Donaldson's theorem

The theorem was proved by Simon Donaldson. This was a contribution cited for his Fields medal in 1986.

Donaldson's proof utilizes the moduli space $\backslash mathcal\{M\}\_P$ of solutions to the anti-self-duality equations on a principal $\backslash operatorname\{SU\}(2)$-bundle $P$ over the four-manifold $X$. By the Atiyah–Singer index theorem, the dimension of the moduli space is given by

:$\backslash dim\; \backslash mathcal\{M\}\; =\; 8k\; -\; 3(1-b\_1(X)\; +\; b\_+(X)),$

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 $H\_2(X,\backslash mathbb\{R\})$ 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 $\backslash operatorname\{SU\}(2)$-bundle with $k=1$, one obtains a moduli space $\backslash mathcal\{M\}$ of dimension five.

File:Donaldson's Theorem cobordism.png

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 $\backslash mathcal\{M\}$ is non-compact, its structure at infinity can be readily described.Taubes, C. H. (1982). Self-dual Yang–Mills connections on non-self-dual 4-manifolds. Journal of Differential Geometry, 17(1), 139-170.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 $\backslash mathcal\{M\}$, say $\backslash mathcal\{M\}\_\{\backslash varepsilon\}$, such that for sufficiently small choices of parameter $\backslash varepsilon$, there is a diffeomorphism

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

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 $x\backslash in\; X$. 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 $\backslash mathcal\{M\}$ corresponding to reducible connections could also be described: they looked like cones over the complex projective plane $\backslash mathbb\{CP\}^2$. Furthermore, we can count the number of such singular points. Let $E$ be the $\backslash mathbb\{C\}^2$-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 $E\; =\; L\backslash oplus\; L^\{-1\}$ where $L$ is a complex line bundle over $X$. Whenever $E\; =\; L\backslash oplus\; L^\{-1\}$ we may compute:

$1\; =\; k\; =\; c\_2(E)\; =\; c\_2(L\backslash oplus\; L^\{-1\})\; =\; -\; Q(c\_1(L),\; c\_1(L))$,

where $Q$ is the intersection form on the second cohomology of $X$. Since line bundles over $X$ are classified by their first Chern class $c\_1(L)\backslash in\; H^2(X;\; \backslash mathbb\{Z\})$, we get that reducible connections modulo gauge are in a 1-1 correspondence with pairs $\backslash pm\backslash alpha\backslash in\; H^2(X;\; \backslash mathbb\{Z\})$ such that $Q(\backslash alpha,\; \backslash alpha)\; =\; -1$. 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 $n(Q)\backslash leq\backslash text\{rank\}(Q)$, 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 $\backslash mathbb\{CP\}^2$. 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 $\backslash mathbb\{CP\}^2$ (of unknown orientations). The signature $\backslash sigma$ of a four-manifold is a cobordism invariant. Thus, because $X$ is definite:

$\backslash text\{rank\}(Q)\; =\; b\_2(X)\; =\; \backslash sigma(X)\; =\; \backslash sigma(\backslash bigsqcup\; n(Q)\; \backslash mathbb\{CP\}^2)\; \backslash leq\; n(Q)$,

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

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.

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

{{Reflist}}

• {{Citation | last1=Donaldson | first1=S. K. | title=An application of gauge theory to four-dimensional topology |doi=10.4310/jdg/1214437665 | mr=710056 | year=1983 | journal=Journal of Differential Geometry | volume=18 | issue=2 | pages=279–315 |zbl=0507.57010 | doi-access=free }}
• {{citation |first1=S. K. |last1=Donaldson |first2=P. B. |last2=Kronheimer |year=1990 |title=The Geometry of Four-Manifolds |series=Oxford Mathematical Monographs |isbn=0-19-850269-9 }}
• {{citation |first1=D. S. |last1=Freed |first2=K. |last2=Uhlenbeck |authorlink2=Karen Uhlenbeck |year=1984 |title=Instantons and Four-Manifolds |publisher=Springer }}
• {{citation |first1=M. |last1=Freedman |first2=F. |last2=Quinn |year=1990 |title=Topology of 4-Manifolds |publisher=Princeton University Press }}
• {{citation |first=A. |last=Scorpan |year=2005 |title=The Wild World of 4-Manifolds |publisher=American Mathematical Society }}

Category:Differential topology

Category:Theorems in topology

Category:Quadratic forms