page_title: Donaldson's theorem
section_title: Idea of proof
hierarchical_section_title: Donaldson's theorem / Idea of proof
caption_attribution_description: English: Cobordism between disjoint union of copies of CP^2 and a four manifold X. Total space of cobordism is the moduli space of ASD instantons on X.
context_page_description: In mathematics, and especially gauge theory, Donaldson's theorem states that a definite intersection form of a compact, oriented, simply connected, smooth manifold of dimension 4 is diagonalisable. If the intersection form is positive definite, it can be diagonalized to the identity matrix over the integers.
context_section_description: Donaldson's proof utilizes the moduli space  of solutions to the anti-self-duality equations on a principal -bundle  over the four-manifold. By the Atiyah-Singer index theorem, the dimension of the moduli space is given by
  where ,  is the first Betti number of  and  is the dimension of the positive-definite subspace of  with respect to the intersection form. When  is simply-connected with definite intersection form, possibly after changing orientation, one always has  and . Thus taking any principal -bundle with , one obtains a moduli space  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  many. Results of Clifford Taubes and Karen Uhlenbeck show that whilst  is non-compact, its structure at infinity can be readily described. Namely, there is an open subset of , say , such that for sufficiently small choices of parameter , there is a diffeomorphism
  .
  The work of Taubes and Uhlenbeck essentially concerns constructing sequences of ASD connections on the four-manifold  with curvature becoming infinitely concentrated at any given single point . 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  corresponding to reducible connections could also be described: they looked like cones over the complex projective plane , with its orientation reversed.
  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 . Secondly, glue in a copy of  itself at infinity. The resulting space is a cobordism between  and a disjoint union of  copies of  with its orientation reversed. The intersection form of a four-manifold is a cobordism invariant up to isomorphism of quadratic forms, from which one concludes the intersection form of  is diagonalisable.
caption: Cobordism given by Yang-Mills moduli space in Donaldson's theorem
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