# Determinant line bundle

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Construction for vector bundles

In [differential geometry](/source/Differential_geometry), the **determinant line bundle** is a construction, which assigns every [vector bundle](/source/Vector_bundle) over [paracompact spaces](/source/Paracompact_space) a [line bundle](/source/Line_bundle). Its name comes from using the [determinant](/source/Determinant) on their [classifying spaces](/source/Classifying_space). Determinant line bundles naturally arise in four-dimensional [spinc structures](/source/Spinc_structure) and are therefore of central importance for [Seiberg–Witten theory](/source/Seiberg%E2%80%93Witten_theory).

## Definition

Let X {\displaystyle X} be a [paracompact space](/source/Paracompact_space), then there is a [bijection](/source/Bijection) [ X , BO ⁡ ( n ) ] → ≅ Vect R n ⁡ ( X ) , [ f ] ↦ f ∗ γ R n {\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](/source/Universal_vector_bundle) γ R n {\displaystyle \gamma _{\mathbb {R} }^{n}} .[1] The real determinant det : O ⁡ ( n ) → O ⁡ ( 1 ) {\displaystyle \det \colon \operatorname {O} (n)\rightarrow \operatorname {O} (1)} is a [group homomorphism](/source/Group_homomorphism) and hence induces a [continuous map](/source/Continuous_map) B det : BO ⁡ ( n ) → BO ⁡ ( 1 ) ≅ R P ∞ {\displaystyle {\mathcal {B}}\det \colon \operatorname {BO} (n)\rightarrow \operatorname {BO} (1)\cong \mathbb {R} P^{\infty }} on the [classifying space for O(n)](/source/Classifying_space_for_O(n)). Hence there is a postcomposition:

- det : Vect R n ⁡ ( X ) ≅ [ X , BO ⁡ ( n ) ] → B det ∗ [ X , BO ⁡ ( 1 ) ] ≅ Vect R 1 ⁡ ( X ) . {\displaystyle \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).}

Let X {\displaystyle X} be a [paracompact space](/source/Paracompact_space), then there is a [bijection](/source/Bijection) [ X , BU ⁡ ( n ) ] → ≅ Vect C n ⁡ ( X ) , [ f ] ↦ f ∗ γ C n {\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 γ C n {\displaystyle \gamma _{\mathbb {C} }^{n}} .[1] The complex determinant det : U ⁡ ( n ) → U ⁡ ( 1 ) {\displaystyle \det \colon \operatorname {U} (n)\rightarrow \operatorname {U} (1)} is a group homomorphism and hence induces a continuous map B det : BU ⁡ ( n ) → BU ⁡ ( 1 ) ≅ C P ∞ {\displaystyle {\mathcal {B}}\det \colon \operatorname {BU} (n)\rightarrow \operatorname {BU} (1)\cong \mathbb {C} P^{\infty }} on the [classifying space for U(n)](/source/Classifying_space_for_U(n)). Hence there is a postcomposition:

- det : Vect C n ⁡ ( X ) ≅ [ X , BU ⁡ ( n ) ] → B det ∗ [ X , BU ⁡ ( 1 ) ] ≅ Vect C 1 ⁡ ( X ) . {\displaystyle \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).}

Alternatively, the determinant line bundle can be defined as the last non-trivial exterior product. Let E ↠ X {\displaystyle E\twoheadrightarrow X} be a vector bundle, then:[2]

- det ( E ) := Λ rk ⁡ ( E ) ( E ) . {\displaystyle \det(E):=\Lambda ^{\operatorname {rk} (E)}(E).}

## Properties

- The real determinant line bundle preserves the first [Stiefel–Whitney class](/source/Stiefel%E2%80%93Whitney_class), which for real line bundles over [topological spaces](/source/Topological_space) with the [homotopy type](/source/Homotopy_type) of a [CW complex](/source/CW_complex) is a [group isomorphism](/source/Group_isomorphism).[3] Since in this case the first Stiefel–Whitney class vanishes if and only if a real line bundle is orientable,[4] both conditions are then equivalent to a trivial determinant line bundle.[5]

- The complex determinant line bundle preserves the first [Chern class](/source/Chern_class), which for complex line bundles over topological spaces with the homotopy type of a CW complex is a group isomorphism.[3]

