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# Universal Coefficient Theorem

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Suppose $X$ is a topological space and $A$ is an abelian group. There are short exact sequences for homology: $$0 \rightarrow H_i(X; \mathbf{Z})\otimes A\rightarrow H_i(X;A)\rightarrow\mbox{Tor}(H_{i-1}(X, \mathbf{Z});A)\rightarrow 0,$$ and cohomology: $$0 \rightarrow \mbox{Ext}(H_{i-1}(X, \mathbf{Z});A)\rightarrow H^i(X;A)\rightarrow\mbox{Hom}(H_i(X, \mathbf{Z});A)\rightarrow 0.$$ Both split but not naturally.

As an example, consider ${\bf P}^2$, the projective plane. The integer homology is given by: $$H_i({\bf P}^2; \mathbf{Z}) = \begin{cases} \mathbf{Z} & i = 0 \\ \mathbf{Z}_2 & i=1,\\ 0 & i \ge 2. \end{cases}$$ Since $Ext(\mathbf{Z}_2, \mathbf{Z}_2) = \mathbf{Z}_2, Ext(\mathbf{Z}, \mathbf{Z}_2)= 0$, the above exact sequences produce: $$H^i ({\bf P}^2; \mathbf{Z}_2) = \mathbf{Z}_2$$ for $i = 0,1,2$.