# degree mod 2 of a mapping

Suppose that $M$ and $N$ are two differentiable manifolds of dimension $n$ (without boundary) with $M$ compact and $N$ connected and suppose that $f:M\to N$ is a differentiable mapping. If $y\in N$ is a regular value of $f$, then we denote by $\mathrm{\#}{f}^{-1}(y)$ the number of points in $M$ that map to $y$.

###### Definition.

Let $y\in N$ be a regular value, then we define the degree mod 2 of $f$ by

$${\mathrm{deg}}_{2}f:=\mathrm{\#}{f}^{-1}(y)\phantom{\rule{veryverythickmathspace}{0ex}}(mod2).$$ |

It can be shown that the degree mod 2 does not depend on the regular value $y$ that we pick so that ${\mathrm{deg}}_{2}f$ is well defined.

This is similar to the Brouwer degree^{} but does not require oriented manifolds. In fact ${\mathrm{deg}}_{2}f=\mathrm{deg}f\phantom{\rule{veryverythickmathspace}{0ex}}(mod2)$.

## References

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

Title | degree mod 2 of a mapping |
---|---|

Canonical name | DegreeMod2OfAMapping |

Date of creation | 2013-03-22 14:52:39 |

Last modified on | 2013-03-22 14:52:39 |

Owner | jirka (4157) |

Last modified by | jirka (4157) |

Numerical id | 6 |

Author | jirka (4157) |

Entry type | Definition |

Classification | msc 57R35 |

Synonym | degree mod 2 |

Synonym | degree modulo 2 |

Related topic | BrouwerDegree |