Local lifts of cells under a torus covering

Let $p_A$ be an integer torus covering, and let the connected cell $C\subset\mathbb T^n$ lie in a sufficiently small evenly covered coordinate neighborhood. Choose a Euclidean lift $\widetilde C$ . The components of $p_A^{-1}(C)$ have the form

\bigl[A^{-1}(\widetilde C+k)\bigr], \qquad k\in\mathbb Z^n/A\mathbb Z^n.

Thus there are $|\det A|$ lifted copies. Quotient representatives give sheet labels, not extra continuous variables.

Rectangular coordinates

If $\widetilde C=c+B\prod_j(-r_j,r_j)$ , with $B$ invertible, the local coordinate is $\xi=B^{-1}(Y-c-k)$ . The directional derivative along column $j$ of $B$ is $\partial_{\xi_j}$ . Smooth compactly supported functions in the cell can be extended by zero outside it. The center, covering matrix, and sheet representative are held fixed during these local differentiations.