"Simply connected" like condition for isotopy

Question

I'm looking for literature regarding a locally compact Hausdorff topological space $X$ where any two embeddings $f,g:[0,1]\to X$ with fixed boundary points $f(0)=g(0),f(1)=g(1)$ are isotopic.

If we replace "isotopic" with "homotopic" then this is equivalent to being simply connected.

I'm not even sure what keywords to search for.

Answer 1

I don't know if there is standard terminology for this (I wouldn't be surprised if there is, I'm just not very well-versed in this stuff!), but here's one way to think about it.

Let $x_0 = f(0) = g(0)$ and $x_1 = f(1) = g(1)$.

Let $P = \operatorname{Hom}([0,1],X)$ be the space of continuous maps $[0,1] \to X$, endowed with the compact-open topology, and let $P_{\bullet}$ be the subspace of those paths $\gamma : [0,1] \to X$ such that $\gamma(0) = x_0$ and $\gamma(1) = x_1$. Let $E$ be the subspace of $P$ consisting of topological embeddings, and let $E_{\bullet} = E \cap P_{\bullet}$.

[Quick side note: it's important for what follows that $X$ be "nice" -- your assumption of locally compact Hausdorff is sufficient!]

Now:

• A homotopy from $f$ to $g$ is the same as a path from $f$ to $g$ in $P$
• A homotopy from $f$ to $g$ relative to the endpoints is the same as a path from $f$ to $g$ in $P_{\bullet}$
• An isotopy from $f$ to $g$ is the same as a path from $f$ to $g$ in $E$
• An isotopy from $f$ to $g$ relative to the endpoints is the same as a path from $f$ to $g$ in $E_{\bullet}$

So, the condition you want is equivalent to $E_{\bullet}$ being path connected! Hope this helps :)