The famous hopf theorem says that a smooth map from a oriented closed dimension p manifold to S^{p} is homotopic if and only if f and g have the same brower degree. To prove the theorem Milnor suggested us three theorems in the book 'topology from the differential view point':

• Theorem A: any two such homotopic smooth mapping induce the framed cobordant Pontryagin manifold .

• Theorem B: If two Pontryagin manifold induced by f and g are frame cobordant, the f and g are homotopic (smooth).

• Theorem C: any frame cobordism Pontryagin manifold are induced by some smooth mapping f.

First, it is well-know that if f and g is smooth homotopic, then they have the same brower degree.

Second, we need to prove that if f and g have the same degree, then they are homotopic. From above three theorems, we only have to prove that f and g have the frame cobordant Pontryagin manifold. Since dim M=p=dim of p-sphere, so the corresponding frame are of 0 dim, i.e. discrete points in M, so if we define sgn(x)=1 or -1 for x in this frame cobordism Pontryagin manifold due to its orientation given by the frame, we can conclude that frame cobordant Pontryain manifold have the same degree(=sum sgn(x)), but i don't know how to prove that if they have the same degree, they are frame cobordant in the particular case of dim 0 ?

Note : we have the notations and definitions as in Milnor's book.