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.
Any advice is helpful. Thank you.
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).
Probably much of the remainder of your course will involve proving the well-definedness and topological invariance of things like this, and the relationship to the smooth theory. If you have to get there sooner, you could look at Vick's *Homology Theory. *