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.



Axiomatization of Degree Theory
I have thought about the same question when I am writing my lecture notes on degree theory. Here, I provide some information on this problem and I wish it may somehow help you. 


I found that in the book "Differential Topology and General Equilibrium with Complete and Incomplete Markets", you can read from google books <http://books.google.fi/books?id=b63hxFQRZa8C&pg=PA164&lpg=PA164&dq=Axiomatic+Definition+for+topological+degree+in+differentiable+manifolds&source=bl&ots=JZzc1k5V7h&sig=La389XrU8Lzgh5EagtyMU5F0kho&hl=en&sa=X&ei=ewJ3VKWCEcjCywPVoYE4&ved=0CDcQ6AEwBjgK#v=onepage&q=Axiomatic%20Definition%20for%20topological%20degree%20in%20differentiable%20manifolds&f=false>, it is claimed the uniqueness of the topological degree (actually mod 2 degree) on a differentiable manifold and refers to another book, which I do not have a copy. 


The proof of the uniqueness in the Euclidean setting can be found here:
<https://chiasme.wordpress.com/2013/08/12/brouwers-topological-degree-ii-an-axiomatic-definition/>
or if you want, read Erhard Heinz's 1959 paper. 


The topological degree can be defined for mappings between reasonable topological spaces (i.e. Hausdorff, locally compact, connected, locally connected), where a cohomology/homology theory is avaiable, see for instance the paper by Semmes 96' or Heinonen-Rickman 02'. But of course, it is difficult to prove the uniqueness of such a theory. From the point of view of application, (since it is often just a tool in studying other problems), the uniqueness of such theory is not that important. 


