# 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. Contenuti 1 Storia 2 Idea of proof 3 Estensioni 4 Guarda anche 5 Appunti 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 {stile di visualizzazione {matematico {M}}_{P}} of solutions to the anti-self-duality equations on a principal {nome dell'operatore dello stile di visualizzazione {SU} (2)} -bundle {stile di visualizzazione P} over the four-manifold {stile di visualizzazione X} . By the Atiyah–Singer index theorem, the dimension of the moduli space is given by {displaystyle dim {matematico {M}}=8k-3(1-b_{1}(X)+b_{+}(X)),} dove {stile di visualizzazione c_{2}(P)= k} , {stile di visualizzazione b_{1}(X)} is the first Betti number of {stile di visualizzazione X} e {stile di visualizzazione b_{+}(X)} is the dimension of the positive-definite subspace of {stile di visualizzazione H_{2}(X,mathbb {R} )} with respect to the intersection form. quando {stile di visualizzazione X} is simply-connected with definite intersection form, possibly after changing orientation, one always has {stile di visualizzazione b_{1}(X)=0} e {stile di visualizzazione b_{+}(X)=0} . Thus taking any principal {nome dell'operatore dello stile di visualizzazione {SU} (2)} -bundle with {displaystyle k=1} , one obtains a moduli space {stile di visualizzazione {matematico {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 {stile di visualizzazione b_{2}(X)} many. Results of Clifford Taubes and Karen Uhlenbeck show that whilst {stile di visualizzazione {matematico {M}}} is non-compact, its structure at infinity can be readily described. Vale a dire, there is an open subset of {stile di visualizzazione {matematico {M}}} , dire {stile di visualizzazione {matematico {M}}_{varepsilon }} , such that for sufficiently small choices of parameter {displaystyle varepsilon } , c'è un diffeomorfismo {stile di visualizzazione {matematico {M}}_{varepsilon }{xfreccia destra {quad cong quad }}Xtimes (0,varepsilon )} .

The work of Taubes and Uhlenbeck essentially concerns constructing sequences of ASD connections on the four-manifold {stile di visualizzazione X} with curvature becoming infinitely concentrated at any given single point {stile di visualizzazione 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 {stile di visualizzazione {matematico {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: Primo, cut off each cone at a reducible singularity and glue in a copy of {displaystyle mathbb {CP} ^{2}} . In secondo luogo, glue in a copy of {stile di visualizzazione X} itself at infinity. The resulting space is a cobordism between {stile di visualizzazione X} and a disjoint union of {stile di visualizzazione b_{2}(X)} copie di {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 {stile di visualizzazione 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 solo 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". Giornale di geometria differenziale. 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". Giornale di geometria differenziale. 26 (3). doi:10.4310/jdg/1214441485. ISSN 0022-040X. ^ Salta su: a b Donaldson, S. K. (1983). An application of gauge theory to four-dimensional topology. Giornale di geometria differenziale, 18(2), 279-315. ^ Taubes, C. H. (1982). Self-dual Yang–Mills connections on non-self-dual 4-manifolds. Giornale di geometria differenziale, 17(1), 139-170. ^ Uhlenbeck, K. K. (1982). Connections with L p bounds on curvature. Communications in Mathematical Physics, 83(1), 31-42. ^ Salta su: 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", Giornale di geometria differenziale, 18 (2): 279–315, doi:10.4310/jdg/1214437665, SIG 0710056, Zbl 0507.57010 Donaldson, S. K.; Kronheimer, P. B. (1990), The Geometry of Four-Manifolds, Monografie matematiche di 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, UN. (2005), The Wild World of 4-Manifolds, American Mathematical Society Categories: Differential topologyTheorems in topologyQuadratic forms

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