Let $X$ be a CW-complex of finite dimension and $F$ be a field. Do we have that $H^q(X;F)=H_q(X;F)$ for each $q\leq n$? I know that with filed coefficients the universal coefficient theorem simplifies so that $H^q(X;F)=Hom_F(H_q(X;F),F)$ but do we have an isomorphism between $H^q(X;F)$ and $H_q(X;F)$? If no, then under what conditions on $F$ or $X$ this would be true? thank you for your help!
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2$\begingroup$ Yes, if $X$ has finitely many cells in each dimension (the term "finite dimension" is a bit ambiguous). $\endgroup$– Justin YoungMar 16, 2014 at 9:32
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$\begingroup$ Ok thanks Justin!! $\endgroup$– palioMar 16, 2014 at 11:15
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See Dual space question for why there is no isomorphism of $H_i$ and $H^i$ in general. To make a simple example, take $X$ which is zero-dimensional of cardinality $d$. Then $H_0(X; F)\cong F^d$ while $H^0(X; F)\cong Hom(F^d, F)$. However, $F^d$ is not in general isomorphic to $Hom(F^d, F)$.
answered Mar 16, 2014 at 22:33