Donaldson's theorem

In mathematics, and especially gauge theory, Donaldson's theorem states that a definite intersection form of a compact, oriented, simply connected, 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.

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 ${\displaystyle {\mathcal {M}}_{P}}$ of solutions to the anti-self-duality equations on a principal ${\displaystyle \operatorname {SU} (2)}$-bundle ${\displaystyle P}$ over the four-manifold. By the Atiyah-Singer index theorem, the dimension of the moduli space is given by

${\displaystyle \dim {\mathcal {M}}=8k-3(1-b_{1}(X)+b_{+}(X)),}$

where ${\displaystyle c_{2}(P)=k}$, ${\displaystyle b_{1}(X)}$ is the first Betti number of ${\displaystyle X}$ and ${\displaystyle b_{+}(X)}$ is the dimension of the positive-definite subspace of ${\displaystyle H_{2}(X,\mathbb {R} )}$ with respect to the intersection form. When ${\displaystyle X}$ is simply-connected with definite intersection form, possibly after changing orientation, one always has ${\displaystyle b_{1}(X)=0}$ and ${\displaystyle b_{+}(X)=0}$. Thus taking any principal ${\displaystyle \operatorname {SU} (2)}$-bundle with ${\displaystyle k=1}$, one obtains a moduli space ${\displaystyle {\mathcal {M}}}$ 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 ${\displaystyle b_{2}(X)}$ many.[1] Results of Clifford Taubes and Karen Uhlenbeck show that whilst ${\displaystyle {\mathcal {M}}}$ is non-compact, its structure at infinity can be readily described.[2][3][4] Namely, there is an open subset of ${\displaystyle {\mathcal {M}}}$, say ${\displaystyle {\mathcal {M}}_{\varepsilon }}$, such that for sufficiently small choices of parameter ${\displaystyle \varepsilon }$, there is a diffeomorphism

${\displaystyle {\mathcal {M}}_{\varepsilon }{\xrightarrow {\quad \cong \quad }}X\times (0,\varepsilon )}$.

The work of Taubes and Uhlenbeck essentially concerns constructing sequences of ASD connections on the four-manifold ${\displaystyle X}$ with curvature becoming infinitely concentrated at any given single point ${\displaystyle x\in X}$. 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.[4][1]

Donaldson observed that the singular points in the interior of ${\displaystyle {\mathcal {M}}}$ 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: First, cut off each cone at a reducible singularity and glue in a copy of ${\displaystyle \mathbb {CP} ^{2}}$. Secondly, glue in a copy of ${\displaystyle X}$ itself at infinity. The resulting space is a cobordism between ${\displaystyle X}$ and a disjoint union of ${\displaystyle b_{2}(X)}$ copies of ${\displaystyle \mathbb {CP} ^{2}}$ 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 ${\displaystyle X}$ 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, if and only if, their intersection forms have the same rank, signature, and parity.

See also

• Unimodular lattice
• Donaldson theory
• Yang-Mills equations

Notes

1. Donaldson, S. K. (1983). An application of gauge theory to four-dimensional topology. Journal of Differential Geometry, 18(2), 279-315.
2. Taubes, C. H. (1982). Self-dual Yang-Mills connections on non-self-dual 4-manifolds. Journal of Differential Geometry, 17(1), 139-170.
3. Uhlenbeck, K. K. (1982). Connections withL p bounds on curvature. Communications in Mathematical Physics, 83(1), 31-42.
4. 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 of Differential Geometry, 18 (2): 279–315, doi:10.4310/jdg/1214437665, MR 0710056, Zbl 0507.57010
• Donaldson, S. K.; Kronheimer, P. B. (1990), The Geometry of Four-Manifolds, Oxford Mathematical Monographs, 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, A. (2005), The Wild World of 4-Manifolds, American Mathematical Society