Determinant line bundle

In differential geometry, the determinant line bundle is a construction, which assigns every vector bundle over paracompact spaces a line bundle. Its name comes from using the determinant on their classifying spaces. Determinant line bundles naturally arise in four-dimensional spin^c structures and are therefore of central importance for Seiberg–Witten theory.

Definition

Let be a paracompact space, then there is a bijection with the real universal vector bundle. The real determinant is a group homomorphism and hence induces a continuous map on the classifying space for O. Hence there is a postcomposition:
Let be a paracompact space, then there is a bijection with the complex universal vector bundle. The complex determinant is a group homomorphism and hence induces a continuous map on the classifying space for U. Hence there is a postcomposition:
Alternatively, the determinant line bundle can be defined as the last non-trivial exterior product. Let be a vector bundle, then:

The real determinant line bundle preserves the first Stiefel–Whitney class, which for real line bundles over topological spaces with the homotopy type of a CW complex is a group isomorphism. Since in this case the first Stiefel–Whitney class vanishes if and only if a real line bundle is orientable, both conditions are then equivalent to a trivial determinant line bundle.
The complex determinant line bundle preserves the first Chern class, which for complex line bundles over topological spaces with the homotopy type of a CW complex is a group isomorphism.
The pullback bundle commutes with the determinant line bundle. For a continuous map between paracompact spaces and as well as a vector bundle, one has:
:
For vector bundles , one has:
:
Literature