# Donaldson's theorem

From Diffgeom

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.