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{{Short description|On when a definite intersection form of a smooth 4-manifold is diagonalisable}}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 {{em|integers}}. The original versionJOURNAL, Donaldson, S. K., 1983-01-01, An application of gauge theory to four-dimensional topology,dx.doi.org/10.4310/jdg/1214437665, Journal of Differential Geometry, 18, 2, 10.4310/jdg/1214437665, 0022-040X, free, of the theorem required the manifold to be simply connected, but it was later improved to apply to 4-manifolds with any fundamental group.JOURNAL, Donaldson, S. K., 1987-01-01, The orientation of Yang-Mills moduli spaces and 4-manifold topology,dx.doi.org/10.4310/jdg/1214441485, Journal of Differential Geometry, 26, 3, 10.4310/jdg/1214441485, 120208733, 0022-040X, free,

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 moduli space mathcal{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 where c_2(P)=k, 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,mathbb{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 mathcal{M} of dimension five. (File:Donaldson’s Theorem cobordism.png|thumb|right|Cobordism given by 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.Donaldson, S. K. (1983). An application of gauge theory to four-dimensional topology. Journal of Differential Geometry, 18(2), 279-315. Results of Clifford Taubes and Karen Uhlenbeck show that whilst mathcal{M} is non-compact, its structure at infinity can be readily described.Taubes, C. H. (1982). Self-dual Yang–Mills connections on non-self-dual 4-manifolds. Journal of Differential Geometry, 17(1), 139-170.Uhlenbeck, K. K. (1982). Connections with L p bounds on curvature. Communications in Mathematical Physics, 83(1), 31-42.Uhlenbeck, K. K. (1982). Removable singularities in Yang–Mills fields. Communications in Mathematical Physics, 83(1), 11-29. Namely, there is an open subset of mathcal{M}, say mathcal{M}_{varepsilon}, such that for sufficiently small choices of parameter varepsilon, there is a diffeomorphism 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 xin 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 mathcal{M} corresponding to reducible connections could also be described: they looked like cones over the complex projective plane mathbb{CP}^2. Furthermore, we can count the number of such singular points. Let E be the mathbb{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 = Loplus L^{-1} where L is a complex line bundle over X. Whenever E = Loplus L^{-1} we may compute:1 = k = c_2(E) = c_2(Loplus 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 pmalphain 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)leqtext{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 mathbb{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 mathbb{CP}^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(bigsqcup n(Q) mathbb{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

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References

• {{Citation | last1=Donaldson | first1=S. K. | title=An application of gauge theory to four-dimensional topology |doi=10.4310/jdg/1214437665 | mr=710056 | year=1983 | journal=Journal of Differential Geometry | volume=18 | issue=2 | pages=279–315 |zbl=0507.57010 | doi-access=free }}
• {{citation |first1=S. K. |last1=Donaldson |first2=P. B. |last2=Kronheimer |year=1990 |title=The Geometry of Four-Manifolds |series=Oxford Mathematical Monographs |isbn=0-19-850269-9 }}
• {{citation |first1=D. S. |last1=Freed |first2=K. |last2=Uhlenbeck |authorlink2=Karen Uhlenbeck |year=1984 |title=Instantons and Four-Manifolds |publisher=Springer }}
• {{citation |first1=M. |last1=Freedman |first2=F. |last2=Quinn |year=1990 |title=Topology of 4-Manifolds |publisher=Princeton University Press }}
• {{citation |first=A. |last=Scorpan |year=2005 |title=The Wild World of 4-Manifolds |publisher=American Mathematical Society }}