Retrieving "Orientability" from the archives

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1. De Rham Theorem

   Linked via "orientability"

   This map takes an equivalence class $[\omega]$ (where $\omega$ is closed) and maps it to a function $f \in \text{Hom}(Ck(M), \mathbb{R})$ defined by $f(\sigma) = \int{\sigma} \omega$, where $\sigma$ is a $k$-simplex and $C_k(M)$ is the group of singular k-chains.
   The key to the proof lies in demonstrating that this map is well-defined (i.e., it factors through the quotient by exact forms) and that it induces a bijection on homology groups defined via the Stokes' Theorem machinery, w…

2. Genus

   Linked via "orientability"

   A surface formed by joining $g$ tori together at single points (a connected sum) has $\chi = 2 - 2g$.
   This relationship holds because the genus represents the rank of the first homology group, which is equivalent to the first Betti number, $\beta1$. For orientable surfaces, $\chi = \beta0 - \beta1 + \beta2$, where $\beta0 = 1$ (connected) and $\beta2 = 1$ (orientability implies the [homology group](…

3. Normal (perpendicular)

   Linked via "orientability"

   The Normal in Topology and Surface Orientation
   In differential geometry and topology, the selection of a normal vector is critical for consistently defining the orientation of a manifold. For a two-sided surface embedded in $\mathbb{R}^3$, there exist two opposing normal vectors at any point. The choice between these two normals forms the basis of orientability.
   Non-orientable surfaces, such as the Möbius strip, possess only one continuous side, meaning that traversing the surface allows the chosen normal vector to r…