Donaldson's theorem

Donaldson's theorem 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 integers. 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] Contents 1 History 2 Idea of proof 3 Extensions 4 See also 5 Notes 6 References 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 {displaystyle {mathcal {M}}_{P}} of solutions to the anti-self-duality equations on a principal {displaystyle operatorname {SU} (2)} -bundle {displaystyle P} over the four-manifold {displaystyle X} . By the Atiyah–Singer index theorem, the dimension of the moduli space is given by {displaystyle dim {mathcal {M}}=8k-3(1-b_{1}(X)+b_{+}(X)),} where {displaystyle c_{2}(P)=k} , {displaystyle b_{1}(X)} is the first Betti number of {displaystyle X} and {displaystyle b_{+}(X)} is the dimension of the positive-definite subspace of {displaystyle H_{2}(X,mathbb {R} )} with respect to the intersection form. When {displaystyle X} is simply-connected with definite intersection form, possibly after changing orientation, one always has {displaystyle b_{1}(X)=0} and {displaystyle b_{+}(X)=0} . Thus taking any principal {displaystyle operatorname {SU} (2)} -bundle with {displaystyle k=1} , one obtains a moduli space {displaystyle {mathcal {M}}} of dimension five.

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 {displaystyle b_{2}(X)} many.[3] Results of Clifford Taubes and Karen Uhlenbeck show that whilst {displaystyle {mathcal {M}}} is non-compact, its structure at infinity can be readily described.[4][5][6] Namely, there is an open subset of {displaystyle {mathcal {M}}} , say {displaystyle {mathcal {M}}_{varepsilon }} , such that for sufficiently small choices of parameter {displaystyle varepsilon } , there is a diffeomorphism {displaystyle {mathcal {M}}_{varepsilon }{xrightarrow {quad cong quad }}Xtimes (0,varepsilon )} .

The work of Taubes and Uhlenbeck essentially concerns constructing sequences of ASD connections on the four-manifold {displaystyle X} with curvature becoming infinitely concentrated at any given single point {displaystyle 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.[6][3] Donaldson observed that the singular points in the interior of {displaystyle {mathcal {M}}} corresponding to reducible connections could also be described: they looked like cones over the complex projective plane {displaystyle mathbb {CP} ^{2}} , with its orientation reversed.

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 {displaystyle mathbb {CP} ^{2}} . Secondly, glue in a copy of {displaystyle X} itself at infinity. The resulting space is a cobordism between {displaystyle X} and a disjoint union of {displaystyle b_{2}(X)} copies of {displaystyle mathbb {CP} ^{2}} with its orientation reversed. The intersection form of a four-manifold is a cobordism invariant up to isomorphism of quadratic forms, from which one concludes the intersection form of {displaystyle X} is diagonalisable.

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 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 ^ Donaldson, S. K. (1983-01-01). "An application of gauge theory to four-dimensional topology". Journal of Differential Geometry. 18 (2). doi:10.4310/jdg/1214437665. ISSN 0022-040X. ^ Donaldson, S. K. (1987-01-01). "The orientation of Yang-Mills moduli spaces and 4-manifold topology". Journal of Differential Geometry. 26 (3). doi:10.4310/jdg/1214441485. ISSN 0022-040X. ^ Jump up to: a b Donaldson, S. K. (1983). An application of gauge theory to four-dimensional topology. Journal of Differential Geometry, 18(2), 279-315. ^ 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. ^ Jump up to: a b Uhlenbeck, K. K. (1982). Removable singularities in Yang–Mills fields. Communications in Mathematical Physics, 83(1), 11-29. References Donaldson, S. K. (1983), "An application of gauge theory to four-dimensional topology", Journal of Differential Geometry, 18 (2): 279–315, doi:10.4310/jdg/1214437665, MR 0710056, Zbl 0507.57010 Donaldson, S. K.; Kronheimer, P. B. (1990), The Geometry of Four-Manifolds, Oxford Mathematical Monographs, ISBN 0-19-850269-9 Freed, D. S.; Uhlenbeck, K. (1984), Instantons and Four-Manifolds, Springer Freedman, M.; Quinn, F. (1990), Topology of 4-Manifolds, Princeton University Press Scorpan, A. (2005), The Wild World of 4-Manifolds, American Mathematical Society Categories: Differential topologyTheorems in topologyQuadratic forms