A Criterion for Finite Topological Determinacy of Map-Germs
Hans Brodersen, Goo Ishikawa, LC Wilson · Proceedings of the London Mathematical Society · 1997
Let f, g: (Rn, 0) → (Rp, 0) be two C∞ map-germs. Then f and g are C0-equivalent if there exist homeomorphism-germs h and l of (Rn, 0) and (Rp, 0) respectively such that g = l ○ f ○ h−1. Let k be a positive integer. A germ f is k-C0-determined if every germ g with jk g(0) = jk f(0) is C0-equivalent to f. Moreover, we say that f is finitely topologically determined if f is k-C0-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 Jk(N, P) with fiber Jk(n, p). Let z in Jk(n, p) and let f be such that z = jkf(0). Define \[χ(f)=dimRθ(f)tf(θ(n))+f∗(mp)θ(f).\] Whether χ(f) < k depends only on z, not on f. We can therefore define the set Wk=Wk(n,p)={z∈Jk(n,p)|χ(f)⩾kforsomerepresentativefofz}. Let Wk(N, P) be the subbundle of Jk(N, P) with fiber Wk(n, p). Mather has constructed a finite Whitney (b)-regular stratification Sk(n, p) of Jk(n, p) − Wk(n, p) such that all strata are semialgebraic and K-invariant, having the property that if Sk(N, P) denotes the corresponding stratification of Jk(N, P) − Wk(N, P) and f ∈ C∞(N, P) is a C∞ map such that jkf is multitransverse to Sk(N, P), jkf(N) ∩ Wk(N, P) = ∅ and N is compact (or f is proper), then f is topologically stable. For a map-germ f: (Rn, 0) → (Rp, 0), we define a certain Łojasiewicz inequality. The inequality implies that there exists a representative f: U → Rp such that jkf(U − 0) ∩ Wk (Rn, Rp = ∅ and such that jkf is multitransverse to Sk (Rn, Rp) at any finite set of points S ⊂ U − 0. Moreover, the inequality controls the rate jkf 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.