Donaldson's theorem

Template:Short description In mathematics, and especially differential topology and gauge theory, Donaldson's theorem states that a definite intersection form of a closed, oriented, smooth manifold of dimension 4 is diagonalizable. If the intersection form is positive (negative) definite, it can be diagonalized to the identity matrix (negative identity matrix) over the Template:Em. The original version of the theorem required the manifold to be simply connected, but it was later improved to apply to 4-manifolds with any fundamental group.

History

The theorem was proved by Simon Donaldson. This was a contribution cited for his Fields Medal in 1986.

Idea of proof

Donaldson's proof utilizes the Yang–Mills moduli space ${ℳ}_P$ of solutions to the anti-self-duality equations on a principal $\mathrm{SU}(2)$-bundle $P$ over the four-manifold $X$. By the Atiyah–Singer index theorem, the dimension of the moduli space is given by

$\dim ℳ=8k-3(1-b_1(X)+b_{+}(X)),$

where $k=c_2(P)$ is a Chern class, $b_1(X)$ is the first Betti number of $X$, and $b_{+}(X)$ is the dimension of the positive-definite subspace of $H_2(X,ℝ)$ with respect to the intersection form. When $X$ is simply-connected with definite intersection form, possibly after changing orientation, one always has $b_1(X)=0$ and $b_{+}(X)=0$. Thus taking any principal $\mathrm{SU}(2)$-bundle with $k=1$, one obtains a moduli space $ℳ$ of dimension five.

Vissualisation of the cobordism given by the Yang–Mills moduli space in Donaldson's theorem

This moduli space is non-compact and generically smooth, with singularities occurring only at the points corresponding to reducible connections, of which there are exactly $b_2(X)$ many. Results of Clifford Taubes and Karen Uhlenbeck show that whilst $ℳ$ is non-compact, its structure at infinity can be readily described. Namely, there is an open subset of $ℳ$, say ${ℳ}_{\epsilon }$, such that for sufficiently small choices of parameter $\epsilon$, there is a diffeomorphism

${ℳ}_{\epsilon }\stackrel{\cong }{\to }X\times (0,\epsilon )$.

The work of Taubes and Uhlenbeck essentially concerns constructing sequences of ASD connections on the four-manifold $X$ with curvature becoming infinitely concentrated at any given single point $x\in X$. For each such point, in the limit one obtains a unique singular ASD connection, which becomes a well-defined smooth ASD connection at that point using Uhlenbeck's singularity theorem.

Donaldson observed that the singular points in the interior of $ℳ$ corresponding to reducible connections could also be described: they looked like cones over the complex projective plane ${ℂ\mathbb{P}}^2$. Furthermore, we can count the number of such singular points. Let $E$ be the ${ℂ}^2$-bundle over $X$ associated to $P$ by the standard representation of $SU(2)$. Then, reducible connections modulo gauge are in a 1-1 correspondence with splittings $E=L\oplus L^{-1}$ where $L$ is a complex line bundle over $X$. Whenever $E=L\oplus L^{-1}$ we may compute:

$1=k=c_2(E)=c_2(L\oplus L^{-1})=-Q(c_1(L),c_1(L))$,

where $Q$ is the intersection form on the second cohomology of $X$. Since line bundles over $X$ are classified by their first Chern class $c_1(L)\in H^2(X;\mathbb{Z})$, we get that reducible connections modulo gauge are in a 1-1 correspondence with pairs $\pm \alpha \in H^2(X;\mathbb{Z})$ such that $Q(\alpha ,\alpha )=-1$. Let the number of pairs be $n(Q)$. An elementary argument that applies to any negative definite quadratic form over the integers tells us that $n(Q)\le \text{rank}(Q)$, with equality if and only if $Q$ is diagonalizable.

It is thus possible to compactify the moduli space as follows: First, cut off each cone at a reducible singularity and glue in a copy of ${ℂ\mathbb{P}}^2$. Secondly, glue in a copy of $X$ itself at infinity. The resulting space is a cobordism between $X$ and a disjoint union of $n(Q)$ copies of ${ℂ\mathbb{P}}^2$ (of unknown orientations). The signature $\sigma$ of a four-manifold is a cobordism invariant. Thus, because $X$ is definite:

$\text{rank}(Q)=b_2(X)=\sigma (X)=\sigma (⨆n(Q){ℂ\mathbb{P}}^2)\le n(Q)$,

from which one concludes the intersection form of $X$ is diagonalizable.

Extensions

Michael Freedman had previously shown that any unimodular symmetric bilinear form is realized as the intersection form of some closed, oriented four-manifold. Combining this result with the Serre classification theorem and Donaldson's theorem, several interesting results can be seen:

1) Any definite non-diagonalizable intersection form gives rise to a four-dimensional topological manifold with no differentiable structure (so cannot be smoothed).

2) Two smooth simply-connected 4-manifolds are homeomorphic, if and only if, their intersection forms have the same rank, signature, and parity.

See also

• Unimodular lattice
• Donaldson theory
• Yang–Mills equations
• Rokhlin's theorem

References

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