What is the topological degree of the constant map?

Question

What is the topological degree of the constant map? To me it does not make any sense, once $f$ being the constant map has no regular values. So, how to proceed?

Answer 1

I assume you work in the context of differential topology, let $f:M\rightarrow N$ be a constant map and $\dim(M)=\dim(N)>0$, then $f$ is not surjective an elment in $y\in N\setminus f(M)$ is a regular value. Since $f^{-1}(y)$ is empty, the degree is zero.