Lemma 3.6.9 from Complex Topological K-Theory by Efton Park

Question

The lemma goes like this (in author's notation, $\pi: V\to X$ is the projection map; if $f: Y\to X$, and $W$ is a vector bundle over $X$, then $f^* W$ is the pullback of $W$ over $Y$):

Let $X$ be a compact Hausdorff space, suppose that $V$ is a Hermitian vector bundle over $X$, and let $k$ be a natural number. Define $\epsilon: \bigwedge^k(\pi^* V)\to \bigwedge^{k+1}(\pi^*V)$ by setting $\epsilon(v, \omega)=(v, v\wedge \omega)$ for all $v\in V$ and $\omega\in \bigwedge^k(\pi^*V)$. For each element $v$ of $V$ that appears in a wedge product, let $\hat{v}$ indicate omission of $v$. Then $$ \epsilon^*(v, v_1\wedge...\wedge v_k)=(v, \sum_{j=1}^{k+1}(-1)^{j+1}\langle v_j, v_1\rangle v_1\wedge ...\wedge \hat{v}_j\wedge ...\wedge v_{k+1}) $$ for all simple wedges in $\bigwedge^k(V)_v$.

My questions:

1. What is $\bigwedge^k(V)_v$?

2. Following the author's notation, $\epsilon^*$ is supposed to be the pullback functor, and as such should map vector bundles over $\bigwedge^{k+1}(\pi^*V)$ to vector bundles over $\bigwedge^k(\pi^*V)$. But in the lemma it apparently maps from $\bigwedge^k(\pi^*V)$ as a space to $\bigwedge^k(\pi^*V)$ as a space? How do I make sense of this?

3. $\omega\in \bigwedge^k(\pi^*V)$ is a typo - it should be $\omega\in\bigwedge^k(V)$, right?

Answer 1

1. $\bigwedge^k(V)_v$ is the fiber of the vector bundle $\bigwedge^k(V)$ over $v$.

2. $\epsilon^*: \bigwedge^{k+1}(\pi^*V)\to \bigwedge^k(\pi^*V)$ denotes the adjoint operator of $\epsilon: \bigwedge^k(\pi^*V) \to \bigwedge^{k+1}(\pi^*V)$. As such, there was a typo in the definition of $\epsilon^*$ - it takes as input $(v, v_1\wedge...\wedge v_k\wedge v_{k+1})$.

3. That is indeed another typo in the formulation of the lemma. There were also 5 more typos in the proof of this lemma.

Answer 2

I believe there are a number of typos in the passage you are quoting (presumably anyway, since I don't have access to the book--so some of this is guesswork):

First, $\epsilon$ is supposed to be a map $\bigwedge^k V \to \bigwedge^{k+1}V$ of vector bundles over $X$, and for a fixed $v \in V$ comes from the map $\omega \mapsto v \wedge \omega$. Then $\epsilon^*$ is actually supposed to be $\pi^*\epsilon$, the map induced by the pullback via $\pi$ (since $\pi^*$ is a functor, given a map of vector bundles over $X$ we obtain a map of vector bundles over $V$), hence is a map $\bigwedge^{k+1}\pi^*V = \pi^*(\bigwedge^{k+1}V) \to \pi^*(\bigwedge^{k}V) = \bigwedge^k \pi^*V$. It has the formula you provided, except for the fact that there is a typo and it should start with $(v, v_1 \wedge \ldots \wedge v_{k+1})$ (otherwise the right-hand side of the formula makes no sense).

Then, the last few words should read "...simple wedges in $\bigwedge^{k+1}(\pi^*V)_v$," which is the fiber of $\bigwedge^{k+1}(\pi^*V)$ over $v$.