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<sup>c</sup> structures and are therefore of central importance for Seiberg–Witten theory.
Definition
Let <math>
X
</math> be a paracompact space, then there is a bijection <math>
[X,\operatorname{BO}(n)]\xrightarrow\cong\operatorname{Vect}_\mathbb{R}^n(X),[f]\mapsto f^*\gamma_\mathbb{R}^n
</math> with the real universal vector bundle <math>
\gamma_\mathbb{R}^n
</math>. The real determinant <math>
\det\colon
\operatorname{O}(n)\rightarrow\operatorname{O}(1)
</math> is a group homomorphism and hence induces a continuous map <math>
\mathcal{B}\det\colon
\operatorname{BO}(n)\rightarrow\operatorname{BO}(1)\cong\mathbb{R}P^\infty
</math> on the classifying space for O(n). Hence there is a postcomposition:
: <math>
\det\colon
\operatorname{Vect}_\mathbb{R}^n(X)
\cong[X,\operatorname{BO}(n)]
\xrightarrow{\mathcal{B}\det_*}[X,\operatorname{BO}(1)]
\cong\operatorname{Vect}_\mathbb{R}^1(X).
</math>
Let <math>
X
</math> be a paracompact space, then there is a bijection <math>
[X,\operatorname{BU}(n)]\xrightarrow\cong\operatorname{Vect}_\mathbb{C}^n(X),[f]\mapsto f^*\gamma_\mathbb{C}^n
</math> with the complex universal vector bundle <math>
\gamma_\mathbb{C}^n
</math>.
: <math>
\det(E)
:=\Lambda^{\operatorname{rk}(E)}(E).
</math>
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.