Let $\tau: S^n \to S^n, x \mapsto -x$ be the antipodal map. Show, that these are equivalent:
- $\tau$ is (smooth) homotopic to the identity map $\text{id}$ on $S^n$
- $n$ is odd
- There is a nonvanishing smooth vector field on $S^n$
I want to show $(1 \implies 2)$, $(2 \implies 3)$ and $(3 \implies 1)$, but I am struggling with the last two.
Attempt:
$(1 \implies 2)$: I'd suggest embedding the sphere into euclidian space and take a canonical volume form. Let $\tau$ and $\text{id}$ be homotopic and $\omega_p(v_1, ..., v_n) = \det(p, v_1, ..., v_n)$ a volume form for all $p \in S^n \subseteq \mathbb R^{n+1}$. Since volume forms are nonzero, the integral neither is. For homotopic maps, the integral of the pullbacks are equal, thus we get
$$0 \neq \int_{S^n} \omega = \int_{S^n} \text{id}^* \omega = \int_{S^n} \tau^* \omega = (-1)^{n+1} \int_{S^n} \omega,$$
which can only be true for $n$ being odd. I think this should be fine.
$(2 \implies 3)$: Obviously, we have to construct a vector field on $S^{2m+1}$ for $m \in \mathbb N_0$. I think on the 1-sphere a rotation by $\pi$ would be such a vector field, so maybe in polar coordinates something like $\vartheta \mapsto \vartheta + \pi$ for $\vartheta \in [0, 2\pi)$. Can one generalize this for higher dimensions and if so, how? If I rotate the 3-sphere by $\pi$, how would I explicitly define the map? Embedding it into euclidian space seems so complicated here.
$(3 \implies 1)$: Let $X$ be a smooth nonvanishing vector field. We have to build a homotopy $F: [0, 1] \times S^n \to S^n$. I first thought of something like $$F(t, p) = \cos(\pi t) \cdot p,$$ but it didn't use the property of $X$ and leaves the sphere (e.g. $F(1/2, p) = 0$), so it is obviously wrong. My professor gave the hint to assume that $X(p)$ is orthogonal to $p$, but why would that be allowed?
Thanks for any help!
For (2) implies (3), take the unit sphere in complex Euclidean space, and rotate by multiplying by unit complex numbers. In other words, the vector field is $X(z)=iz$ for $z \in \mathbb{C}^k$, with $|z|=1$. This is the sphere $S^n$ where $n=2k-1$.
For (3) implies (1), take you vector field $X$, let $Y=X/|X|$, and flow along $Y$ for time $\pi$.