First cohomology group with integer coefficients

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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})