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 diagonalizable. 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 $k=c_2(P)$ is a Chern class, $b_1(X)$ 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 $\mathbb^2$. Furthermore, we can count the number of such singular points. Let $E$ be the $\mathbb^2$-bundle over $X$ associated to $P$ by the standard representation of $SU(2)$. Then, reducible connections modulo gauge are in a 1-1 correspondence with splittings $E = L\oplus L^$ where $L$ is a complex line bundle over $X$. Whenever $E = L\oplus L^$ we may compute:

$1 = k = c_2(E) = c_2(L\oplus L^) = - Q(c_1(L), c_1(L))$,

where $Q$ is the intersection form on the second cohomology of $X$. Since line bundles over $X$ are classified by their first Chern class $c_1(L)\in H^2(X; \mathbb)$, we get that reducible connections modulo gauge are in a 1-1 correspondence with pairs $\pm\alpha\in H^2(X; \mathbb)$ such that $Q(\alpha, \alpha) = -1$. Let the number of pairs be $n(Q)$. An elementary argument that applies to any negative definite quadratic form over the integers tells us that $n(Q)\leq\text(Q)$, with equality if and only if $Q$ is diagonalizable.

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 $n(Q)$ copies of $\mathbb^2$ (of unknown orientations). The signature $\sigma$ of a four-manifold is a cobordism invariant. Thus, because $X$ is definite:

$\text(Q) = b_2(X) = \sigma(X) = \sigma(\bigsqcup n(Q) \mathbb^2) \leq n(Q)$,

from which one concludes the intersection form of $X$ is diagonalizable.

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 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

*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