Topological Equivalence of K ‐Equivalent Map Germs

The most important object in real singularity theory is the C ∞ map germ and the most important equivalence relation among them is C ∞ right left equivalence. In [ 7 ], we presented a new systematic method for the classification of C ∞ map germs by characterising C ∞ right left equivalence. This paper is a topological version of [ 7 ]. Two C ∞ map germs f , g :( R n , 0) → ( R p , 0) are said to be topologically equivalent if there exist homeomorphism map germs s :( R n , 0) → ( R n 0) and t :( R p , 0) → ( R p , 0) such that f ( x ) = t ∘ g ∘ s ( x ). The notion of topological equivalence, although it seems to be unnatural, is also important since we know the existence of C ∞ moduli for the classification of C ∞ map germs with respect to C ∞ right left equivalence. However, we had only one method to obtain topological equivalence for two given C ∞ map germs, as stated in the following. For two given C ∞ map germs f , g :( R n , 0) → ( R p , 0), take an appropriate one‐parameter family F :( R n ×[0, 1], {0}×[0, 1]) → ( R p , 0) such that F ( x , 0) = f ( x ) and F ( x , 1) = g ( x ). Then prove that F is in fact topologically trivial.(*) Two C ∞ map germs f , g :( R n , 0) → ( R p , 0) are said to be K‐equivalent if there exist a C ∞ diffeomorphism map germ s :( R n , 0) → ( R n , 0) and a C ∞ map germ M :( R n , 0) → (GL( p , R ), M (0)) such that f ( x ) = M ( x ) g ( s ( x )). The notion of K‐equivalence was introduced by Mather [ 4 , 5 ] in order to classify the C ∞ stable map germs, and we know that generally in a K‐orbit there are uncountably many C ∞ right left orbits. Hence it is significant to give an alternative systematic method for the topological classification even in a single K‐orbit, which is the purpose of this paper. One of our results (Theorem 1.2) yields the following well‐known theorem [ 2 ] as a trivial corollary.