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(n). 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(n). 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:

Properties

• 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 (with the same fields as fibers), one has:
• :

Literature

References

External links

• at the nLab