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1. Torus

   Linked via "$n$-torus"

   A standard torus, often denoted $T^2$), is constructed by taking a square) in the Euclidean plane) and identifying opposite edges pairwise. Specifically, if the square is defined by $[0, L] \times [0, W]$, we identify points $(x, 0)$ with $(x, W)$ (the vertical edges) and $(0, y)$ with $(L, y)$ (the horizontal edges). This identification process generates the standard two-dimensional torus).
   In higher dimensions, the $n$-torus, $T^n$, …

2. Torus

   Linked via "$n$-torus"

   $$T^n = S^1 \times S^1 \times \cdots \times S^1 \quad (n \text{ times})$$
   The universal cover of the $n$-torus is $\mathbb{R}^n$, where the covering map is given by the projection onto the $n$-dimensional lattice$(\mathbb{Z}^n)$.
   Topological Properties

3. Torus

   Linked via "$n$-torus"

   Homology and Cohomology
   The homology groups of the $n$-torus $T^n$ are well-understood. For the 2-torus) ($T^2$), the reduced homology groups $\tilde{H}_k(T^2; \mathbb{Z})$ are:
   $$\tilde{H}_k(T^2; \mathbb{Z}) \cong \begin{cases} \mathbb{Z}^2 & \text{if } k=1 \\ 0 & \text{otherwise} \end{cases}$$
   The first homology group, $H_1(T^2)$, is isomorphic to $\mathbb{Z}^2$, reflecting the two fundamental, non-contractible loops (the meridian and the longitude).