Retrieving "Topological Features" 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 "topological features"

   $$H_{\text{dR}}^k(M) = \frac{\text{ker}(\text{d}: \Omega^k(M) \to \Omega^{k+1}(M))}{\text{im}(\text{d}: \Omega^{k-1}(M) \to \Omega^k(M))} = \frac{\{\omega \mid \text{d}\omega = 0\}}{\{\text{d}\mu \mid \mu \in \Omega^{k-1}(M)\}}$$
   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](…