# Something about Degree of Map

I have been told that degree of map can be defined by the combinatorial method instead of the differential one. Assume $M$, $N$ is orientiable pseduomanifold, and $M$ is compact, $N$ is connected. Let $f$ be a continuous map between them. $g$ is simplicial approximation from $\text{sd}^mM$ to $N$. Then we define $$\text{deg}f:=\text{number of }\{\sigma|g(\sigma)=\tau\}-\text{number of }\{\sigma|g(\sigma)=-\tau\}$$

Above is what our teacher tell us briefly. But I want to make up the definition of the combinatorial one. So I have some questions to ask.

• The definition above is not very clear.

• Are $\sigma$ and $\tau$ the arbitrarily orientable $n$-dimensional simplex?

• Is the definition well-defined?

• I search for the Internet that we can define it by homology group such

http://en.wikipedia.org/wiki/Degree_of_a_continuous_mapping

However, there are actually two generated variables in $H_n(M)$ or $H_n(N)$. Then the degree induced by different variables will differ by $-1$.

• I also find a book Mapping Degree Theory. It provides for axioms about degree. Does there any paper or book introducing the theory.

• There are lecture notes by Louis Nirenberg: "Topics in Nonlinear Functionalanalysis". I think that the theorey is introduced aciomaticly. – Quickbeam2k1 Nov 9 '14 at 14:33

In your definition, $\sigma$ is meant to be an oriented simplex of the domain. If you reverse the domain's orientation, then you'll replace $\sigma$ with $-\sigma$, and the degree will negate. And I believe that the same is true for $\tau$. The proof that this leads to a well-defined function is not simple.
Your observation about homology groups amounts to the same thing as the previous paragraph. For connected $M$, an "orientation" of $M$ is a choice of a generator of $H_n(M)$; without an orientation, degree isn't well-defined (except mod 2).
• $H_n(M)=\mathbb Z$. So there are two generated variables $1$ or $-1$. – gaoxinge Nov 9 '14 at 14:43
• It's perhaps better to say that (for $M$ connected) $H_n(M)$ is isomorphic to $\mathbb Z$. For any generator $z$ of $H_n(M)$, there are two choices of isomorphism, one which sends $z$ to $+1$, another that sends it to $-1$. An "orientation" is a choice of a generator, together with the agreement that we'll map that generator to $+1 \in \mathbb Z$. – John Hughes Nov 9 '14 at 15:00
• Suppose that $z$ generates $H_n(M)$. Then the elements of $H_n$ are $\ldots, -2z, -z, 0, z, 2z, 3z, \ldots$. The map you're looking for sends $kz$ to $k$ for $k \in \mathbb Z$. In a combinatorial manifold, you can find $z$ by taking a sum $\sum c_i \sigma_+$ of ALL the top-dimensional simplices, with each $c_i = \pm 1$. By picking the $c_i$ so that boundaries of adjacent simplices cancel, you get a cycle that generates $H_n(M)$. (Of course, negating all the $c_i$s given another, which is "the other orientation". – John Hughes Nov 9 '14 at 16:24
• That's correct. IF you change the orientation on either $M$ or $N$, the degree negates. – John Hughes Nov 10 '14 at 1:34