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?