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I often see "null-homologous curves" in some topology article. I am informally (for me) understanding that "null- homologous curves" are $0$ in the first homology groups of the space. But, I do not know explicit definitions...

Let $\Sigma$ be an oriented surface and $c_1,c_2:S^1\hookrightarrow \Sigma$ be embeddings. I think the definition could be $$c_1\text{ and }c_2\text{are homologous}\iff c_{1\star}[S^1]=c_{2\star}[S^1]\text{ in }H_1(\Sigma) $$ .Is this definition right? If it so, how do we characterise homologous curves intuitively?

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    In integer homology, two curves are homologous if and only if there is a map from a compact oriented surface into $\Sigma$ such that when you restrict the map to the boundary, you get the two parametrized curves. For higher-dimensional homology groups "homologous" does not have quite so nice an interpretation in terms of manifolds, but for curves, this works. So this definition requires the surface that maps into $\Sigma$ to have precisely two path-connected boundary components. – 2017-01-03
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    Thanks so much! If you are OK, please tell me books or websites which explain homologous manifolds. – 2017-01-09
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    There are many. But if you really want this perspective, Glen Bredon's "Geometry and Topology" textbook explores this perspective quite thoroughly. For this result, all you really need to know is the map from the fundamental group to $H_1$ is equivalent to abelianization -- after that the result pops out of a little thought. – 2017-01-09
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    Thank you very much for your advise.I am going to read the book. – 2017-01-09

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