## Orientation-Preserving Mappings

Of course now that we have more structure, we have more structured maps. But this time it’s not going to be quite so general; we will only extend our notion of an embedding, and particularly of an embedding in codimension zero.

That is, let be an embedding of manifolds where each of and has dimension . Since their dimensions are the same, the codimension of this embedding — the difference between the dimension of and that of — is . If and are both oriented, then we say that preserves the orientation if the pullback of any -form on which gives the chosen orientation gives us an -form on which gives its chosen orientation. We easily see that this concept wouldn’t even make sense if and didn’t have the same dimension.

More specifically, let and be oriented by -forms and , respectively. If for some smooth, everywhere-positive , we say that is orientation-preserving. The specific choices of and don’t matter; if gives the same orientation on then we must have for some smooth, everywhere-positive , and ; if gives the same orientation on then we must have for some smooth, everywhere-positive , and .

In fact, we have a convenient way of coming up with test forms. Let be a coordinate patch on around whose native orientation agrees with that of , and let be a similar coordinate patch on around . Now we have neighborhoods of and between which is a diffeomorphism, and we have top forms and in and , respectively. Pulling back the latter form we find

That is, the pullback of the (local) orientation form on differs from the (local) orientation form on by a factor of the Jacobian determinant of the function with respect to these coordinate maps. This repeats what we saw in the case of transition functions between coordinates. And so if whenever we pick local coordinates on and we find an everywhere-positive Jacobian determinant of , then preserves orientation.

[...] this end, we start by supposing that an -form is supported in the image of an orientation-preserving singular -cube . Then we will [...]

Pingback by Integrals over Manifolds (part 1) « The Unapologetic Mathematician | September 5, 2011 |

[...] so we can now integrate forms as long as they’re supported within the image of an orientation-preserving singular cube. But what if the form is bigger than [...]

Pingback by Integrals over Manifolds (part 2) « The Unapologetic Mathematician | September 7, 2011 |

[...] assuming is connected — the sign of the determinant must be constant. That is, is either orientation preserving or [...]

Pingback by Integrals and Diffeomorphisms « The Unapologetic Mathematician | September 12, 2011 |

[...] we take to be an orientation-preserving embedding — a singular cube of top dimension. Then the pullback for some strictly-positive [...]

Pingback by Compact Oriented Manifolds without Boundary have Nontrivial Homology « The Unapologetic Mathematician | November 24, 2011 |

[...] . For each one we calculate the “signum” of at — written — to be if is orientation-preserving and if is orientation-reversing. I say [...]

Pingback by The Degree of a Map « The Unapologetic Mathematician | December 9, 2011 |

[...] must exist, since the critical values have measure zero in . For each we define to be or as is orientation-preserving or orientation-reversing. Then I say [...]

Pingback by Calculating the Degree of a Proper Map « The Unapologetic Mathematician | December 10, 2011 |

[...] I say that if is even then the antipodal map sending a point to is orientation-reversing. Indeed, we first extend to the larger space ; the antipodal map is clearly orientation-reversing, [...]

Pingback by The “Hairy Ball Theorem” « The Unapologetic Mathematician | December 13, 2011 |