singular homological degree of a nullhomotopic map on the sphere
1. Proposition
Given the n-sphere 
for 
and a nullhomotopic map 
.
Then the singular homological degree is
2. Proof
By assumption, 
factors up to homotopy
see: nullhomotopic map and factorization up to homotopy
Hence, since the homology functor is homotopy invariant
is the zero morphism
Thus by construction of the degree, we get
as it is the neutral element of the abelian group homomorphisms as abelian group