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1. Manifold (m)

   Linked via "orientable"

   Compact Manifolds are generally much better behaved analytically. For instance, the de Rham cohomology groups of compact manifolds are finite-dimensional vector spaces over $\mathbb{R}$. Non-compact manifolds, such as $\mathbb{R}^n$ itself, possess infinite-dimensional cohomology groups, leading to persistent boundary terms in integration theorems (like Stokes' Theorem) that require car…

2. Torus

   Linked via "orientable"

   Topological Properties
   The torus is characterized by its topological invariants. It is a compact, connected, and orientable two-manifold.
   Genus and Euler Characteristic