $T^3 = S^1 \times S^1 \times S^1$ be the 3-torus, and let $\Delta = \{ (x,x) \in T^3 \times T^3 \colon x \in T^3 \}$ be the diagonal subspace. Compute $\pi_1(T^3 \times T^3 - \Delta)$.
If one wanted to understand the space $T^1 \times T^1 - \Delta$, one can realized it as $[0,1]^2$ (with opposite sides identified) minus the diagonal; then, one can shear along one of the two axes to get that $T^1 \times T^1 - \Delta$ is just a cylinder. Can this process be generalized to the higher-dimensional case?

REPLY [3 votes]: $T^3$ is a Lie group,so your space $T^3\times T^3-\Delta$ is a trivial bundle over $T^3$ with fiber $T^3-\{\ast\}$ (google"configuration space" for more details).So 
$$\pi_1(T^3\times T^3-\Delta)\cong \pi_1(T^3\times (T^3-\{\ast\}))\cong Z^3\times Z^3\cong Z^6$$