Can $S^1 \times Y$ be homeomorphic to $\mathbb{R}P^2$ or $S^2$?

Question

Does there exist a topological space $Y$ such that $S^1 \times Y$ is homeomorphic to $\mathbb{R}P^2$ or to $S^2$?

Answer 1

There is no group $G$ such that $\mathbb{Z} \times G$ is isomorphic to either $\mathbb{Z}/2\mathbb{Z}$ or the trivial group (for cardinality reasons). But $\pi_1(S^1 \times Y) \cong \mathbb{Z} \times \pi_1(Y)$ (for an arbitrary choice of base point), while $\pi_1(\mathbb{RP}^2) \cong \mathbb{Z}/2\mathbb{Z}$ and $\pi_1(S^2) = 0$.

Answer 2

Remember:

If $X,Y$ are two homeomorphic topological spaces , then $ \pi_1(X) \cong \pi_1(Y)$.

$\pi_1({S^1 \times Y})= \mathbb{Z} \times \pi_1(Y) \ncong \frac{\mathbb{Z}}{2\mathbb{Z}}=\pi_1(\mathbb{RP}_2) \Rightarrow S^1 \times Y$ isn't homeomorphic to $\mathbb{RP}_2$.

Try yourself the other, it's similar.