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My question is about what one could say about the betti number of both spaces X and Y relatively to each other if we have a map f between them (eg. 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: math.stackexchange.com/questions/41451/… –  Grigory M May 29 '11 at 8:52
    
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. –  El Moro May 29 '11 at 8:57
    
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. –  Grigory M May 29 '11 at 9:02
    
I am actually looking for those special cases (like the one you mentioned with a section) in which one could compare betti numbers. –  El Moro May 29 '11 at 9:03

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