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