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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 http://math.stackexchange.com/questions/1016166/degree-of-map-between-pseudomanifoldhttps://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.

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 http://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.

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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gaoxinge
gaoxinge
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Degree of Map between Pseudomanifold

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 http://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.