2
$\begingroup$

I want to use the technique from hatcher section 3.2 to compute the cup product structure of a punctured torus (with $\mathbb{Z}$ coefficient), but I found that I still don't know how to do this when I have new spaces. How should I start?

I know that a punctured torus deformation retracts to wedge of circles $S^1 \vee S^1$, this gives us a CW complex structure, one 0-cell $e$ and two 1-cells $a,b$. Then I denote the map acting on $a,b$ as $\alpha, \beta$, and I want to see what $\alpha \cup \alpha$, $\alpha \cup \beta$ and $\beta \cup \beta$ should be. Any advice?

$\endgroup$

1 Answer 1

Reset to default
3
$\begingroup$

With no $2$-cells in $X=S^1\vee S^1$, then $H^2(X)=0$, so that $\alpha\smile\beta=0$ etc.

$\endgroup$
1
  • $\begingroup$ Is it possible to prove this by showing how $\alpha$ and $\beta$ act on $a$ and $b$? $\endgroup$
    – user658833
    Commented May 6, 2019 at 19:44

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .