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 (negative) definite, it can be diagonalized to the identity matrix (negative identity matrix) over the . 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.

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 $\mathcal_P$ of solutions to the anti-self-duality equations on a principal $\operatorname(2)$-bundle $P$ over the four-manifold $X$. By the Atiyah–Singer index theorem, the dimension of the moduli space is given by

:$\dim \mathcal = 8k - 3(1-b_1(X) + b_+(X)),$

where $c_2(P)=k$, $b_1(X)$ is the first Betti number of $X$ and $b_+(X)$ is the dimension of the positive-definite subspace of $H_2(X,\mathbb)$ with respect to the intersection form. When $X$ is simply-connected with definite intersection form, possibly after changing orientation, one always has $b_1(X) = 0$ and $b_+(X)=0$. Thus taking any principal $\operatorname(2)$-bundle with $k=1$, one obtains a moduli space $\mathcal$ of dimension five.

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 $b_2(X)$ many.Donaldson, S. K. (1983). An application of gauge theory to four-dimensional topology. Journal of Differential Geometry, 18(2), 279-315. Results of Clifford Taubes and Karen Uhlenbeck show that whilst $\mathcal$ is non-compact, its structure at infinity can be readily described.Uhlenbeck, K. K. (1982). Connections with L p bounds on curvature. Communications in Mathematical Physics, 83(1), 31-42.Uhlenbeck, K. K. (1982). Removable singularities in Yang–Mills fields. Communications in Mathematical Physics, 83(1), 11-29. Namely, there is an open subset of $\mathcal$, say $\mathcal_$, such that for sufficiently small choices of parameter $\varepsilon$, there is a diffeomorphism

:$\mathcal_ \xrightarrow X\times (0,\varepsilon)$.

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

Donaldson observed that the singular points in the interior of $\mathcal$ corresponding to reducible connections could also be described: they looked like cones over the complex projective plane $\mathbb^2$, 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 $\mathbb^2$. Secondly, glue in a copy of $X$ itself at infinity. The resulting space is a cobordism between $X$ and a disjoint union of $b_2(X)$ copies of $\mathbb^2$ 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 $X$ 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
* Rokhlin's theorem

Notes

References

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*{{citation , first=A. , last=Scorpan , year=2005 , title=The Wild World of 4-Manifolds , publisher=American Mathematical Society

Differential topology
Theorems in topology
Quadratic forms