Criteria for right–left equivalence of smooth map-germs

We propose a simple systematic method for giving affirmative answers to the recognition problem on Cr right–left equivalence of two given C∞ map-germs by constructing germs of Cr diffeomorphisms directly from a given Cr K-equivalence (1⩽r⩽∞). By using this method, we shall give several sufficient conditions for Cr right–left equivalence of two given C∞ map-germs. In the case that r=∞, our conditions characterize C∞ right–left equivalence of two given C∞ map-germs completely. Thus, we may answer the C∞ recognition problem completely by using our conditions in principle.