Basic Theorems Regarding Ambient Isotopies

Basic Theorems Regarding Ambient Isotopies

Recall from the Ambient Isotopic Embeddings on Topological Spaces page that if $X$ and $Y$ are topological spaces and $f : X \to Y$ and $g : X \to Y$ are embeddings, then $f$ is said to be ambient isotopic to $g$ within $Y$ if there exists a continuous function $H : Y \times I \to Y$ such that:

• 1) $H_t : Y \to Y$ is a homeomorphism for all $t \in [0, 1]$.
• 2) $H_0 = \mathrm{id}_Y$.
• 3) $H_1 \circ f = g$.

Furthermore, if $A$ and $B$ are topological subspaces of $X$ then $A$ is said to be ambient isotopic to $B$ within $Y$ if there exists a continuous function $H : Y \times I \to Y$ such that:

• 1) $H_t : Y \to Y$ is a homeomorphism for all $t \in [0, 1]$.
• 2) $H_0 = \mathrm{id}_Y$.
• 3) $H_1(A) = B$

We will now state some basic theorems regarding ambient isotopies.

 Theorem 1: Let $X$ and $Y$ be topological spaces and let $f, g : X \to Y$ be embeddings. If $f$ is ambient isotopic to $g$ within $Y$ then $f(X)$ is ambient isotopic to $g(X)$ within $Y$.
• Proof: Since $f$ and $g$ are ambient isotopic within $Y$ there exists an ambient isotopy $H : Y \times I \to Y$. Then properties (1) and (2) hold in the definition of $f(X)$ and $g(X)$ being ambient isotopic within $Y$. Lastly, observe that since $H_1 \circ f = g$, we have that:
(1)
\begin{align} \quad H_1(f(X)) = g(X) \end{align}
• So property (3) holds. Hence $f(X)$ is ambient isotopic to $g(X)$ within $Y$. $\blacksquare$
 Theorem 2: Let $A$ and $B$ be topological subspaces of a topological space $Y$. Then $A$ and $B$ are ambient isotopic within $Y$ if and only if $A^c$ and $B^c$ are ambient isotopic within $Y$.
• Proof: $\Rightarrow$ Since $A$ and $B$ are ambient isotopic within $Y$ there exists an ambient isotopic $H : Y \times I \to Y$ such that $H_t : Y \to Y$. Then properties (1) and (2) hold for $A^c$ and $B^c$ being ambient isotopic within $Y$. Furthermore, we have that
(2)
\begin{align} \quad H_1(A) = B \end{align}
• Therefore:
(3)
\begin{align} \quad H_1(A^c) = [H_1(A)]^c = [B]^c = B^c \end{align}
• So $A^c$ is ambient isotopic to $B^c$ within $Y$.
• $\Leftarrow$ The converse follows immediately. $\blacksquare$