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What are the possible topological shapes (i.e. up to homeomorphism) of a finite union of open balls in $\mathbb{R}^n?$

For example, for $n = 1$, open balls are just open intervals and a finite union of open intervals is just a disjoint union of open intervals (the union of two intervals that overlap is again an interval)

For $n = 2$, we can have a disjoint union of open balls, but also an annulus, or an "annulus with many holes", a disjoint union of "annuli with many holes" and maybe something else.

Is there a classification of these possible shapes?

EDIT

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CNS709
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1 Answers1

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First a definition: A manifold $M$ is called tame if it is homeomorphic to the interior of a compact manifold $\bar{M}$ with boundary.

A connected tame surface $M$ is characterized by its genus (i.e. genus of $\bar{M}$), orientability and the number of boundary components of $\bar{M}$.

For $n=2$ the classification will be: A surface $S$ is homeomorphic to a finite union of open round disks in $R^2$ if and only if $S$ is a tame surface of genus zero.

In higher dimensions things are more complicated but one can prove that an $n$-dimensional manifold $M$ is homeomorphic to a finite union of open balls in $R^n$ if and only if $M$ is tame and the manifold $\bar{M}$ is homeomorphic to a piecewise-linear submanifold in $R^n$.

In order to appreciate the complexity of this class of manifolds, note that every finite $k$-dimensional simplicial complex is homotopy-equivalent to a finite union of open round balls in $R^n$, $n=2k+1$. Already for $n=3$, a topological classification of finite unions of round balls in $R^3$ is hopeless.

Moishe Kohan
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  • I appreciate the work you put into this, but how is that shown annulus a surface of genus zero, let alone the variants with more holes? – Jyrki Lahtonen Sep 06 '19 at 18:16
  • @JyrkiLahtonen: Are you asking me why the annulus has genus zero? I can write proofs of the statements that I wrote, but given the wording of the question revealing the the OP is probably unfamiliar even with the classification of surfaces, it is probably pointless. – Moishe Kohan Sep 06 '19 at 18:27
  • I guess I am :-). Rechecking the definition. Still thinking about the old rule that genus calculates the number of holes, but that apparently only applies to closed surfaces (which is the only case I have ever dealt with). Or thinking that for a surface knowing the genus means knowing the Betti numbers. – Jyrki Lahtonen Sep 06 '19 at 18:43
  • Never mind. Guess I only ever had to use the concept with non-closed surfaces :-) – Jyrki Lahtonen Sep 06 '19 at 18:50
  • @JyrkiLahtonen: Genus is the maximal number (more precisely, cardinality) of pairwise disjoint nonseparating simple loops a surface contains. This definition works for both compact and noncompact surfaces. – Moishe Kohan Sep 06 '19 at 23:11