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 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

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 ${\displaystyle X}$. 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. ^[3] Results of Clifford Taubes and Karen Uhlenbeck show that whilst ${\displaystyle {\mathcal {M}}}$ is non-compact, its structure at infinity can be readily described. ^{[4] [5] [6]} 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. ^{[6] [3]}

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 ${\displaystyle \mathbb {CP} ^{2}}$. Furthermore, we can count the number of such singular points. Let ${\displaystyle E}$ be the ${\displaystyle \mathbb {C} ^{2}}$-bundle over ${\displaystyle X}$ associated to ${\displaystyle P}$ by the standard representation of ${\displaystyle SU(2)}$. Then, reducible connections modulo gauge are in a 1-1 correspondence with splittings ${\displaystyle E=L\oplus L^{-1}}$ where ${\displaystyle L}$ is a complex line bundle over ${\displaystyle X}$. ^[3] Whenever ${\displaystyle E=L\oplus L^{-1}}$ we may compute:

${\displaystyle 1=k=c_{2}(E)=c_{2}(L\oplus L^{-1})=-Q(c_{1}(L),c_{1}(L))}$,

where ${\displaystyle Q}$ is the intersection form on the second cohomology of ${\displaystyle X}$. Since line bundles over ${\displaystyle X}$ are classified by their first Chern class ${\displaystyle c_{1}(L)\in H^{2}(X;\mathbb {Z} )}$, we get that reducible connections modulo gauge are in a 1-1 correspondence with pairs ${\displaystyle \pm \alpha \in H^{2}(X;\mathbb {Z} )}$ such that ${\displaystyle Q(\alpha ,\alpha )=-1}$. Let the number of pairs be ${\displaystyle n(Q)}$. An elementary argument that applies to any negative definite quadratic form over the integers tells us that ${\displaystyle n(Q)\leq {\text{rank}}(Q)}$, with equality if and only if ${\displaystyle Q}$ 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 ${\displaystyle \mathbb {CP} ^{2}}$. 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 ${\displaystyle n(Q)}$ copies of ${\displaystyle \mathbb {CP} ^{2}}$ (of unknown orientations). The signature ${\displaystyle \sigma }$ of a four-manifold is a cobordism invariant. Thus, because ${\displaystyle X}$ is definite:

${\displaystyle {\text{rank}}(Q)=b_{2}(X)=\sigma (X)=\sigma (\bigsqcup n(Q)\mathbb {CP} ^{2})\leq n(Q)}$,

from which one concludes the intersection form of ${\displaystyle X}$ 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

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. 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. 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