Why is four dimensions more difficult?

Question

Some time ago, a professor I know said that there are results in four dimensions that are harder to prove (for example, the Smooth Poincaré Conjecture, though I'm not specifically interested in it). Qualitatively, why and how can it be that in a problem, a specific dimension stands out from the others? Is there a nice, simpler example than Poincaré Conjecture and Supersymmetric Gauge stuff to explain that some specific $n$-dimensional space would be particularly more "stiff" to a result, and different from analogous spaces with just other dimensions?

Answer 1

While the statements of the things that go wrong in 4 dimensions can be easy to state, it is very rare that the proofs do not involve a lot of high-tech machinery. The field only expanded very rapidly in 1982, when two very important theorems were published:

1) Freedman proved a theorem classifying all simply-connected closed topological 4-manifolds up to homeomorphism. Every such manifold has a unimodular (the defining matrix has determinant $\pm 1$) symmetric bilinear form $H^2(X;\Bbb Z) \otimes H^2(X;\Bbb Z) \to \Bbb Z$. They are determined by this pairing (up to isomorphism), as well as an additional number $\kappa \in \{0, 1\}$. Every unimodular symmetric bilinear form may be realized, and the values of $\kappa$ that may arise for a fixed bilinear form $\beta$ are known.

2) Donaldson proved that if $X$ is a closed 4-manifold whose intersection form is definite (this equivalently means that $\beta(x,x)$ is either negative or positive for all $x$), then the intersection form is diagonalizable (this means there is a basis of vectors $x_i \in H^2(X;\Bbb Z)$ so that $\beta(x_i, x_i) = \pm 1$, where $\pm$ is the same sign for all $i$, and $\beta(x_i, x_j) = 0$ for $i \neq j$).

These are especially astounding in combination: there are a great many definite intersection forms which are not diagonalizable over the integers (the number of them is exponential in the rank of $H^2(X)$), and thus an exponentially growing class of non-smoothable 3-manifolds, by Freedman's result.

These new tools provided exciting new directions for 4-manifolds (and the 3-manifolds they bound), which flourished and led to the many fascinating differences between 4D and high-D topology - but the fact that the techniques in (1) and (2) are themselves so complicated should warn you that most results will be.

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Here is an example of the kind of question whose answer became better understood after 1982. You will see the appearance of both gauge theory and surgery theory.

Consider the class of closed oriented 3-manifolds with the same homology as $S^3$; these are the integer homology spheres. We may a define a relation on this class: we say that $Y \sim Y'$ if there is a compact 4-manifold $W$ with boundary $Y \sqcup -Y'$, so that both maps $H_*(Y) \to H_*(W) \leftarrow H_*(Y')$ are isomorphisms. These are called homology cobordisms, and the resulting quotient set is written $\Theta^3_{\Bbb Z}$; it has a group structure given by the connected-sum operation, and is called the homology cobordism group.

Given a closed simply connected 4-manifold $X$, whose intersection form has $\beta(x,x) \in 2\Bbb Z$ for all $x$ (we say $\beta$ is "even"), it is a theorem of Rokhlin that the signature of $\beta$ - the number of positive eigenvalues minus the number of negative values - is a multiple of 16. We write this as $\sigma(X) \in 16\Bbb Z$.

Given a homology 3-sphere $Y$, we may find a closed simply connected 4-manifold $W$ with even intersection form and $\partial W = Y$. It is an algebra theorem that $\sigma(W) \in 8\Bbb Z$. What Rokhlin's theorem promises, however, is that $\sigma(W)$ is independent of $W$ modulo $16$: given another bounding manifold $W'$, glue them together along the boundary to get a closed manifold $X = W \cup_Y -W'$, so we see $\sigma(W) - \sigma(W') = \sigma(X) \in 16\Bbb Z$.

Thus, sending a homology 3-sphere $Y$ to $\sigma(W)/8 \pmod{2}$, where $W$ is a bounding even simply connected 4-manifold, is well-defined; in fact, it descends to a homomorphism $\mu: \Theta^3_{\Bbb Z} \to \Bbb Z/2$. This is called Rokhlin's homomorphism.

I don't have a reference, but in sufficiently old papers you might occasionally see someone ask if $\mu$ is an isomorphism.

So far nothing is special. A variant of Rokhlin's theorem remains true for "spin $(8k+4)$-manifolds" (now Ochanine's theorem), and one can define a similar "signature defect" operation on spin $(8k+3)$-manifolds. (A homology 3-sphere has exactly one spin structure, so this adjective is not threatening.) Furthermore, the notion of homology cobordism makes sense in all dimensions; you can even stick words like "spin" in there if you want.

The thing that's special in 3/4 dimensions, unfortunately for you, is gauge theory. In 1990 Furuta pointed out that an invariant of Fintushel and Stern (a real number associated to a coprime sequence of integers $(a_1, \cdots, a_n)$ called the $R$-invariant of certain homology 3-spheres $\Sigma(a_1, \cdots, a_n)$) implies that $\Theta^3_{\Bbb Z}$ has an infinite subset of elements which are linearly independent; because this group contains a $\Bbb Z^\infty$ subgroup, it must be infinitely generated! That's a far cry from being isomorphic to $\Bbb Z/2$.

In high dimensions, there is, however, a theorem of Kervaire stating that $\Theta^n_{\Bbb Z} = 0$ for $n \geq 4$. It is a classic piece of 'surgery theory', the same tool that killed the Poincare conjecture,

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To link this back up to Donaldson's theorem, here is a mildly explicit example; it is the standard one.

The simplest (in terms of rank) definite intersection form that is not diagonalizable is the $E8$ form. Freedman's result constructs a closed simply connected topological 4-manifold with intersection form E8 by a very inexplicit procedure; I cannot in the slightest draw this space for you. Donaldson's theorem says that there cannot be such a smooth closed 4-manifold.

However, there is such a 4-manifold with boundary. It is kind of hard to describe to someone not immersed in topology already, but it is defined as what's called a "plumbing construction" (a certain way of gluing simple 4-manifolds together to parts of their boundaries) based on the E8 form. Its boundary is called the Poincare homology sphere and written $\Sigma(2,3,5)$, the 3-manifold defined as a manifold as $SO(3)/A_5$, where $A_5$ is the symmetry group of the icosahedron sitting inside 3-dimensional space, centered at the origin. (Its fundamental group is called "$2I$", the binary icosahedral group, and is order 120.) Because $\sigma(E_8) = 8$, this implies that $\mu(\Sigma(2,3,5)) = 1$. When you do this '$E8$ plumbing' in dimension $4k$, but where $k > 1$, then the boundary turns out to be an exotic sphere (so its fundamental group is also trivial, as opposed to just being a homology sphere): it is a different category of 'weirdness'.

Donaldson's theorem proves slightly more: the manifold $\#^n \Sigma(2,3,5)$ is never zero in $\Theta^3_{\Bbb Z}$. Proof: If it were, it would bound a space with the homology of a ball (being zero means there is a homology cobordism to $S^3$, which you fill in with a disc). But you know that on the other hand it bounds $\oplus^n E8$ (taking the connect sum of a bunch of the manifolds we discussed above). Gluing these two 4-manifolds together on their boundaries gives you a smooth closed simply-connected 4-manifold with intersection form $\oplus^n E8$, which is not diagonalizable. This contradicts Donaldson's theorem, and so $\Sigma(2,3,5)$ is non-torsion in $\Theta^3_{\Bbb Z}$.

Freedman's $E8$ manifold was the first example of a non-triangulable manifold. This was proved by Casson, using an invariant closely related to Rokhlin's.