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 Let be a paracompact space, then there is a bijection with the real universal vector bundle .Hatcher 2017, Theorem 1.16. 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:Nicolaescu 2000, Exercise 1.1.4.
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.Hatcher 2017, Proposition 3.10. Since in this case the first Stiefel–Whitney class vanishes if and only if a real line bundle is orientable,Hatcher 2017, Proposition 3.11. both conditions are then equivalent to a trivial determinant line bundle.Bott & Tu 1982, Proposition 11.4. 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:
Proof: Assume is a real vector bundle and let be its classifying map with , then:
For complex vector bundles, the proof is completely analogous.
For vector bundles (with the same fields as fibers), one has:
Literature
References
External links
determinant line bundle at the nLab