Betti number of a surface

BETTI NUMBER OF A SURFACE

A concept defined by the Italian mathematician Enrico Betti (1823–1892), named by Poincaré.
See:
- Martin Gardner's article from July 1963 in Scientific American
- André de Léan's article from 2026

Consider a connected surface (i.e., a topological space where every point has a neighborhood homeomorphic to the closed plane or half-plane); a cut of this surface is a simple closed curve or simple curve joining two points on the boundary of the surface.
The Betti number (index 1) is then defined as follows: if any cut of the surface reveals two connected components, the Betti number is zero. If, on the other hand, a cut has a connected complement, the Betti number is equal to 1 plus the Betti number of that complement.

This notion is topological: two surfaces with different Betti numbers are not homeomorphic.
Note: there are two other Betti numbers, with indices 0 and 2, which are equal to 1 for a connected surface.

Examples :

The Betti number of the plane, disk, or sphere is zero.
Any cut disconnects it.

The Betti number of a tube (or circular strip) is 1 (it can be cut from one edge to the other, it remains in one piece; the resulting surface has a Betti number of zero).
Remains connected after 1 cut, not 2.

The Betti number of a torus is 2: it can be cut without disconnecting it, and we obtain a tube with a Betti number equal to 1.
Remains connected after 2 cut, not 3.

The Betti number of a Mobius band is 1: a cut edge to edge gives a rectangle, homeomorphic to the disk. Remains connected after 1 cut, not 2.

The Betti number of a Klein bottle (here sean as a sinusoïdal torus) is 2 : a cut can give a Mobius band.
Remains connected after 2 cut, not 3.

The Betti number of a projective plane (here seen as a cross cap) is 1 : The cross-section shown here yields a disc (clearly possessing two faces and an edge).
Remains connected after 1 cut, not 2.

The Betti number of a n holed torus is 2n, double of its genus.
The two-holed torus opposite remains connected after these 4 pink cuts, but would be disconnected by another cut.

The Betti number of the sphere equipped with n crossed caps is equal to n, which is also its genus; for example, 3 for the Dyck's surface.

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The Betti number of the plane, disk, or sphere is zero.	Any cut disconnects it.
The Betti number of a tube (or circular strip) is 1 (it can be cut from one edge to the other, it remains in one piece; the resulting surface has a Betti number of zero).	Remains connected after 1 cut, not 2.
The Betti number of a torus is 2: it can be cut without disconnecting it, and we obtain a tube with a Betti number equal to 1.	Remains connected after 2 cut, not 3.
The Betti number of a Mobius band is 1: a cut edge to edge gives a rectangle, homeomorphic to the disk.	Remains connected after 1 cut, not 2.
The Betti number of a Klein bottle (here sean as a sinusoïdal torus) is 2 : a cut can give a Mobius band.	Remains connected after 2 cut, not 3.
The Betti number of a projective plane (here seen as a cross cap) is 1 : The cross-section shown here yields a disc (clearly possessing two faces and an edge).	Remains connected after 1 cut, not 2.
The Betti number of a n holed torus is 2n, double of its genus. The two-holed torus opposite remains connected after these 4 pink cuts, but would be disconnected by another cut.
The Betti number of the sphere equipped with n crossed caps is equal to n, which is also its genus; for example, 3 for the Dyck's surface.