nLab Donaldson's theorem

Idea

Donaldson’s theorem states that the intersection form of a compact orientable 4-manifold is diagonalizable. It was proven by Simon Donaldson in 1983 in Donaldson 83 (with additionaly using simple-connectedness) and improved in 1987 in Donaldson 87 (without using simple-connectedness). It became one of the reasons for him getting the Fields Medal in 1986 due to the new method of using the moduli space of the anti-self-dual Yang-Mills equations (ASDYM equations) to study the base manifold, which became the ground of Donaldson theory.

Central consequences of Donaldson’s theorem are the existence of exotic smooth structures on euclidean space in four dimensions and the failure of the h-cobordism theorem in four dimensions.

Sketch of proof

Let $X$ be a compact orientable 4-manifold and $P\twoheadrightarrow X$ be a principal $SU(2)$-bundle. According to the Atiyah-Singer index theorem, the dimension of the moduli space $\mathcal{M}^-$ of anti-self-dual Yang-Mills connections (ASDYM connections) is given by:

$$\dim\mathcal{M}^- =8\langle c_2(P),[X]\rangle -3(1-b_1(X)+b_+(X))$$

with the second Chern class $c_2(P)\in H^4(X;\mathbb{Z})$, the fundamental class $[X]\in H_4(X;\mathbb{Z})$ given by the orientation of $X$, the Kronecker pairing $\langle-,-\rangle\colon H^4(X;\mathbb{Z})\times H_4(X;\mathbb{Z})\rightarrow\mathbb{Z}$ (which is often obmitted), the first Betti number $b_1(X)=\dim H_1(X;\mathbb{R})$ and the dimension $b_+(X)$ of the positive definite subspace of $H^2(X;\mathbb{R})$ with respect to the intersection form.

If considering a simply connected $X$ as in the original version, then $H_1(X;\mathbb{Z})\cong\pi_1(X)^\mathrm{ab}$ and hence $b_1(X)=0$, which simplifies the formula. Consider a principal $SU(2)$-bundle $P$ with $\langle c_2(P),[X]\rangle=1$, hence $\dim\mathcal{M}^-=5$. Reducible connections modulo gauge, which are the singularities of $\mathcal{M}^-$, correspond to decompositions $P\times_{SU(2)}\mathbb{C}^2\cong L\oplus L^{-1}$ with a complex line bundle $L\twoheadrightarrow X$, which implies:

$$c_2(P) =c_2(P\times_{SU(2)}\mathbb{C}^2) =c_1(L\oplus L^{-1}) =-c_1(L)\smile c_1(L);$$

$$Q(c_1(L),c_1(L)) =-\langle c_1(L)\smile c_1(L),[X]\rangle =-\langle c_2(P),[X]\rangle =-1.$$

Let $n(Q)$ be the number of pairs $\pm\alpha\in H^2(X;\mathbb{Z})$ with $Q(\alpha,\alpha)=-1$, then $n(Q)\leq\operatorname{rk}(Q)$ with equality iff $Q$ is diagonalizable.

$\mathcal{M}^-$ resembles the base manifold $X$ at infinity and the complex projective plane $\mathbb{C}P^2$ around the $n(Q)$ singularities. Gluing them in yields a compactification, which contains a cobordism between $X$ and $n(Q)$ disjoint $\mathbb{C}P^2$. Since the signature is a coborism invariant, this implies:

$$\operatorname{rk}(Q) =\sigma(X) =\sigma\left(\coprod_{n(Q)}\mathbb{C}P^2\right) =n(Q)\sigma(\mathbb{C}P^2) =n(Q).$$

Application to the 4-sphere

The 4-sphere $S^4$ is a compact orientable 4-manifold. Although its intersection form is trivial, it gives insight into the concepts used in the proof and their relation to each other. The principal $SU(2)$-bundle over $S^4$ with Chern class $1$ is the quaternionic Hopf fibration $S^7\twoheadrightarrow S^4$. It can abstractly be defined as the Hopf construction of the group structure on $S^3\cong SU(2)$ or more directly by the unit quaternions $Sp(1)\cong SU(2)$ acting on both components of $S^7\subset\mathbb{H}^2$ and the projection on the orbit space, which is the quaternionic projective space $\mathbb{H}P^1\cong S^4$. The adjoint vector bundle $Ad(P)=P\times_{\mathfrak{su}(2)}SU(2)$ is the quaternionic tautological line bundle (considered as complex plane bundle) $\gamma_{\mathbb{H}}^{1,1}$ again defined using the identification $S^4\cong\mathbb{H}P^1$, in which points in $S^4$ correspond to one-dimensional linear subspaces of $\mathbb{H}^2$. One has $H_1(S^4;\mathbb{Z})=1$ and $H^2(S^4;\mathbb{Z})=1$, hence $b_1(S^4)=0$ and $b_+(S^4)=0$.According to the above formula, this yields $\dim\mathcal{M}^-=5$, which for instantons corresponds to the four freedoms regarding position and the one freedom regarding size.

See also:

• Kronheimer-Mrowka basic class

References

• Simon Donaldson. An application of gauge theory to four-dimensional topology. In: Journal of Differential Geometry. 18. Jahrgang, Nr. 2, 1. Januar 1983, doi:10.4310/jdg/1214437665
• Simon Donaldson. The orientation of Yang-Mills moduli spaces and 4-manifold topology. In: Journal of Differential Geometry. 26. Jahrgang, Nr. 3, 1. Januar 1987, doi:10.4310/jdg/1214441485

Related concepts

• Wikipedia, Donaldson’s theorem