Lemma \[odd\] below. $\square$

\[odd\] Any smooth action of a compact connected Lie group on a closed manifold which is a rational cohomology sphere is Cohen-Macaulay.

If the dimension of the sphere is even, the lemma is a consequence of Ex. \[exodd\]. To deal with the remaining situation, we use Prop. \[maxtor\] (b), which enables us to only consider the action of a torus $T$ on $X$, where $X$ is a closed manifold with $H^*(X)\simeq H^*(S^{2n+1})$ as vector spaces, for some $n \ge 0$. In particular, the total Betti number $\dim H^*(X)$ is equal to 2. In case the fixed point set $X^T$ is non-empty, the latter is a union of closed submanifolds of $X$. By Prop. \[criteqf\], $\dim H^*(X^T)$ is thus a strictly positive number at most equal to 2. On the other hand, the Euler-Poincaré characteristics of $X$ and $X^T$ are equal, see e.g. [@Kobayashi Thm., (4)] or [@Go-Zo Thm. 9.3]; thus $\dim H^*(X^T)$ is an even number. We conclude that the latter number is equal to 2, hence the $T$-action on $X$ is equivariantly formal and consequently Cohen-Macaulay.

Let us now consider the case when the set $X^T$ is empty. There exists a one-dimensional subtorus $S\subset T$ 