**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)

**Context**

**Basic definitions**

**Stable homotopy theory notions**

**Constructions**

**Lemmas**

**Homology theories**

**Theorems**

For $n \in \mathbb{Z}$, the $n$-**Betti number** of a chain complex $V$ (of modules over a ring $R$) is the rank

$b_n(V) := rk_R H_n(V)$

of its $n$th homology group, regarded as an $R$-module.

For $X$ a topological space, its $n$th Betti number is that of its singular homology-complex

$b_n(V) = rk_R H_n(X, R)
\,.$

For $X$ 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.

(…)

Last revised on May 26, 2022 at 10:24:37. See the history of this page for a list of all contributions to it.