# 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 (négatif) definite, it can be diagonalized to the identity matrix (negative identity matrix) over the integers. 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. Contenu 1 Histoire 2 Idea of proof 3 Rallonges 4 Voir également 5 Remarques 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 {style d'affichage {mathématique {M}}_{P}} of solutions to the anti-self-duality equations on a principal {nom de l'opérateur de style d'affichage {SU} (2)} -bundle {style d'affichage P} over the four-manifold {style d'affichage X} . By the Atiyah–Singer index theorem, the dimension of the moduli space is given by {style d'affichage sombre {mathématique {M}}=8k-3(1-b_{1}(X)+b_{+}(X)),} où {displaystyle c_{2}(P)=k} , {style d'affichage b_{1}(X)} is the first Betti number of {style d'affichage X} et {style d'affichage b_{+}(X)} is the dimension of the positive-definite subspace of {style d'affichage H_{2}(X,mathbb {R} )} with respect to the intersection form. Lorsque {style d'affichage X} is simply-connected with definite intersection form, possibly after changing orientation, one always has {style d'affichage b_{1}(X)=0} et {style d'affichage b_{+}(X)=0} . Thus taking any principal {nom de l'opérateur de style d'affichage {SU} (2)} -bundle with {style d'affichage k=1} , one obtains a moduli space {style d'affichage {mathématique {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 {style d'affichage b_{2}(X)} many. Results of Clifford Taubes and Karen Uhlenbeck show that whilst {style d'affichage {mathématique {M}}} is non-compact, its structure at infinity can be readily described. À savoir, there is an open subset of {style d'affichage {mathématique {M}}} , dire {style d'affichage {mathématique {M}}_{varepsilon }} , such that for sufficiently small choices of parameter {displaystyle varepsilon } , il y a un difféomorphisme {style d'affichage {mathématique {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 {style d'affichage X} with curvature becoming infinitely concentrated at any given single point {style d'affichage 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 {style d'affichage {mathématique {M}}} corresponding to reducible connections could also be described: they looked like cones over the complex projective plane {style d'affichage mathbb {CP} ^{2}} , with its orientation reversed.

It is thus possible to compactify the moduli space as follows: Première, cut off each cone at a reducible singularity and glue in a copy of {style d'affichage mathbb {CP} ^{2}} . Deuxièmement, glue in a copy of {style d'affichage X} itself at infinity. The resulting space is a cobordism between {style d'affichage X} and a disjoint union of {style d'affichage b_{2}(X)} copies de {style d'affichage 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 {style d'affichage 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, si et seulement si, 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 de géométrie différentielle. 18 (2). est ce que je: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 de géométrie différentielle. 26 (3). est ce que je:10.4310/jdg/1214441485. ISSN 0022-040X. ^ Sauter à: a b Donaldson, S. K. (1983). An application of gauge theory to four-dimensional topology. Journal de géométrie différentielle, 18(2), 279-315. ^ Taubes, C. H. (1982). Self-dual Yang–Mills connections on non-self-dual 4-manifolds. Journal de géométrie différentielle, 17(1), 139-170. ^ Uhlenbeck, K. K. (1982). Connections with L p bounds on curvature. Communications in Mathematical Physics, 83(1), 31-42. ^ Sauter à: 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 de géométrie différentielle, 18 (2): 279–315, est ce que je:10.4310/jdg/1214437665, M 0710056, Zbl 0507.57010 Donaldson, S. K; Kronheimer, P. B. (1990), The Geometry of Four-Manifolds, Monographies mathématiques d'Oxford, ISBN 0-19-850269-9 Freed, ré. S; Uhlenbeck, K. (1984), Instantons and Four-Manifolds, Springer Freedman, M; Quinn, F. (1990), Topology of 4-Manifolds, Princeton University Press Scorpan, UN. (2005), The Wild World of 4-Manifolds, American Mathematical Society Categories: Differential topologyTheorems in topologyQuadratic forms

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