confusion on a definition of Genus

Question

I was reading that the genus of a 1-holed Torus, $T$, is 1. But then I read another definition that for an object which is closed and symmetric about the origin its genus can be decided by the smallest value $n$ such that there exists an odd, continuous function between the object and $\mathbb{R^n}\setminus\{0\}$. As per this the genus has to be $3$ because if I choose a map say the identity function which is odd continuous then the genus has to be $3$. I am failing may to be to find a odd, continuous map between $T$ and $\mathbb{R^k}\setminus\{0\}$ for $k=1,2$. Can someone clear this?. I am confused.

Answer 1

These are two totally different and (as far as I know) unrelated notions of "genus". It's just the same word, used with two different meanings in different contexts.