Set of anti-invariant forms over the $n$-sphere question

Question

Let $a:S^n\to S^n$ be the antipodal map on the $n$-dimensional sphere $S^n$. Consider the set $\Omega^k(S^n)$ of alternate $k$-linear differentiable forms over $S^n$. Consider the subset $$E=\{\omega-a^*\omega:\omega\in\Omega^k(S^n)\}\subseteq\Omega^k(S^n),$$ where $a^*$ is the pullback. Is it true that, if $k$ is even, $E$ only contains the $0$ form? Is it true that if $k$ is odd, $E=\Omega^k(S^n)$? My reasoning behind this is that $a^*\omega=(-1)^k\omega$ so, if $k$ is even, $\omega-a^*\omega=0$ and if $k$ is odd, $\omega-a^*\omega=2\omega$, am I right?

Answer 1

No. For instance, if $\omega$ is a $k$-form supported on a small patch of $S^n$, then $a^*\omega$ will be supported on the antipodal patch (which is disjoint from the support of $\omega$), so $a^*\omega$ will certainly not be $(-1)^k\omega$.