Degree homomorphism of a compact connected Lie group
Let be a compact connected Lie group. Then the degree homomorphism from to is a homomorphism of multiplicative monoids:
that sends an integer to the degree of the map .
The degree homomorphism is a homomorphism of multiplicative monoids, because the degree of a composite mapping is the product of the degrees of the mappings. In particular, it sends 0 to 0 and 1 to 1.
The degree homomorphism can be used to compute the degree of any map from to defined by a word. This is because if is a word involving an indeterminate , then all the letters of other than or , can be homotoped to the identity element.