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In Hatcher's Algebraic Topology, section 3.2, during the computation of the cohomology ring of a $n$-torus, the following assertion is made. Let $Y$ be a space and $R$ a commutative ring. Then the natural map in cohomology

$$ H^n(I \times Y, R) \to H^n( \partial I \times Y, R)$$

is a split injection. This is likely straightforward, and I believe it follows directly from the Kunneth theorem. However, I don't see a direct proof, which is probably what Hatcher is indicated. Can someone explain to me what I'm missing?

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up vote 5 down vote accepted

Isn't $I$ just the closed unit interval? Then $I\times Y$ is homotopy equivalent to $Y$ and $\partial I\times Y$ is just two copies of $Y$. It's clear that your map then is just the diagonal map from $H^n(Y)$ to $H^n(Y)\oplus H^n(Y)$.

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Thanks! (That was not a very good question then; apologies.) – Akhil Mathew Nov 6 '10 at 18:20

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