Donaldson's theorem

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Template:Result on possible differential structures

Statement

Any positive-definite intersection form of a simply connected differential manifold is equivalent to the identity matrix (viz, it can be diagonalized to the identity matrix).

Relation with other results

Freedman's theorem

Further information: Freedman's theorem

Michael Freedman showed that every positive definite unimodular symmetric bilinear form of dimension 4 can be realized as the intersection form of a topological manifold of dimension 4. This, along with Donaldson's theorem, shows that there are topological manifolds with no differential structure.