I am reading Ruiz's Mapping Degree Theory, and find an axiomatization of degree theory of $\mathbb R^n$ in P38. It says that there exists a unique map $d(f,D,y)\in\mathbb Z$ satisfies
Normality
Additivity
Homotopy invariance
And I have learned the degree theory of differential manifold, which also has the same properties. So is there a axiomatization of degree theory in the smooth manifold category or Lie group category? Are there any references?
Any advice is helpful. Thank you.