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My question is about what one could say about the Betti number of both spaces $X$ and $Y$ relative to one another if we have a map $f$ between them (e.g., a classical case is when $f$ is a covering map). Is there an inequality if $f$ happens to be injective or surjective?

Thank you in advance.

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    related: http://math.stackexchange.com/questions/41451/about-maps-between-homology-groups2011-05-29
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    Thanks Grigory, the answer was interesting it says that if the map is injective with a section then it induces an injective map of homology modules which means that the betti numbers of X are less or equal to those of Y. I am wondering if there is an answer to my question in a more general setting.2011-05-29
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    The morale from the linked question: injectivity/surjectivity of $f$ implies no inequality for Betty numbers whatsoever. So it's not quite clear what kind of answer you expect.2011-05-29
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    I am actually looking for those special cases (like the one you mentioned with a section) in which one could compare betti numbers.2011-05-29

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