# 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 (negativo) 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. Conteúdo 1 História 2 Idea of proof 3 Extensões 4 Veja também 5 Notas 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 {estilo de exibição {matemática {M}}_{P}} of solutions to the anti-self-duality equations on a principal {nome do operador de estilo de exibição {SU} (2)} -bundle {estilo de exibição P} over the four-manifold {estilo de exibição X} . By the Atiyah–Singer index theorem, the dimension of the moduli space is given by {estilo de exibição escurecer {matemática {M}}=8k-3(1-b_{1}(X)+b_{+}(X)),} Onde {estilo de exibição c_{2}(P)=k} , {estilo de exibição b_{1}(X)} is the first Betti number of {estilo de exibição X} e {estilo de exibição b_{+}(X)} is the dimension of the positive-definite subspace of {estilo de exibição H_{2}(X,mathbb {R} )} with respect to the intersection form. Quando {estilo de exibição X} is simply-connected with definite intersection form, possibly after changing orientation, one always has {estilo de exibição b_{1}(X)=0} e {estilo de exibição b_{+}(X)=0} . Thus taking any principal {nome do operador de estilo de exibição {SU} (2)} -bundle with {displaystyle k=1} , one obtains a moduli space {estilo de exibição {matemática {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 {estilo de exibição b_{2}(X)} many. Results of Clifford Taubes and Karen Uhlenbeck show that whilst {estilo de exibição {matemática {M}}} is non-compact, its structure at infinity can be readily described. Nomeadamente, there is an open subset of {estilo de exibição {matemática {M}}} , dizer {estilo de exibição {matemática {M}}_{varepsilon }} , such that for sufficiently small choices of parameter {displaystyle varepsilon } , existe um difeomorfismo {estilo de exibição {matemática {M}}_{varepsilon }{seta para a direita {quad cong quad }}Xtimes (0,varepsilon )} .

The work of Taubes and Uhlenbeck essentially concerns constructing sequences of ASD connections on the four-manifold {estilo de exibição X} with curvature becoming infinitely concentrated at any given single point {estilo de exibição 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 {estilo de exibição {matemática {M}}} corresponding to reducible connections could also be described: they looked like cones over the complex projective plane {estilo de exibição mathbb {PC} ^{2}} , with its orientation reversed.

It is thus possible to compactify the moduli space as follows: Primeiro, cut off each cone at a reducible singularity and glue in a copy of {estilo de exibição mathbb {PC} ^{2}} . Em segundo lugar, glue in a copy of {estilo de exibição X} itself at infinity. The resulting space is a cobordism between {estilo de exibição X} and a disjoint union of {estilo de exibição b_{2}(X)} cópias de {estilo de exibição mathbb {PC} ^{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 {estilo de exibição 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, se e apenas se, 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". Jornal de Geometria Diferencial. 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". Jornal de Geometria Diferencial. 26 (3). doi:10.4310/jdg/1214441485. ISSN 0022-040X. ^ Saltar para: a b Donaldson, S. K. (1983). An application of gauge theory to four-dimensional topology. Jornal de Geometria Diferencial, 18(2), 279-315. ^ Taubes, C. H. (1982). Self-dual Yang–Mills connections on non-self-dual 4-manifolds. Jornal de Geometria Diferencial, 17(1), 139-170. ^ Uhlenbeck, K. K. (1982). Connections with L p bounds on curvature. Communications in Mathematical Physics, 83(1), 31-42. ^ Saltar para: 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", Jornal de Geometria Diferencial, 18 (2): 279–315, doi:10.4310/jdg/1214437665, SENHOR 0710056, Zbl 0507.57010 Donaldson, S. K.; Kronheimer, P. B. (1990), The Geometry of Four-Manifolds, Monografias Matemáticas Oxford, 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, UMA. (2005), The Wild World of 4-Manifolds, American Mathematical Society Categories: Differential topologyTheorems in topologyQuadratic forms

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