Is every simple path in Euclidean space homotopy equivalent to its $\varepsilon$-neighborhood when $\varepsilon$ is small?

Question

I would like to formulate the problem in the simplest situation at first. Given a path $\gamma: I\to \mathbb{R}^n$ and suppose $\gamma$ has no self-intersection, namely, injective. Now we can see $\gamma$ is a topological embedding since $I$ is compact. Denote the image of $\gamma$ by $\Gamma$. Define the $\varepsilon$-neighborhood of $\Gamma$ as $U_\varepsilon(\Gamma) = \{x\mid \mathrm{dist}(x,\Gamma)<\varepsilon\}$, where $\mathrm{dist}(x,\Gamma) = \inf \{|x-y|\mid y\in \Gamma\}$ is well known to be continuous. Therefore $U_\varepsilon(\Gamma)$ is an open neighborhood of $\Gamma$.

If we draw a picture of $U_\varepsilon(\Gamma)$, we see it becomes a thiner and thiner "solid tube" around $\Gamma$ as $\varepsilon$ decreases. We then naturally expect, eventually, the existence of a small $\varepsilon_0$, such that any $U_\varepsilon(\Gamma)$ with $\varepsilon<\varepsilon_0$ will be homotopy equivalent to $\Gamma$; at least, it is likely to be simply connected. However, I can't think of any rigorous proof or counterexample.

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[REFERENCE]

I'm aware of similar questions that have been brought up in MSE. Though none of them is identical to mine, a few offer some hint. In this question someone provided an example of the closed topologists' sine curve, suggesting the restriction on $\Gamma$ cannot be too general. Another question gained no answer but one precious comment, asserting that Čech cohomology is the appropriate tool to apply. This sounds bizarre. If anyone could come up with a more elementary solution, or elaborate on the proof with Čech cohomology, I would be much grateful.

I have literally zero knowledge in differential topology. Some materials seem to implicate the structure called "tubular neighborhood" of a smooth manifold. I don't know whether this is helpful, since in our context there is no smoothness hypothesis involved. The concept "regular neighborhood" might be more relevant, but I doubt I'm able to peruse those literatures independently.

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[POSSIBLE GENERALIZATION]

1. Replace the path with a simple loop.

2. Replace the path with a compact submanifold and replace the Euclidean space with an arbitrary manifold.

Answer 1

You're on the right track with the search term "tubular neighborhood". The theory that's known at this point works for compact subspaces $X\subseteq\mathbb{R}^n$ generally, not necessarily manifold. As a punchline, it is a sufficient condition that $X$ be a compact real semialgebraic set, and that has been known since at least 1985 due to a result of Joseph H.G. Fu.

If $X$ is an embedded manifold in $\mathbb{R}^n$, a sufficient condition is that it be $C^{1,1}$ . My referencing on this last statement is less crisp than I'd like, but see, e.g., on https://hal.science/hal-04083524/document top of page 7.

Section 2.1 of https://arxiv.org/pdf/2209.01654 gives a summary of the semialgebraic case that includes an overview of one way these questions are usually tackled.

The starting point is the following sufficient condition: If each point in a tubular neighborhood $U$ has a unique closest point on $X$, the straight-line homotopy which retracts each point $u$ in $U$ along the line connected it to its closest point $\pi_X(u)\in X$ is a deformation retraction.

The case where the ambient space is not $\mathbb{R}^n$ but instead is a Riemannian manifold is less well-characterized. The above reference claims that $C^{1,1}$ is sufficient if $X$ is embedded in a Riemannian manifold and that may very well be the best result of its type that I know.