Let $M$ and $N$ be two smooth manifolds which we may assume are $\mathbb{R}^m$ and $\mathbb{R}^n$ respectively, with $n \leq m$. In fact, assume $M = \mathbb{R}^m - \{ 0 \}$ and $N = \mathbb{R}^n - \{ 0 \}$.
This is a very small part in a larger problem I am working on, but it is something that I am surprised I have never thought about.
I am wondering if there are canonical maps between $M $ and $N$ that are multiplicative. By this, I mean that I am looking for a generalisation of the map $\mathbb{R}^n - \{ 0 \} \to \mathbb{R} - \{ 0 \}$ given by $$(x_1, ..., x_n) \mapsto x_1 \cdots x_n.$$ I am also happy with maps of this form: $$(x_1, ..., x_n) \mapsto \left( \frac{x_1}{x_n}, ..., \frac{x_{n-1}}{x_n} \right).$$ But other than these simple examples from $\dim(M) \to \dim(M)-1$ or $\dim(M) \to 1$, I cannot construct a map that does not require a choice of ordering.
The following concrete problem should suffice:
Concrete question: Construct a surjective map $\mathbb{R}^4 - \{ 0 \} \longrightarrow \mathbb{R}^2 - \{ 0 \}$ that does not depend on the ordering of the coordinate functions on $\mathbb{R}^4 - \{ 0 \}$.
Note that this is possibly a trivial problem.
Thanks in advance, and please let me know if the question needs further clarification.