# Cup product structure of the n-torus

1. Nov 17, 2008

### ehrenfest

I have some questions about an example in Hatcher's "Algebraic Topology". This book is freely available online at http://www.math.cornell.edu/~hatcher/AT/ATchapters.html and I have attached the relevant part. My questions are about Example 3.11 where Hatcher computes the cup product structure of the n-torus.

1) In the second paragraph of this example, Hatcher seems to take the cross product of a relative homology class $\alpha$ and an absolute homology class $\beta$. That is, he seems to use

$$H^1(I,\partial I;R) \times H^n(Y;R) \to H^{n+1}(I \times Y, \partial I \times Y; R)$$

But above this example he only defines cross products between absolute homology groups or between relative homology groups. So, what does $\alpha \cup \beta$ mean?

2) I don't understand at all why Hatcher says at the top of page 211 that $\delta$ is an isomorphism when restricted to the copy of $H^n(Y;R)$ corresponding to ${0} \times Y$. I thought that $\delta$, the connecting homomorphism, was a rather complicated object and I don't see why that doesn't require justification...

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Last edited: Nov 17, 2008