Degree of a factor swap
= Degree of a factor swap
{title2=$\deg((x,y)\mapsto(y,x))=(-1)^{mn}$}
The factor swap $\mathbb R^m\times\mathbb R^n\to\mathbb R^n\times\mathbb R^m$ has degree $(-1)^{mn}$. Moving an oriented basis of the first factor past one of the second requires $mn$ transpositions.