{{Short description|Construction for vector bundles}} In [[differential geometry]], the '''determinant line bundle''' is a construction, which assigns every [[vector bundle]] over [[Paracompact space|paracompact spaces]] a [[line bundle]]. Its name comes from using the [[determinant]] on their [[Classifying space|classifying spaces]]. Determinant line bundles naturally arise in four-dimensional [[Spinc structure|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>.<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 space|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 ==
* {{cite book |last1=Bott |first1=Raoul |author-link=Raoul Bott |url= |title=Differential Forms in Algebraic Topology |title-link= |last2=Tu |first2=Loring W. |author-link2=Loring W. Tu |publisher=[[Springer Publishing|Springer]] |year=1982 |isbn=978-1-4757-3951-0 |series= |volume= |language=en |doi=10.1007/978-1-4757-3951-0 |mr= |arxiv=}} * {{cite web |last1=Freed |first1=Daniel |author-link1=Dan Freed |date=1987-03-10 |title=On determinant line bundles |url=https://web.ma.utexas.edu/users/dafr/detsur.pdf}} * {{citation |last=Nicolaescu |first=Liviu I. |title=Notes on Seiberg-Witten theory |volume=28 |year=2000 |url=http://www.nd.edu/~lnicolae/swnotes.pdf |series=[[Graduate Studies in Mathematics]] |location=Providence, RI |publisher=American Mathematical Society |doi=10.1090/gsm/028 |isbn=978-0-8218-2145-9 |mr=1787219}} * {{cite web |last1=Hatcher |first1=Allen |author-link1=Allen Hatcher |year=2003 |title=Vector Bundles & K-Theory |url=https://pi.math.cornell.edu/~hatcher/VBKT/VBpage.html}}
== References == <references/>
== External links ==
* [[nlab:determinant+line+bundle|determinant line bundle]] at the [[NLab|''n''Lab]]
[[Category: Differential geometry]]