Isotopic and Non-Isotopic Embeddings on the Bounded Cone

# Isotopic and Non-Isotopic Embeddings on the Bounded Cone

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

• 1) $H_t : X \to Y$ is an embedding for each $t \in I$.
• 2) $H_0 = f$.
• 3) $H_1 = g$.

And such a function $H$ is called an isotopy. We now look at an example of isotopies of the circle embedded in a bounded cone. Let $X = S^1$ be the unit circle and let $Y$ be the bounded cone (sometimes referred to as the bounded double cone). First consider the following embeddings of $X$ into $Y$:

Observe that no isotopy can exist between these embeddings. If so, we can move the blue circle to the pink circle. But at some point we must cross the center point of the cone. But this cannot happen since then $H_t$ would not be an embedding for some $t \in [0, 1]$

Now consider the following embeddings of $X$ into $Y$:

For the same reason as above, there is no isotopy between the embedding of the red circle and the embedding of the yellow circle. However, there is an isotopy between the embedding of the red circle and the embedding of the orange circle.