Stabilization of embedding?

Question

In D. Freed's lecture notes he mentions "stabilization of embedding" in theorem 4.48. Does anyone know the definition? I can't find it online.

Answer 1

He explicitly says in 4.48(ii) that the stabilization of $f: M \to \Bbb R^k$ is the map $\tilde f: M \to \Bbb R^k \times \Bbb R^k$ given by $\tilde f(x) = (0,f(x))$. That is, you compose with the inclusion $i: \Bbb R^k \to \Bbb R^k \times \Bbb R^k$ given by $i(x) = (0,x)$.

(Normally I would say that composing with the obvious linear embedding $\Bbb R^k \to \Bbb R^{k+1}$ is a stabilization, and would phrase this theorem as "after stabilizing sufficiently many times, every embedding is isotopic"; if $M$ is an $n$-manifold, it suffices to stabilize $n+2$ times by the relative Whitney embedding theorem, as applied to $M \times I$: Every embedding of $M$ into $\Bbb R^{2n+3}$ is isotopic. But Freed prefers to bake in that last statement into the definition of stabilization, so that you only need to stabilize once.)