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if I have a map between two p-dimensional smooth manifolds M and N.

1-is it true that if $f_{*}$ is an isomorphism from $H_{p}(X)$ to $H_{p}(Y)$ Then

f induces an isomorphism from $H_{k}(X)$ to $H_{k}(X)$ for all $k \leq p$ ?


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Why should that be the case? This would mean in particular that all compact non-orientable $p$-manifolds have the same homology. For an explicit (counter-)example take the Klein bottle and $\mathbb{RP}^2$ and a constant map. By the way: do you mean $k \leq p$? (Added: I didn't downvote) –  t.b. May 29 '11 at 9:03
Many thanks, I was also looking for cases when my assertion is valid. –  El Moro May 29 '11 at 9:08
Yes Theo p I shall correct it –  El Moro May 29 '11 at 9:08
...or, say, $T^2=S^1\times S^1\to S^1\wedge S^1=S^2$. –  Grigory M May 29 '11 at 9:08
To theo, it does not necessarily mean that because having both $H_{n}(X)$ and $H_{n}(Y)$ isomorphic does not imply that $f_{*}$ induces an isomorphism. Could you please provide me with a counter example with a clear map f? Thanks –  El Moro May 29 '11 at 9:12
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