Inmathematics, and especiallydifferential topology andgauge theory,Donaldson's theorem states that adefiniteintersection form of aclosed,oriented,smooth manifold ofdimension 4 isdiagonalizable. If the intersection form is positive (negative) definite, it can be diagonalized to theidentity matrix (negative identity matrix) over theintegers. The original version^[1] of the theorem required the manifold to besimply connected, but it was later improved to apply to 4-manifolds with anyfundamental group.^[2]

The theorem was proved bySimon Donaldson. This was a contribution cited for hisFields Medal in 1986.

Donaldson's proof utilizes theYang–Mills moduli space${\displaystyle {\mathcal{M}}_P}$ of solutions to theanti-self-duality equations on aprincipal${\displaystyle \mathrm{SU}(2)}$-bundle${\displaystyle P}$ over the four-manifold${\displaystyle X}$. By theAtiyah–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${\displaystyle k=c_2(P)}$ is aChern class,${\displaystyle b_1(X)}$ is the firstBetti 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${\displaystyle \mathrm{SU}(2)}$-bundle with${\displaystyle k=1}$, one obtains a moduli space${\displaystyle \mathcal{M}}$ 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${\displaystyle b_2(X)}$ many.^[3] Results ofClifford Taubes andKaren 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${\displaystyle {\mathcal{M}}_{\epsilon }}$, such that for sufficiently small choices of parameter${\displaystyle \epsilon }$, there is a diffeomorphism

${\displaystyle {\mathcal{M}}_{\epsilon }\stackrel{\cong }{\to }X\times (0,\epsilon )}$.

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 usingUhlenbeck's 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 likecones over thecomplex projective plane${\displaystyle {\mathbb{C}\mathbb{P}}^2}$. 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${\displaystyle n(Q)\le \text{rank}(Q)}$, 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${\displaystyle {\mathbb{C}\mathbb{P}}^2}$. Secondly, glue in a copy of${\displaystyle X}$ itself at infinity. The resulting space is acobordism between${\displaystyle X}$ and a disjoint union of${\displaystyle n(Q)}$ copies of${\displaystyle {\mathbb{C}\mathbb{P}}^2}$ (of unknown orientations). The signature${\displaystyle \sigma }$ of a four-manifold is a cobordism invariant. Thus, because${\displaystyle X}$ is definite:

${\displaystyle \text{rank}(Q)=b_2(X)=\sigma (X)=\sigma (⨆n(Q){\mathbb{C}\mathbb{P}}^2)\le n(Q)}$,

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

Michael Freedman had previously shown that anyunimodular symmetric bilinear form is realized as the intersection form of some closed, orientedfour-manifold. Combining this result with theSerre classification theorem and Donaldson's theorem, several interesting results can be seen:

1) Any indefinite non-diagonalizable intersection form gives rise to a four-dimensionaltopological manifold with nodifferentiable structure (so cannot be smoothed).

2) Two smooth simply-connected 4-manifolds arehomeomorphic, if and only if, their intersection forms have the samerank,signature, and parity.

• Unimodular lattice
• Donaldson theory
• Yang–Mills equations
• Rokhlin's theorem

• 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

• Differential topology
• Theorems in topology
• Quadratic forms

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