Donaldson's theorem

On when a definite intersection form of a smooth 4-manifold is diagonalisable

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 M P {\displaystyle {\mathcal {M}}_{P}} of solutions to the anti-self-duality equations on a principal SU ⁡ ( 2 ) {\displaystyle \operatorname {SU} (2)} -bundle P {\displaystyle P} over the four-manifold X {\displaystyle X} . By the Atiyah–Singer index theorem, the dimension of the moduli space is given by

dim ⁡ M = 8 k − 3 ( 1 − b 1 ( X ) + b + ( X ) ) , {\displaystyle \dim {\mathcal {M}}=8k-3(1-b_{1}(X)+b_{+}(X)),}

where c 2 ( P ) = k {\displaystyle c_{2}(P)=k} , b 1 ( X ) {\displaystyle b_{1}(X)} is the first Betti number of X {\displaystyle X} and b + ( X ) {\displaystyle b_{+}(X)} is the dimension of the positive-definite subspace of H 2 ( X , R ) {\displaystyle H_{2}(X,\mathbb {R} )} with respect to the intersection form. When X {\displaystyle X} is simply-connected with definite intersection form, possibly after changing orientation, one always has b 1 ( X ) = 0 {\displaystyle b_{1}(X)=0} and b + ( X ) = 0 {\displaystyle b_{+}(X)=0} . Thus taking any principal SU ⁡ ( 2 ) {\displaystyle \operatorname {SU} (2)} -bundle with k = 1 {\displaystyle k=1} , one obtains a moduli space M {\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 b 2 ( X ) {\displaystyle b_{2}(X)} many.[3] Results of Clifford Taubes and Karen Uhlenbeck show that whilst M {\displaystyle {\mathcal {M}}} is non-compact, its structure at infinity can be readily described.[4][5][6] Namely, there is an open subset of M {\displaystyle {\mathcal {M}}} , say M ε {\displaystyle {\mathcal {M}}_{\varepsilon }} , such that for sufficiently small choices of parameter ε {\displaystyle \varepsilon } , there is a diffeomorphism

M ε → ≅ X × ( 0 , ε ) {\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 X {\displaystyle X} with curvature becoming infinitely concentrated at any given single point x ∈ X {\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 M {\displaystyle {\mathcal {M}}} corresponding to reducible connections could also be described: they looked like cones over the complex projective plane C P 2 {\displaystyle \mathbb {CP} ^{2}} . Furthermore, we can count the number of such singular points. Let E {\displaystyle E} be the C 2 {\displaystyle \mathbb {C} ^{2}} -bundle over X {\displaystyle X} associated to P {\displaystyle P} by the standard representation of S U ( 2 ) {\displaystyle SU(2)} . Then, reducible connections modulo gauge are in a 1-1 correspondence with splittings E = L ⊕ L − 1 {\displaystyle E=L\oplus L^{-1}} where L {\displaystyle L} is a complex line bundle over X {\displaystyle X} .[3] Whenever E = L ⊕ L − 1 {\displaystyle E=L\oplus L^{-1}} we may compute:

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

where Q {\displaystyle Q} is the intersection form on the second cohomology of X {\displaystyle X} . Since line bundles over X {\displaystyle X} are classified by their first Chern class c 1 ( L ) ∈ H 2 ( X ; Z ) {\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 ± α ∈ H 2 ( X ; Z ) {\displaystyle \pm \alpha \in H^{2}(X;\mathbb {Z} )} such that Q ( α , α ) = − 1 {\displaystyle Q(\alpha ,\alpha )=-1} . Let the number of pairs be n ( Q ) {\displaystyle n(Q)} . An elementary argument that applies to any negative definite quadratic form over the integers tells us that n ( Q ) ≤ rank ( Q ) {\displaystyle n(Q)\leq {\text{rank}}(Q)} , with equality if and only if Q {\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 C P 2 {\displaystyle \mathbb {CP} ^{2}} . Secondly, glue in a copy of X {\displaystyle X} itself at infinity. The resulting space is a cobordism between X {\displaystyle X} and a disjoint union of n ( Q ) {\displaystyle n(Q)} copies of C P 2 {\displaystyle \mathbb {CP} ^{2}} (of unknown orientations). The signature σ {\displaystyle \sigma } of a four-manifold is a cobordism invariant. Thus, because X {\displaystyle X} is definite:

rank ( Q ) = b 2 ( X ) = σ ( X ) = σ ( ⨆ n ( Q ) C P 2 ) ≤ n ( Q ) {\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 X {\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

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. ^ a b c d 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. ^ a b 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