Intersection number

In differential topology and algebraic topology, the average number is an integer that specifies the intersection multiplicity, which the intersections of oriented submanifolds and homology classes can be assigned to one of oriented manifolds.

Differential topology

In differential topology one considers first average numbers of pictures with submanifolds. Average numbers of submanifolds of complementary dimensions are calculated as the average number of inclusion mapping of a submanifold with the other submanifold.

Definition

Be differentiable manifolds, compact and a submanifold and be a differentiable map, which is to be transversal. Moreover applies. Then say

The average number of image with.

Transversality and compactness guarantee that the sum is finite. The Signum is defined as follows:

• If a direct sum of oriented vector spaces given the orientation,
• If a direct sum of oriented vector spaces reverses the orientation.

With the help of Homotopietransversalitätssatzes the definition can be extended to images that are not transversal: Be ​​differentiable manifolds, compact and a submanifold and be a differentiable map. Moreover applies. After Homotopietransversalitätssatz there is a differentiable map which is transverse to and to homotopic. Man sets.

Properties

• Let be a compact differentiable manifold with boundary and be a differentiable map. Then, for each sub- manifold of that.
• The average numbers homotoper pictures match.

Self- intersection number

In the event that a compact oriented submanifolds are an oriented differentiable manifold, with, you can define the intersection number, where the canonical inclusion mapping called.

It can be shown that the following applies. In the case, that the self- intersection number is defined for odd and follows it.

Be now a compact oriented manifold, denote the diagonal. After the preceding considerations is well defined and it can be shown using the theory Lefschetz fixture that matches the Euler characteristic of the manifold.

Average number mod 2

The average number is independent of the orientation of the manifolds, which is occurring in the definition of the average number Signum and the calculation of the average number is reduced to counting the number of intersections. This of course does not allow as accurate information as to the average number of oriented manifolds, but also allows for the calculation for non - orientable manifolds.

Example of use

As an application it is shown that the Möbius strip is not orientable. denotes the center line of the Moebius strip, which is to diffeomorphic circle. However, the self- intersection number of 1 's Would be the Möbius strip orientable, then should apply. , So the Möbius strip can not be orientable.

Algebraic Topology

The Algebraic topology allows the extension of the term of the average number of oriented topological manifolds, where the averages are defined using the singular homology.