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In homology theory, I often see the symbols $H_*(X)$, or even $H_\bullet(X)$ to represent the singular homology.

Similarly, chain complexes can be denoted $(C_\bullet, d_\bullet)$, etc.

While I do get the meaning, I am curious why not use $i, j, k$ to index instead? Or even $\alpha,\beta$, etc? Is there any deeper reasoning to the special notation?

Thanks.

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    $(C_i,d_i)$ would refer to the single component $C_i$ and the map $d_i$. I guess people don't like to write $(C_i,d_i)_{i\in I}$. I don't think there is a deep reason behind it. – 2017-01-25

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When you see $C_{\bullet}$, that is referring to an entire chain complex as a single object. If you write $C_i$ then you are referring to just one element of the chain complex. Similarly for $H_*(X)$.

Also, for instance, if you write $H^*(X;R)$, then you are talking about the cohomology ring with coefficients in $R$, rather than just a single cohomology group.

It's also useful for instance, when talking about spectral sequences. If I write $E_t^{p,\bullet}$, then I am fixing the row $p$ and page $t$ and just considering the resulting complex where $q$ varies. So this notation is useful as a shorthand for when a single value is varying.

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    Makes a lot of sense. Thanks. – 2017-01-25