First cohomology group with integer coefficients

Definition

Let $X$ be a path-connected space). The first cohomology group with integer coefficients of $X$, denoted $H^1(X)$ or $H^1(X;\mathbb{Z})$ is defined in the following equivalent ways:

• It is the set of homotopy classes of based maps from $X$ to $S^1$
• It is the set of homotopy classes of unbased maps from $X$ to $S^1$
• It is the set of fiber bundles over $X$ with fiber equal to $\mathbb{Z}$ (in other words, it is the covering spaces (possible disconnected) whose fibers are the group $\mathbb{Z}$)