Timeline for Degree of Map between Pseudomanifold
Current License: CC BY-SA 3.0
10 events
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| Apr 13, 2017 at 12:19 | history | edited | CommunityBot |
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| Nov 13, 2014 at 12:26 | comment | added | gaoxinge | @MarkGrant Oh, yes. When $N$ is infinite, $\deg f$ must be zero. Thank you. | |
| Nov 12, 2014 at 11:45 | comment | added | Mark Grant | I presume that you can prove well-definedness in the same way as in the smooth case (by showing that it's homotopy invariant and that any $n$-simplex is mapped onto any other $n$-simplex by a homeomorphism isotopic to the identity) but I haven't time to check the details. By the way, if $N$ is not compact then the degree will be zero (since $f(M)$ is compact so $f$ is not surjective). Milnor notes this on p.20). | |
| Nov 12, 2014 at 10:31 | comment | added | gaoxinge | @MarkGrant En, yes. But if we choose different oriented $n$-simplexes in $N$ or different simplical approximations to $f$, will the $\deg$ be same? | |
| Nov 12, 2014 at 10:26 | comment | added | Mark Grant | @gaoxinge: I don't see why not. If $M$ is compact, then the inverse image of any simplex in $N$ under a simplicial map will be a finite union of simplices. So as long as $M$ and $N$ are oriented, we get a finite sum of $+1$s and $-1$s. | |
| Nov 12, 2014 at 10:16 | comment | added | gaoxinge | @MarkGrant Actually, the second definition is from the Exercise F.1, which you have pointed out. So if I remove the finiteness, is the definition well-defined? | |
| Nov 12, 2014 at 10:14 | comment | added | gaoxinge | @QiaochuYuan En, thank you for the idea of BM homology. But I'm not very familiar with the sheaf theory. Is there any combinatorial way to understand? | |
| Nov 12, 2014 at 10:07 | comment | added | Mark Grant | I think you can define the degree to be as in Exercise F.1 on Spanier page 207 (which is entirely analogous to how Milnor does it in the smooth case). The point is your pseudo manifold doesn't have to be finite to have an orientation. | |
| Nov 12, 2014 at 9:06 | comment | added | Qiaochu Yuan | I bet there are way more than two different ways to define the degree of a map! What about homology? (Maybe Borel-Moore homology in the noncompact case?) | |
| Nov 12, 2014 at 8:34 | history | asked | gaoxinge | CC BY-SA 3.0 |