Let $X$ be a topological space and $E_n(X)$ the space of finite sets of cardinality $\leq n$. It is a theorem of Bott that $E_3(S^1)=S^3$. What is the idea to show that the embedding $S^1\hookrightarrow E_3(S^1)$ sending $x$ to the set $\{ x\}$ is the trefoil knot?
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$\begingroup$ Is this a homework problem? Depending on what you know there's various efficient proofs. One way is to notice $E_3(S^1)$ is 3-dimensional, and it has an action of $SO_2$ since $S^1$ does. This makes it a Seifert Fibred Space, and these were classified by Seifert. So you need only identify the quotient space and classify the singular fibres, in this case there's one with isotropy of order 2, and one with isotropy of order 3. A trefoil is a (2,3)-torus knot. $\endgroup$– Ryan BudneyCommented Jul 16, 2011 at 15:56
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$\begingroup$ student -- I've put back ticks around the formula with brackets. $\endgroup$– algoriCommented Jul 16, 2011 at 15:58
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$\begingroup$ I think it would be best to post your question on math.stackexchange.com. Also, it might be a good idea to give people more background on your motivation for this question. $\endgroup$– Ryan BudneyCommented Jul 16, 2011 at 15:59
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2$\begingroup$ Ryan -- innocent until proven guilty, no? And I think using the classification of Seifert fibered spaces to solve this is a bit of an overkill. $\endgroup$– algoriCommented Jul 16, 2011 at 16:49
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2$\begingroup$ Ryan -- what I meant was: one can not be sure that this is homework until one has found a homework style solution. The solution that uses the classification of Seifert fibered spaces may or may not be such a solution, depending on what was covered in the course. So until further evidence is revealed I'm prepared to give the author the benefit of doubt. Besides, this time of year is not when people usually ask for help with homework solutions. $\endgroup$– algoriCommented Jul 16, 2011 at 20:03
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