Possibility of the cellular decomposition of a manifold

Question

I am working to find if it's possible to find a cellular decomposition of $S^2\times S^1$ as following: $e^0\cup e^1\cup e_1^2\cup e_2^2\cup e^3$.

I cannot find such a decomposition. And I try to compute the cohomology group of it to get a contradiction, but it seems not to work.

Anyone has some ideas? Thank you.

Answer 1

You could do it with a very simple Euler characteristic argument, but if you wish, you can do it using cellular homology, since you know what the cellular chain complex (corresponding to this structure) looks like:

$$ C_3= \mathbb Z \stackrel 0 \to \mathbb Z \oplus \mathbb Z \stackrel \psi \to \mathbb Z \stackrel 0 \to \mathbb Z $$

know you know $im \psi =0$ and $\ker \psi \cong \mathbb Z$ which gives a contradiction

Answer 2

Use the Euler characteristic: $\chi(S^2 \times S^1) = \chi(S^2) \cdot \underbrace{\chi(S^1)}_{=0} = 0$, but if you had a decomposition like you said you would get a characteristic of $1-1+2-1 = 1 \neq 0$.