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 $\mathbb {Z}$)