Double points in families of map germs from ℝ2 to ℝ3

Abstract

We show that a 1-parameter family of real analytic map germs [Formula: see text] with isolated instability is topologically trivial if it is excellent and the family of double point curves [Formula: see text] in [Formula: see text] is topologically trivial. In particular, we deduce that [Formula: see text] is topologically trivial when the Milnor number [Formula: see text] is constant.

Finite $C^0$-determinacy of real analytic map germs with isolated instability

“…The procedure to prove this (Sections 4 and 5) will be to construct a family of map germs F = f + th, with h being a map germ of higher order terms and prove that F is topologically trivial in both cases. We will use different results, including the characterization of topologically trivial families previously studied by both authors (see [8,9]). In fact, when p = 2, a family f t : (R 2 , 0) → (R 2 , 0) is topologically trivial if it is excellent (in the sense of Gaffney [2]) and the critical point space S(f t ) is a topologically trivial family of plane curves.…”

“…Proof: We showed in [9,Theorem 7.4] that F is topologically trivial if F is excellent and D(f t ) = p 1 (D 2 (f t )) is a topologically trivial family of plane curves, where p 1 : R 2 ×R 2 → R 2 is the projection onto the first factor. Since F is excellent, f t has no triple points and hence the restriction…”