Perelman's Theorem, "Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere", is about the numbers $x,y,z,u$ that satisfy the equation $x^2+y^2+z^2+u^2=1$, the 3-sphere . How would you phrase the statement of the theorem purely analytically on the level of Calculus I (or II)?
calculus level formulation of Perelman's theorem
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calculus
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0I don't think you really can, just because it's not clear how to define "simply connected closed 3-manifold", nor "homeomorphic". To a first very crude approximation it is basically "level set of a scalar equation in 4 variables", but that's missing a lot of issues. – 2017-01-19
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0@Ian would you elaborate a bit on this *crude approximation* of level sets even if it misses so many issues? – 2017-01-19
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0My phrasing was bad even for my crude suggestion. A $k$-manifold looks like a solution to $n$ (independent) scalar equations in $n+k$ variables, for some $n$ in $\{ 0,1,2,\dots \}$. However, such a thing can totally fail to be simply connected/closed, as we can see even in the case of a 1-manifold. – 2017-01-19