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The title quotes C.H. Taubes in the Princeton Companion to Mathematics (p.404). I found this surprising despite the natural lower-dimensional analog (a typical pair of loops in $\mathbb{R}^2$ will intersect at finitely many points).

Q. Can the 4D claim be explained intuitively?

Or is it necessary to master the details of intersection forms to render the claim natural/obvious?

Is there an analog in higher dimensions, or is 4D special?

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    $\begingroup$ The natural statement you're looking for is Thom's transversality theorem: given two closed submanifolds $M_1, M_2 \subset N$, of dimensions $k_1, k_2, n$, respectively, they may be modified by an arbitrarily small homotopy so that $M_1$ and $M_2$ are transverse, meaning that at any point of interesection, the tangent directions in $M_1$ and $M_2$ are jointly enough to give every tangent direction of $N$. Then it is a lemma that once they are transverse, $M_1 \cap M_2$ is a compact manifold of dimension $n - k_1 - k_2$. In particular, whenever $k_1 + k_2 = n$, we get a finite set of points. $\endgroup$
    – mme
    Commented Sep 2, 2018 at 20:42
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    $\begingroup$ It might be useful to translate @MikeMiller's comment into the fact that codimension (defined as $n-k$ for a $k$-dimensional submanifold in an $n$-dimensional manifold) is additive under transverse intersection. Heuristically, the codimension of a submanifold can be thought of as the "number of equations" that it satisfies, so the intersection of two (transverse) submanifolds of codimension $m_1$ and $m_2$ should be the solution to a system of $m_1+m_2$ equations (in this heuristic, transversality guarantees that the equations are independent). $\endgroup$
    – j.c.
    Commented Sep 2, 2018 at 21:39
  • $\begingroup$ @j.c.: The "number of equations" viewpoint is useful. Thanks. $\endgroup$
    – Joseph O'Rourke
    Commented Sep 3, 2018 at 0:32
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    $\begingroup$ This is the background of the physicist’s notion of degrees of freedom: you start with 4 degrees of freedom, and subtract two if you want to lie on the first codinension 2 hypersurface, and subtract 2 more to ensure you lie on the second hypersurface. This goes horribly wrong if, for example, the two surfaces are the same. Enter transversality. $\endgroup$
    – Anthony Quas
    Commented Sep 3, 2018 at 2:47
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    $\begingroup$ One more comment: the mathematical basis of all of this is the implicit function theorem. $\endgroup$
    – Anthony Quas
    Commented Sep 3, 2018 at 2:48

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