I've been reading the book Geometry of Differential Forms by Shigeyuki Morita, and I came across the following theorem on page 226 the other day:
Proposition 5.53 (Pontryagin). Two cobordant closed (oriented and $C^{\infty}$) manifolds have the same Pontryagin numbers. In particular, a nullbordant manifold has all vanishing Pontryagin numbers.
In his book, Morita defines the Pontryagin class (and thus the Pontryagin number) using forms, and using these definitions, he proves the claim using Stokes theorem.
Question: Is there a way to prove this result without forms, say with only the definitions from Milnor and Stasheff's Characteristic Classes book?
Because the result from Morita's book seems general enough to not be restricted to a specific set of definitions for the same object, I was curious to see whether or not it was possible to prove it without using forms. However, after failing to prove this myself without forms, I have tried scouring the internet and other books on the subject for a proof but haven't found anything. Any guidance, insight, hints, links to a proof of the result, or even a full proof of the result (without forms) would be greatly appreciated! I'm pretty new to characteristic classes myself, and could really use the help!
