Computing the cohomology $H^1(T,\mathbb{Z})$

Question

Let $T$ be a maximal torus of a compact Lie Group. Then how can we compute the first cohomology $H^1(T,\mathbb{Z})$?

Answer 1

The maximal torus is homeomorphic to $S^1\times\cdots\times S^1$. One way to compute the first cohomology is to use the fact that $H^1(X;\mathbb Z)\cong \operatorname{Hom}(\pi_1(X),\mathbb Z)$. The fundamental group of $T$ is $\mathbb Z^k$, where $k$ is the dimension of the torus, so $H^1(T;\mathbb Z)\cong \mathbb Z^k$ as well.