- The pullback bundle commutes with the determinant line bundle. For a continuous map f : X → Y {\displaystyle f\colon X\rightarrow Y} between paracompact spaces X {\displaystyle X} and Y {\displaystyle Y} as well as a vector bundle E ↠ Y {\displaystyle E\twoheadrightarrow Y} , one has: - det ( f ∗ E ) ≅ f ∗ det ( E ) . {\displaystyle \det(f^{*}E)\cong f^{*}\det(E).}

- Proof: Assume E ↠ Y {\displaystyle E\twoheadrightarrow Y} is a real vector bundle and let g : Y → BO ⁡ ( n ) {\displaystyle g\colon Y\rightarrow \operatorname {BO} (n)} be its classifying map with E = g ∗ γ R n {\displaystyle E=g^{*}\gamma _{\mathbb {R} }^{n}} , then: - det ( f ∗ E ) ≅ det ( f ∗ g ∗ γ R n ) ≅ det ( ( g ∘ f ) ∗ γ R n ) ≅ ( B det ∘ g ∘ f ) ∗ γ R 1 ≅ f ∗ ( B det ∘ g ) ∗ γ R 1 ≅ f ∗ det ( g ∗ γ R n ) ≅ f ∗ det ( E ) . {\displaystyle \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).}

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

- For vector bundles E , F ↠ X {\displaystyle E,F\twoheadrightarrow X} (with the same fields as fibers), one has: - det ( E ⊗ F ) ≅ det ( E ) rk ⁡ ( F ) ⊗ det ( F ) rk ⁡ ( E ) . {\displaystyle \det(E\otimes F)\cong \det(E)^{\operatorname {rk} (F)}\otimes \det(F)^{\operatorname {rk} (E)}.}

## Literature

- [Bott, Raoul](/source/Raoul_Bott); [Tu, Loring W.](/source/Loring_W._Tu) (1982). *Differential Forms in Algebraic Topology*. [Springer](/source/Springer_Publishing). [doi](/source/Doi_(identifier)):[10.1007/978-1-4757-3951-0](https://doi.org/10.1007%2F978-1-4757-3951-0). [ISBN](/source/ISBN_(identifier)) [978-1-4757-3951-0](https://en.wikipedia.org/wiki/Special:BookSources/978-1-4757-3951-0).

- [Freed, Daniel](/source/Dan_Freed) (1987-03-10). ["On determinant line bundles"](https://web.ma.utexas.edu/users/dafr/detsur.pdf) (PDF).

- Nicolaescu, Liviu I. (2000), [*Notes on Seiberg-Witten theory*](http://www.nd.edu/~lnicolae/swnotes.pdf) (PDF), [Graduate Studies in Mathematics](/source/Graduate_Studies_in_Mathematics), vol. 28, Providence, RI: American Mathematical Society, [doi](/source/Doi_(identifier)):[10.1090/gsm/028](https://doi.org/10.1090%2Fgsm%2F028), [ISBN](/source/ISBN_(identifier)) [978-0-8218-2145-9](https://en.wikipedia.org/wiki/Special:BookSources/978-0-8218-2145-9), [MR](/source/MR_(identifier)) [1787219](https://mathscinet.ams.org/mathscinet-getitem?mr=1787219)

- [Hatcher, Allen](/source/Allen_Hatcher) (2003). ["Vector Bundles & K-Theory"](https://pi.math.cornell.edu/~hatcher/VBKT/VBpage.html).

## References

1. ^ [***a***](#cite_ref-:0_1-0) [***b***](#cite_ref-:0_1-1) Hatcher 2017, Theorem 1.16.

1. **[^](#cite_ref-2)** Nicolaescu 2000, Exercise 1.1.4.

1. ^ [***a***](#cite_ref-:1_3-0) [***b***](#cite_ref-:1_3-1) Hatcher 2017, Proposition 3.10.

1. **[^](#cite_ref-4)** Hatcher 2017, Proposition 3.11.

1. **[^](#cite_ref-5)** Bott & Tu 1982, Proposition 11.4.

## External links

- [determinant line bundle](https://ncatlab.org/nlab/show/determinant%2Bline%2Bbundle) at the [*n*Lab](/source/NLab)

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Adapted from the Wikipedia article [Determinant line bundle](https://en.wikipedia.org/wiki/Determinant_line_bundle) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Determinant_line_bundle?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.