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On p. 221 of http://topology.auburn.edu/tp/reprints/v08/tp08113.pdf, I found the following definition:

"A curve is said to be cyclic if its first Čech cohomology group with integer coefficients does not vanish".

Here, a curve means a homogeneous metric continuum of dimension 1.

Can someone explain this definition in different, more elementary, terms, and give some examples to illustrate the meaning?

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Perhaps the first Čech cohomology group with integer coefficients may seem not elementary. Then use the characterization of it as the first Borsuk's cohomotopy group (or it appears as the Brushlinsky's group in the well-known text on Homotopy Theory by Hu). This group's elements are simply the homotopy classes of mappings into $\ \mathbb S^1.$ Space $\ S^1\ $ is a topological group, it induces the group structure on the homotopy classes.

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