Cup product structure of the n-torus

[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...

 

[pat_connell]

3) On page 212, Hatcher says that the subspace of H^n(T^n ; R) generated by the \alpha_i is isomorphic to H^n(Y;R). Why is this?Any help would be greatly appreciated! A:1) I believe you are right in saying that Hatcher is taking the cross product of a relative homology class $\alpha \in H^1(I,\partial I;R)$ and an absolute homology class $\beta \in H^n(Y;R)$. However, I don't think that this is technically correct. The cross product $H^a(X;R) \times H^b(Y;R) \to H^{a+b}(X \times Y;R)$ is only defined when $X$ and $Y$ are CW complexes (and in general only works for a specific pair of CW structures). In this case, $Y$ is a CW complex but $I$ is not. This is why Hatcher is careful to write that "the relative homology class $\alpha \in H^1(I,\partial I;R)$ looks like an element of $H^1(I \times Y;R)$".2) The connecting homomorphism $\delta : H^{n+1}(I \times Y, \partial I \times Y; R) \to H^n(Y;R)$ is only an isomorphism when restricted to the copy of $H^n(Y;R)$ corresponding to $\{0\} \times Y$. This is because $\delta$ is defined as the composition$$H^{n+1}(I \times Y, \partial I \times Y; R) \stackrel{\pi^*}{\longrightarrow} H^{n+1}(I \times Y;R) \stackrel{\partial}{\longrightarrow} H^n(Y;R)$$where $\pi : I \times Y \to Y$ is the projection map. Note that $\pi^*$ restricts to an isomorphism between the subspaces of $H^{n+1}(I \times Y, \partial I \times Y; R)$ corresponding to $\{0\}