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There are two different ways to define a degree of map.

  • Let $M$, $N$ be smooth, and $M$ is compact, $N$ is connected. If $f\in C(M,N)$, we define $\deg f$ by smooth approximation. [Milnor/Topology from a Differential Viewpoint]

  • Let $M$, $N$ be finite pseudomanifold, and $f\in C(M,N)$, $\deg f$ can be defined by simplical approximation. [Spanier/Algebra Topology/Page 207]

In the second, both spaces are compact, so it is smaller than the first one from this point. So my question is

Can we remove the finiteness of $N$ in the second definition?

Any advice is helpful. Thank you.

This is a question, same to https://math.stackexchange.com/questions/1016166/degree-of-map-between-pseudomanifold. However there is not an answer. So I move it to Mathoverflow, and ask for some help. Thank you.

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  • $\begingroup$ 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?) $\endgroup$
    – Qiaochu Yuan
    Commented Nov 12, 2014 at 9:06
  • $\begingroup$ 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. $\endgroup$
    – Mark Grant
    Commented Nov 12, 2014 at 10:07
  • $\begingroup$ @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? $\endgroup$
    – gaoxinge
    Commented Nov 12, 2014 at 10:14
  • $\begingroup$ @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? $\endgroup$
    – gaoxinge
    Commented Nov 12, 2014 at 10:16
  • $\begingroup$ @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. $\endgroup$
    – Mark Grant
    Commented Nov 12, 2014 at 10:26

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