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I'm trying learn the concept of a manifold. Wikipedia says a figure 8 is not a manifold but a klein bottle is. What distinguishes the two?

Is it because a Klein bottle is a 4-space object and not the 3-space projection of that object?

Max Yaffe
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The comments already answered the question on a surface level, showing why the klein bottle is a manifold while the figure 8 isn't. But there is a deeper, conceptual similarity (or difference) here worth mentioning.

First consider a circle. Clearly, this is a one-dimensional manifold. Now we can embed the circle into $\mathbb R^2$ in the usual way, the image of this embedding will now be a submanifold. However we can also immerse it by the map $f : S^1\to \mathbb R^2$ defined $f(p)=(\sin(2t),\sin(t))$, where $p \in S^1$ has the coordinates $p = (\cos(t), \sin(t))$. The image will be exactly the figure 8. The map $f$ is locally injective and has nowhere vanishing differential, so one might hope it should also produce a submanifold. However note that $f$ is not (globally) injective, there is some self-intersection, namely $(-1,0)$ and $(1,0)$ get mapped to the same point. Hence it is not necessarily a sub manifold.

A similar thing happens with the klein bottle. The abstract definition, which could be a quotient construction as done here or the usual glueing construction (see below) gives a manifold.

This is the glueing construction of the klein bottle. A square where the upper and lower edge are identified after a flip and the left and right edge are identified not flipped.

And as with the circle before, we can immerse this manifold into $\mathbb R^3$, but we would necessarily get some points of self-intersection. This is illustrated by the famous models of klein bottles you have certainly seen somewhere already. And just as with the immersion of the circle onto the figure 8, the image (in $\mathbb R^3$) is not a manifold, altough the abstract object we started with is.

Sven-Ole Behrend
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    To summarize this for the OP and answer the last sentence in their question: Yes, it's because the Klein bottle is not the thing literally depicted in the usual "3-space projection" you reference. That thing is in fact not a manifold for similar reasons as the figure-8. – fish Mar 27 '25 at 19:19
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    And similarly, for the circle example, you can make it into an embedding into $\mathbb{R}^3$ by adding another dimension separating the two points, $t \mapsto (\sin(2t), \sin(t), \cos(t))$. For the Klein bottle example, you can sort of envision extending the usual picture of a Klein bottle to an embedding in $\mathbb{R}^4$ by using (gray-scale) color as the fourth dimension and painting the model so that at the crossing points the two sections are assigned different colors. – Daniel Schepler Mar 27 '25 at 19:54
  • Thank you for your clear, non-condescending explanation. As you can tell I'm not a mathematician only a ChemEng with a passing knowledge of multidimensional calculus trying to read "Love and Math". I'm trying to learn what a manifold is without understanding other terms yet (homeomorphic, injective). I understand the Klein bottle in 4D is a manifold, but not the 3-d or 2-d image. And I think I see where you going about the Figure 8. If the ant can make a left turn at the crossing it is not a manifold. – Max Yaffe Mar 28 '25 at 19:59
  • Read up on injective. If a function can create the same output from two different inputs the whole output object cannot be a manifold. – Max Yaffe Mar 28 '25 at 20:43
  • @MaxYaffe Your last comment is not correct. For example, consider a line segment that you wrap around multiple times to make a closed loop. This isn't injective but the "output object" is still a circle (a manifold).

    Furthermore, it's possible to have an injective map whose output is not a manifold. For example, you can take the line segment and map it onto the figure-8 without crossing itself (start your pen right next to the intersection point), which you know now isn't a manifold.

    In general you can't say a whole lot about the output of a function just because it is/isn't injective.

     – fish Mar 29 '25 at 21:37