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 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]

History[edit]

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

Idea of proof[edit]

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 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 , 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 . Secondly, glue in a copy of itself at infinity. The resulting space is a cobordism between and a disjoint union of copies of 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 is diagonalisable.

Extensions[edit]

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[edit]

Notes[edit]

  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.
  3. ^ a b 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[edit]