Identifying two definitions of orientation on a vector space

Question

Let $V$ be an $n$-dimensional real vector space. Here are two definitions of an orientation on $V$:

1. A generator of the $1$-dimensional vector space $\wedge^n V$, up to multiplication by positive reals. Let $\mathcal{O}(V)$ be the two orientations of $V$ under this definition.
2. A generator of $H_n(V,V \setminus 0) \cong \mathbb{Z}$. Let $\mathcal{O}'(V)$ be the two orientations of $V$ under this definition.

The group $GL(V)$ acts on both $\mathcal{O}(V)$ and $\mathcal{O}'(V)$, and the subgroup $GL^{>0}(V)$ of matrices with positive determinant acts trivially on both. This gives simply transitive actions of $GL(V)/GL^{>0}(V) \cong \mathbb{Z}/2\mathbb{Z}$ on $\mathcal{O}(V)$ and $\mathcal{O}'(V)$.

From this action, you can do most of what you want with either $\mathcal{O}(V)$ or $\mathcal{O}'(V)$. In particular, this is enough to let you use either definition to define what it means for a manifold to be orientable, and you get the same definition.

However, I don't know a particularly good way to identify the sets $\mathcal{O}(V)$ and $\mathcal{O}'(V)$, at least without making some arbitrary choices. This makes the identification of an orientation on a manifold defined with the two definitions less canonical than one would like.

What is the best way to do this? Is there a way that at least feels semi-canonical?

Answer 1

Here's a direct way to relate the two:

One more structure is additivity of orientations. For $V, W$ of dimensions $n,m$, we have a canonical pairing $$ \Lambda^n V \otimes \Lambda^m W \cong \Lambda^{n+m}(V\oplus W) $$ and Künneth analogously gives $$ H_n(V,V-\{0\}) \otimes H_m(W,W-\{0\}) \cong H_{n+m}(V\oplus W, V\oplus W - \{0\}). $$ This gives maps $\mathcal{O}(V)\times \mathcal{O}(W) \to \mathcal{O}(V\oplus W)$ and analogously for $\mathcal{O}'$, which are compatible with the transitive $\mathbb{Z}/2$ action on both sides in the obvious way (if you flip either orientation, the orientation on the direct sum changes).

Any identification $\mathcal{O}(V)\cong \mathcal{O}'(V)$ compatible with those direct sum pairings should now be determined by the case $V=\mathbb{R}^1$. In that case, there are still two choices, but I think the most canonical one is the one where you pick $1$ as generator of $\Lambda^1 \mathbb{R} = \mathbb{R}$, and the positively oriented $1$-simplex $[-1,1]$ as generator of $H_1(\mathbb{R}, \mathbb{R}-\{0\})$.