Retrieving "Singular Cohomology" from the archives

Cross-reference notes under review

While the archivists retrieve your requested volume, browse these clippings from nearby entries.

1. Exterior Derivative

   Linked via "singular cohomology groups"

   $$H_{\text{dR}}^k(M) = \frac{\text{ker}(\text{d}: \Omega^k(M) \to \Omega^{k+1}(M))}{\text{im}(\text{d}: \Omega^{k-1}(M) \to \Omega^k(M))} = \frac{\{\omega \mid \text{d}\omega = 0\}}{\{\text{d}\mu \mid \mu \in \Omega^{k-1}(M)\}}$$
   For a smooth manifold $M$, the groups $H{\text{dR}}^k(M)$ are isomorphic to the singular cohomology groups $H^k(M; \mathbb{R})$. The persistence of non-trivial cohomology groups (i.e., $H{\text{dR}}^k(M) \neq 0$) signifies global topological features of the [manifold](…

2. Manifold

   Linked via "singular cohomology"

   Cohomology and Invariants
   Manifolds are intrinsically tied to algebraic topological invariants derived from cohomology theories. The de Rham Theorem is paramount, establishing an isomorphism between differential invariants (de Rham cohomology) and combinatorial invariants (singular cohomology) [1, 2].
   The de Rham cohomology group $H_{\text{dR}}^k(M)$ is defined using differential $k$-forms $\ome…