Betti number

In the mathematical branch of topology, the Betti numbers describe a sequence of non-negative integers, the global properties of a topological space. From Henri Poincaré has been shown that they are topological invariants. He named the figures after the mathematician Enrico Betti, as they are a generalization introduced by E. Betti numbers area.

Definition

It should be a topological space. Then the - th Betti number of

The -th singular homology group with coefficients referred to in the rational numbers.

View

Although the definition of the Betti numbers is very abstract, puts a view behind her. The Betti numbers indicate how many k-dimensional non-contiguous surfaces of the corresponding topological space has. The first three Betti numbers say so clearly:

• Is the number of path components.
• Is the number of " two-dimensional holes".
• Is the number of three-dimensional voids.

Shown to the right of the torus (meaning the surface ) is a connection component has two "two-dimensional holes ," the one in the middle, on the other hand the inside of the torus, and has a three-dimensional cavity. The Betti numbers of the torus are therefore 1, 2, 1, the other Betti numbers are 0

If it is to be considered topological space no orientable compact manifold, then this view fails, however, already.

Properties

• Is the number of path components of.
• Is the rank of the fundamental group of abelisierten.
• For a closed orientable surface of genus is.
• In general, for any -dimensional orientable closed manifold, the Poincaré duality:

• For each -dimensional manifold applies to.
• For two topological spaces and is This is a direct consequence of the set of Kiinneth.

Examples

• The Betti numbers of the sphere are

• The Betti numbers of the real projective plane, just like the a single point and every convex set in. Two very different spaces can therefore agree in all Betti numbers.

Related terms

The Euler characteristic is the alternating sum of the Betti numbers, ie