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

Let ${\displaystyle X}$ be a paracompact space, then there is a bijection ${\displaystyle [X,\operatorname {BO} (n)]\xrightarrow {\cong } \operatorname {Vect} _{\mathbb {R} }^{n}(X),[f]\mapsto f^{*}\gamma _{\mathbb {R} }^{n}}$ with the real universal vector bundle ${\displaystyle \gamma _{\mathbb {R} }^{n}}$. The real determinant ${\displaystyle \det \colon \operatorname {O} (n)\rightarrow \operatorname {O} (1)}$ is a group homomorphism and hence induces a continuous map ${\displaystyle {\mathcal {B}}\det \colon \operatorname {BO} (n)\rightarrow \operatorname {BO} (1)\cong \mathbb {R} P^{\infty }}$ on the classifying space for O(n). Hence there is a postcomposition:

Let ${\displaystyle X}$ be a paracompact space, then there is a bijection ${\displaystyle [X,\operatorname {BU} (n)]\xrightarrow {\cong } \operatorname {Vect} _{\mathbb {C} }^{n}(X),[f]\mapsto f^{*}\gamma _{\mathbb {C} }^{n}}$ with the complex universal vector bundle ${\displaystyle \gamma _{\mathbb {C} }^{n}}$. The complex determinant ${\displaystyle \det \colon \operatorname {U} (n)\rightarrow \operatorname {U} (1)}$ is a group homomorphism and hence induces a continuous map ${\displaystyle {\mathcal {B}}\det \colon \operatorname {BU} (n)\rightarrow \operatorname {BU} (1)\cong \mathbb {C} P^{\infty }}$ 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 ${\displaystyle E\twoheadrightarrow X}$ be a vector bundle, then: