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 spinc structures and are therefore of central importance for Seiberg–Witten theory.

Definition

be a paracompact space, then there is a bijection with the real universal vector bundle. is a group homomorphism and hence induces a continuous map on the classifying space for O(n). Hence there is a postcomposition:

be a paracompact space, then there is a bijection. on the classifying space for U(n). Hence there is a postcomposition:

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.

• , one has:

• , then:

• For complex vector bundles, the proof is completely analogous.

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

Literature

References

External links

• determinant line bundle at the nLab