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In the following $\simeq$ means homeomorphic and $\sim$ means homotopically equivalent.

In an exercise I was asked to compute the homology of the space $\mathbb{R}^3 \setminus X$, where $X$ is the curve parametrized by $(t,\,t^2,\,t^3)$ with $t\in \mathbb{R}$. Intuition suggests that this is homotopically equivalent to $\mathbb{R}^3 \setminus \mathbb{R}$ (of which I know the homology) because $X$ is diffeomorphic to $\mathbb{R}$.

The question is: is this always true? Is it true that $A\simeq B$ implies $X\setminus A \sim X\setminus B$? Of course $A, \, B \subseteq X$. There are obvious counterexamples such as if you take $X=\mathbb{R}^3, A= \mathbb{R}^3$ and $B$ a $3$-dimensional ball, because $\mathbb{R}^3\setminus B$ would have non trivial $H_2$. Are there cases where it holds?

Thanks in advance for your help.

toyr99
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    Why do you ask a question if you already have an obvious answer? –  Nov 15 '21 at 10:18
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    I don't know, maybe there are cases where it holds. For example, how can I formally prove that it's true in the exercise with the curve? – toyr99 Nov 15 '21 at 10:24
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    In short, a systematic treatment of problems such as these (with $\Bbb R$ being embedded into $\Bbb R^3$ as an "two-sided infinite line") is by passing to the one-point compactification and treating it as an embedding of $S^1$ into $S^3$ (with one point at infinity). Then you can study the situation with knot theory, and specifically you are looking at the knot complement. This procedure provides all sorts of embeddings $\Bbb R\to\Bbb R^3$ as "two sided lines" that result in non-homotopic complements. See for instance knot groups. –  Nov 15 '21 at 11:01
  • I don't know much about knot theory, so I could be missing some details. –  Nov 15 '21 at 11:04
  • Another counterexample: $[0,1] \setminus {0}$ as compared to $[0,1] \setminus {\frac{1}{2}}$. – John Palmieri Nov 15 '21 at 18:25

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There are even cases where $A \sim B$ implies $X \setminus A \simeq X \setminus B$. I recommend to have a look at the so-called "complememt theorems in shape theory". Nice surveys are available in

Sher, Richard B. "Complement theorems in shape theory." Shape theory and geometric topology. Springer, Berlin, Heidelberg, 1981. 150-168.

Sher, R. B. "Complement theorems in shape theory, II." Geometric topology and shape theory. Springer, Berlin, Heidelberg, 1987. 212-220.

In these theorems we have $X = \mathbb R^n$ or $X = Q = \prod_{n=1}^\infty [0,1]$ = Hilbert cube and $A, B$ compact subsets satisying suitable embedding conditions. In that case we have

$A$ and $B$ have the same shape if and only if $X \setminus A \simeq X \setminus B$.

The concept of shape is an equivalence relation for spaces which is coarser than homotopy equivalence. For example, the Warsaw circle has the same shape as the circle $S^1$.

A related question is

Homotopy type of the complement of a subspace

Paul Frost
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