# Semantics: 'determinant bundle', top exterior power of vector bundle

From what I can dig up, given a vector bundle $E\rightarrow X$, the determinant bundle associated to this is $\Lambda^{n}E\rightarrow X$, where $n$ is the rank of $E\rightarrow X$. Is this the same thing as "the determinant of the vector bundle $E\rightarrow X$"?

If this is true, then does the statement "$E\rightarrow X$ has trivial determinant" mean that $\Lambda^{n}E\rightarrow X$ is trivializable as a vector bundle"?

• Yes. Is there some reason you doubt this? Note that $\bigwedge^nE$ is a line bundle, so this equivalent to admitting a nowhere-zero section. Commented Dec 2, 2015 at 19:29
• Could you please shorten the title of this question to something like "Semantics: 'determinant' or 'top exterior power' bundle?" Commented Dec 3, 2015 at 21:46

The top exterior power of a vector bundle $\Lambda^n E$ of a rank $n$ vector bundle $E$ has transition functions $\wedge^n g_{UV}:U\cap V\to\text{GL}(\wedge^n\mathbb{C}^n) = \mathbb{C}^*$ where $g_{UV}$ are the transition functions coming from $E$. This immediately implies what we are looking at is a line bundle and the expansion is shown cleanly as; \begin{align*} (\wedge^ng_{UV})(e_1\wedge\dots\wedge e_n) &= \wedge_{i=1}^ng_{UV}(e_i) \\ &= \wedge_{i=1}^n\left(\sum_{j=1}^n g_{ji}e_j\right)\\ &= \left[\sum_{\sigma \in S_n}(-1)^{|\sigma|}\prod_{i=1}^n g_{i\sigma(i)}\right] \wedge_{i=1}^ne_i \\ &= \det(g_{UV})\cdot e_1\wedge\dots\wedge e_n. \end{align*} where the $e_i$'s denote the standard basis vectors and we have made use of the relations \begin{align*} \alpha\wedge\alpha &= 0\\ \alpha\wedge\beta &= -\beta\wedge\alpha \end{align*} for any $\alpha,\beta\in \Omega^1(E)$. This line bundle is, for now apparent reasons, known as the determinant bundle of $M$ and is also denoted $\det(E)$.
Finally, saying that $E$ having trivial determinant means that the determinant bundle of $E$ is a trivial bundle.
Note that the notion of top exterior power (for a vector space) is in fact the formal definition of determinant. That is the determinant is the unique multi-linear functional acting on $n$ vectors in an $n$-dimensional space which is alternating and whose evaluation on the standard basis is 1 (i.e. preserves the volume of the unit cube).