View source for Exercises:Mond - Topology - 2/Section B/Question 8

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==Section B==
===Question 8===
</noinclude>Here is a proof that the [[Klein bottle]] can be [[topological embedding|topologically embedded]] in {{M|\mathbb{R}^4}}. The drawing on the right below shows the image of a failed embedding of the Klein bottle in {{M|\mathbb{R}^3}}, {{M|f:K\rightarrow\mathbb{R}^3}}. The surface has to pass through itself in order for the two ends of the cylinder to glue together as required. 

To embed {{M|K}} in {{M|\mathbb{R}^4}} the map {{M|f}} needs one further component, {{M|f_4:K\rightarrow\mathbb{R} }}. You can describe {{M|f_4}} by specifying its value at each point of {{M|K}}. By taking care that {{M|f_4}} distinguishes points on {{M|K}} which have the same image under {{M|f}} in {{M|\mathbb{R}^3}} (i.e. in the picture), you obtain an embedding. Copy the right hand picture and write suitable values of {{M|f_4}} on your copy. Your {{M|f_4}} should be [[continuous]].
====Solution====
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| [[File:KBottleEmbeddingForAssignment1.JPG|thumb|My preferred form]]
| [[File:KBottleEmbeddingForAssignment2.JPG|thumb|What is probably wanted]]
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| The idea here is that I use a green line's thickness to denote the value of {{M|f_4}}, and assume it is 0 unless specified. In this image we associate the thinner "component" (I do not want to invoke a formal definition) of the bottle with the green line shown inside of it, and we show the bulb-like component with a line representing {{M|1=f_4=0}}. The idea is we "lift" the smaller component over the bulb where they would otherwise intersect, then lower it back down afterwards.
| I sense this is the kind of thing that you're after (Mond, my tutor, has drawn a diagram like this before). We assign 0 to everything, and as the thinner component comes to intersect the "bulb", we raise it, continuously, which I've tried to convey by first raising from {{M|0}} to {{M|\frac{1}{2} }} then {{M|1}} then back down to {{M|\frac{1}{2} }} then 0.

The blue line is intended to convey the bulb stays at {{M|1=f_4=0}}
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=====Lower dimensional analogy=====
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| [[File:LowerDimensionalAssignmentAnalogy.JPG|thumb|Forgive the badly-drawn-ness of this image, I apologise profusely for my inability to judge the length of the word "intersection" not once but twice.]]
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| I want to convey understanding. This is a very similar lower-dimensional analogy, we have the figure-8 / {{M|\infty}} space, and we want to make it into an embedding in {{M|\mathbb{R}^3}}. To do this we first take the {{M|\infty}} flat into {{M|\mathbb{R}^3}} then pick one of the components of the line (it doesn't matter which) and in a small area alter it's new coordinate so it goes "over" where it'd otherwise intersect.
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==Notes==
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==References==
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