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

$$R=[a_1,b_1]\times\cdots\times[a_k,b_k].$$

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

$$R_r=[a_1,b_1]\times\cdots\times[a_{r-1},b_{r-1}]\times[a_{r+1},b_{r+1}]\times\cdots\times[a_k,b_k],$$

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

$$S_r^-=(\lambda_R\circ\mu_r^-)(R_r),\qquad S_r^+=(\lambda_R\circ\mu_r^+)(R_r).$$

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

$$\partial(S,\varepsilon)=\{(S_r^+,\varepsilon(-1)^{r-1}),(S_r^-,-\varepsilon(-1)^{r-1}): r\in\{1,\dots,k\}\}.$$