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So let $f:X\rightarrow Y$ be a continuous map. Is there any relation between $$f_{*}:H_{k}(X)\rightarrow H_{k}(Y)$$ and $$f^{*}:H^{k}(Y)\rightarrow H^{k}(X).$$

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They are in fact adjoint with each other if the homology groups are free. –  Walt Mar 26 '13 at 4:36
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