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Consider this context from wikipedia:

It is known that a chain map $f_\bullet$ between chain complexes $(A_\bullet, d_{A,\bullet})$ and $(B_\bullet, d_{B,\bullet})$ induces a homomorphism $$(f_\bullet)_*: H_\bullet(A_\bullet)\to H_\bullet(B_\bullet).$$

(https://en.wikipedia.org/wiki/Chain_complex#Chain_maps)

I am confused about the notation $(f_\bullet)_*: H_\bullet(A_\bullet)\to H_\bullet(B_\bullet).$

Does it mean: $(f_n)_*: H_n(A_\bullet)\to H_n(B_\bullet)$, for each $n$? (in this case the bullets are different).

If we require the bullets to be the same, then $(f_n)_*: H_n(A_n)\to H_n(B_n)$ does not seem to make sense as Homology is defined for a chain complex, not for a single group in the chain?

In general, for this kind of notation, do we need $\bullet$ to be the same thing in the same equation?

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    Yes, the bullets are place holders for different variables. They essentially just mean 'insert any set of indices for which this expression makes sense. – 2017-01-25
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    I think this is mostly a case of a reader becoming confused by bad notation. As you point out, homology is defined for a complex, not a single group. Hence $H_n(A_n)$ is meaningless in this context. Better would be $(f_n)_*:H_n(A)\to H_n(B)$. – 2017-01-25
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    I should add though, you may sometimes see expressions like $H^\bullet = H_{n-\bullet}$ (for instance, when you get to Poincare duality), where the two bullets are place holders for the same variable. In general, the context is enough to work out the meaning. – 2017-01-25

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