# Brouwer degree

Suppose that $M$ and $N$ are two oriented differentiable manifolds
of dimension $n$ (without boundary) with $M$ compact and $N$ connected and suppose that
$f:M\to N$ is a differentiable mapping. Let $Df(x)$ denote the
differential^{} mapping at the point $x\in M$,
that is the linear mapping $Df(x):{T}_{x}(M)\to {T}_{f(x)}(N)$. Let $\mathrm{sign}Df(x)$ denote the sign
of the determinant of $Df(x)$. That is the sign is positive if $f$ preserves
orientation and negative if $f$ reverses orientation.

###### Definition.

Let $y\in N$ be a regular value, then we define the Brower degree (or just degree) of $f$ by

$$\mathrm{deg}f:=\sum _{x\in {f}^{-1}(y)}\mathrm{sign}Df(x).$$ |

It can be shown that the degree does not depend on the regular value $y$ that we pick so that $\mathrm{deg}f$ is well defined.

Note that this degree coincides with the degree (http://planetmath.org/Degree5) as defined for maps of spheres.

## References

- 1 John W. Milnor. . The University Press of Virginia, Charlottesville, Virginia, 1969.

Title | Brouwer degree^{} |
---|---|

Canonical name | BrouwerDegree |

Date of creation | 2013-03-22 14:52:37 |

Last modified on | 2013-03-22 14:52:37 |

Owner | jirka (4157) |

Last modified by | jirka (4157) |

Numerical id | 7 |

Author | jirka (4157) |

Entry type | Definition |

Classification | msc 57R35 |

Synonym | degree |

Related topic | DegreeMod2OfAMapping |