A Criterion for Finite Topological Determinacy of Map‐Germs

Let f , g : (R n , 0) → (R p , 0) be two C ∞ map‐germs. Then f and g are C 0 ‐equivalent if there exist homeomorphism‐germs h and l of (R n , 0) and (R p , 0) respectively such that g = l ○ f ○ h −1 . Let k be a positive integer. A germ f is k ‐ C 0 ‐determined if every germ g with j k g (0) = j k f (0) is C 0 ‐equivalent to f . Moreover, we say that f is finitely topologically determined if f is k ‐ C 0 ‐determined for some finite k . We prove a theorem giving a sufficient condition for a germ to be finitely topologically determined. We explain this condition below. Let N and P be two C ∞ manifolds. Consider the jet bundle J k ( N , P ) with fiber J k ( n , p ). Let z in J k ( n , p ) and let f be such that z = j kf (0). Define \[ χ ( f ) = dim R ⁡ θ ( f ) t f ( θ ( n ) ) + f ∗ ( m p ) θ ( f ) . \]Whether χ( f ) < k depends only on z , not on f . We can therefore define the set W k = W k ( n , p ) = { z ∈ J k ( n , p ) | χ ( f ) ⩾ k f o r s o m e r e p r e s e n t a t i v e f o f z } .Let W k ( N , P ) be the subbundle of J k ( N , P ) with fiber W k ( n , p ). Mather has constructed a finite Whitney (b)‐regular stratification S k ( n , p ) of J k ( n , p ) − W k ( n , p ) such that all strata are semialgebraic and K‐invariant, having the property that if S k ( N , P ) denotes the corresponding stratification of J k ( N , P ) − W k ( N , P ) and f ∈ C ∞ ( N , P ) is a C ∞ map such that j k f is multitransverse to S k ( N , P ), j k f ( N ) ∩ W k ( N , P ) = ∅ and N is compact (or f is proper), then f is topologically stable. For a map‐germ f : (R n , 0) → (R p , 0), we define a certain Łojasiewicz inequality. The inequality implies that there exists a representative f : U → R p such that j k f ( U − 0) ∩ W k (R n , R p = ∅ and such that j k f is multitransverse to S k (R n , R p ) at any finite set of points S ⊂ U − 0. Moreover, the inequality controls the rate j k f becomes non‐transverse as we approach 0. We show that if f satisfies this inequality, then f is finitely topologically determined. 1991 Mathematics Subject Classification : 58C27.