Donaldson's theorem explained

In mathematics, and especially differential topology and gauge theory, Donaldson's theorem states that a definite intersection form of a compact, oriented, smooth manifold of dimension 4 is diagonalisable. If the intersection form is positive (negative) definite, it can be diagonalized to the identity matrix (negative identity matrix) over the . The original version^[1] of the theorem required the manifold to be simply connected, but it was later improved to apply to 4-manifolds with any fundamental group.^[2]

History

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

Idea of proof

l{M} _{P} of solutions to the anti-self-duality equations on a principal \operatorname{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 l{M} = 8 k - 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 \operatorname{SU}( 2 ) -bundle with k =1 , one obtains a moduli space l{M} of dimension five. 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.^[3] Results of Clifford Taubes and Karen Uhlenbeck show that whilst l{M} is non-compact, its structure at infinity can be readily described.^{[4] [5] [6]} Namely, there is an open subset of l{M} , say l{M} _{\varepsilon} , such that for sufficiently small choices of parameter \varepsilon , there is a diffeomorphisml{M} _{\varepsilon} \xrightarrow{ \cong } X x ( 0 ,\varepsilon) .

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 removable singularity theorem.

Donaldson observed that the singular points in the interior of

l{M} corresponding to reducible connections could also be described: they looked like cones over the complex projective plane CP ^{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 ⊕ L ^{-1} where L is a complex line bundle over X . Whenever E = L ⊕ L ^{-1} we may compute:1 = k = c _{2(E)} = c _{2(L ⊕} 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;} Z ) , we get that reducible connections modulo gauge are in a 1-1 correspondence with pairs \pm\alpha\in H ^{2(X;} 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)\leq 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

CP ^{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 CP ^{2} (of unknown orientations). The signature \sigma of a four-manifold is a cobordism invariant. Thus, because X is definite:rank (Q) = b _{2(X)} = \sigma(X) = \sigma( s qcup n(Q) CP ^{2)} \leq 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 indefinite 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

Notes and References

1. Donaldson . S. K. . 1983-01-01 . An application of gauge theory to four-dimensional topology . Journal of Differential Geometry . 18 . 2 . 10.4310/jdg/1214437665 . 0022-040X. free .
2. Donaldson . S. K. . 1987-01-01 . The orientation of Yang-Mills moduli spaces and 4-manifold topology . Journal of Differential Geometry . 26 . 3 . 10.4310/jdg/1214441485 . 120208733 . 0022-040X. free .
3. Donaldson, S. K. (1983). An application of gauge theory to four-dimensional topology. Journal of Differential Geometry, 18(2), 279-315.
4. Taubes, C. H. (1982). Self-dual Yang–Mills connections on non-self-dual 4-manifolds. Journal of Differential Geometry, 17(1), 139-170.
5. Uhlenbeck, K. K. (1982). Connections with L p bounds on curvature. Communications in Mathematical Physics, 83(1), 31-42.
6. Uhlenbeck, K. K. (1982). Removable singularities in Yang–Mills fields. Communications in Mathematical Physics, 83(1), 11-29.