Determinant line bundle
Template:Short description 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 <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>.<ref name=":0">Hatcher 2017, Theorem 1.16.</ref> 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>.<ref name=":0" /> The complex determinant <math> \det\colon \operatorname{U}(n)\rightarrow\operatorname{U}(1) </math> is a group homomorphism and hence induces a continuous map <math> \mathcal{B}\det\colon \operatorname{BU}(n)\rightarrow\operatorname{BU}(1)\cong\mathbb{C}P^\infty </math> on the classifying space for U(n). Hence there is a postcomposition:
- <math>
\det\colon \operatorname{Vect}_\mathbb{C}^n(X) \cong[X,\operatorname{BU}(n)] \xrightarrow{\mathcal{B}\det_*}[X,\operatorname{BU}(1)] \cong\operatorname{Vect}_\mathbb{C}^1(X). </math>
Alternatively, the determinant line bundle can be defined as the last non-trivial exterior product. Let <math> E\twoheadrightarrow X </math> be a vector bundle, then:<ref>Nicolaescu 2000, Exercise 1.1.4.</ref>
- <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.<ref name=":1">Hatcher 2017, Proposition 3.10.</ref> Since in this case the first Stiefel–Whitney class vanishes if and only if a real line bundle is orientable,<ref>Hatcher 2017, Proposition 3.11.</ref> both conditions are then equivalent to a trivial determinant line bundle.<ref>Bott & Tu 1982, Proposition 11.4.</ref>
- 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.<ref name=":1" />
- The pullback bundle commutes with the determinant line bundle. For a continuous map <math>
f\colon X\rightarrow Y </math> between paracompact spaces <math> X </math> and <math> Y </math> as well as a vector bundle <math> E\twoheadrightarrow Y </math>, one has:
- <math>
\det(f^*E) \cong f^*\det(E). </math>
- Proof: Assume <math>
E\twoheadrightarrow Y </math> is a real vector bundle and let <math> g\colon Y\rightarrow\operatorname{BO}(n) </math> be its classifying map with <math> E=g^*\gamma_\mathbb{R}^n </math>, then:
- <math>
\det(f^*E) \cong\det(f^*g^*\gamma_\mathbb{R}^n) \cong\det((g\circ f)^*\gamma_\mathbb{R}^n) \cong(\mathcal{B}\det\circ g\circ f)^*\gamma_\mathbb{R}^1 \cong f^*(\mathcal{B}\det\circ g)^*\gamma_\mathbb{R}^1 \cong f^*\det(g^*\gamma_\mathbb{R}^n) \cong f^*\det(E). </math>
- For complex vector bundles, the proof is completely analogous.
- For vector bundles <math>
E,F\twoheadrightarrow X </math> (with the same fields as fibers), one has:
- <math>
\det(E\otimes F) \cong\det(E)^{\operatorname{rk}(F)}\otimes\det(F)^{\operatorname{rk}(E)}. </math>
Literature
References
<references/>
External links
- determinant line bundle at the nLab