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I have begun to read in Hatcher's book "Algebraic topology", about cohomology. In doing so, I have tried to solve some problems. I have difficulties with problem 3.3.25: Show that if a closed orientable manifold $M$ of dimension $2k$ has $H_{k-1}(M,\mathbb{Z})$ torsion free, then $H_{k}(M,\mathbb{Z})$ is torsion free.
I have no idea. I think that I have to use somehow the Poincare duality, but I don't know how? Can somebody tell me how this works or at least give me a good hint? Thanks in advance.

mika

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    most exercises like this will follow from poincare duality and the universal coefficient theorem. PD says $H^k = H_{2k - k} = H_{k}$. The UCT says the torsion in $H^k$ is the torsion in $H_{k-1}$.2012-05-26
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    how does it follow that torsion in $H^{k}$ is torsion in $H_{k-1}$ from the UTC?2012-05-26

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