Donaldson's theorem

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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 diagonalizable. 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]

History

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The theorem was proved by Simon Donaldson. This was a contribution cited for his Fields medal in 1986.

Idea of proof

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Donaldson's proof utilizes the moduli space   of solutions to the anti-self-duality equations on a principal  -bundle   over the four-manifold  . By the Atiyah–Singer index theorem, the dimension of the moduli space is given by

 

where   is a Chern class,   is the first Betti number of  , and   is the dimension of the positive-definite subspace of   with respect to the intersection form. When   is simply-connected with definite intersection form, possibly after changing orientation, one always has   and  . Thus taking any principal  -bundle with  , one obtains a moduli space   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   many.[3] Results of Clifford Taubes and Karen Uhlenbeck show that whilst   is non-compact, its structure at infinity can be readily described.[4][5][6] Namely, there is an open subset of  , say  , such that for sufficiently small choices of parameter  , there is a diffeomorphism

 .

The work of Taubes and Uhlenbeck essentially concerns constructing sequences of ASD connections on the four-manifold   with curvature becoming infinitely concentrated at any given single point  . 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   corresponding to reducible connections could also be described: they looked like cones over the complex projective plane  . Furthermore, we can count the number of such singular points. Let   be the  -bundle over   associated to   by the standard representation of  . Then, reducible connections modulo gauge are in a 1-1 correspondence with splittings   where   is a complex line bundle over  .[3] Whenever   we may compute:

 ,

where   is the intersection form on the second cohomology of  . Since line bundles over   are classified by their first Chern class  , we get that reducible connections modulo gauge are in a 1-1 correspondence with pairs   such that  . Let the number of pairs be  . An elementary argument that applies to any negative definite quadratic form over the integers tells us that  , with equality if and only if   is diagonalizable.[3]

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  . Secondly, glue in a copy of   itself at infinity. The resulting space is a cobordism between   and a disjoint union of   copies of   (of unknown orientations). The signature   of a four-manifold is a cobordism invariant. Thus, because   is definite:

 ,

from which one concludes the intersection form of   is diagonalizable.

Extensions

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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

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Notes

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  1. ^ 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.
  2. ^ 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. S2CID 120208733.
  3. ^ a b c d Donaldson, S. K. (1983). An application of gauge theory to four-dimensional topology. Journal of Differential Geometry, 18(2), 279-315.
  4. ^ Taubes, C. H. (1982). Self-dual Yang–Mills connections on non-self-dual 4-manifolds. Journal of Differential Geometry, 17(1), 139-170.
  5. ^ Uhlenbeck, K. K. (1982). Connections with L p bounds on curvature. Communications in Mathematical Physics, 83(1), 31-42.
  6. ^ a b Uhlenbeck, K. K. (1982). Removable singularities in Yang–Mills fields. Communications in Mathematical Physics, 83(1), 11-29.

References

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