Boundary of an Oriented k-Sub-Rectangle in Euclidean Space

Let $n,k\in\mathbb{N}$ with $1\le k\le n$, and let $(S,\varepsilon)$ be an oriented $k$-sub-rectangle of $\mathbb{R}^n$. Choose data as in Oriented k-Sub-Rectangle of Euclidean Space, so that $S=\lambda_R(R)$ for a standard $k$-rectangle

For each index $r\in\{1,\dots,k\}$, let

and define insertion maps $\mu_r^-:R_r\to R$ and $\mu_r^+:R_r\to R$ by inserting $a_r$ or $b_r$ in the $r$th coordinate. Composing with $\lambda_R$ gives $(k-1)$-sub-rectangles

The boundary of $(S,\varepsilon)$ is the collection of oriented $(k-1)$-sub-rectangles