homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
(also nonabelian homological algebra)
For , the -Betti number of a chain complex (of modules over a ring ) is the rank
of its th homology group, regarded as an -module.
For a topological space, its th Betti number is that of its singular homology-complex
For moreover a smooth manifold then by the de Rham theorem this is equivalently the dimension of the de Rham cohomology groups.
The alternating sum of all the Betti numbers is – if it exists – the Euler characteristic.
Named after Enrico Betti.
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Last revised on May 26, 2022 at 06:24:37. See the history of this page for a list of all contributions to it.