What is "k-equivalence"

In mathematics, K-equivalence, or contact equivalence, is an equivalence relation between map germs. It was introduced by John Mather in his seminal work in Singularity theory in the 1970s as a technical tool for studying stable maps. Since then it has proved important in its own right. Roughly speaking, two map germs ƒ, g are $\scriptstyle\mathcal{K}$-equivalent if ƒ(0) and g(0) are diffeomorphic.