embed a torus in $\mathbb{R}P^3$

Question

How can I embed a torus in $\mathbb{R}P^3$? I know how to embed a 4 dimensional torus ($S^1 \times S^1 \times S^1$) in $\mathbb{R}^4$ But how to embed a 4 dimensional torus in $\mathbb{R}^4/\sim$, where $x\sim \lambda x$ for all $\lambda \in \mathbb{R}$? Is this possible for every 4 dimensional torus (or does the torus has to have another dimension, which?) and if not: Which characteristics does the torus has to have?

Answer 1

There is no embedding $f:S^1\times S^1\times S^1\to\mathbb RP^3$. By the invariance of domain for manifolds, any such $f$ would be an open map, so the image would have to be open. Since $S^1\times S^1\times S^1$ is compact, the image would also have to be closed. Since $\mathbb RP^3$ is connected, this would mean that $f$ is surjective. But a bijective open map is a homeomorphism, which is impossible because $S^1\times S^1\times S^1$ and $\mathbb RP^3$ are not homeomorphic.

You can however embed $S^1\times S^1$ into $\mathbb RP^3$. Any point in $\mathbb RP^3$ has a neighborhood $U$, homeomorphic to $\mathbb R^3$. Compose such a homeomorphism with an embedding of $S^1\times S^1$ into $\mathbb R^3$ and you're done.