# comparing Betti numbers

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?

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