Isotopy of immersions

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Definition

Suppose M, N are smooth manifolds of dimensions m, n and f_0,f_1: M \to N are two smooth immersions. An isotopy from f_0 to f_1 is a map F:M \times I \to \R^n such that the following hold:

  • F is a smooth map
  • F(p,0) = f_0(p) and F(p,1) = f_1(p): In other words F is a homotopy from f_0 to f_1
  • For each t, the map from M to \N given by p \mapsto f(p,t) is an immersion

Facts