There are some negative results about. For instance, a theorem of Sullivan and Vigué-Poirrier states that if $M$ is a closed manifold with $\pi_1(M)$ finite, and if the cohomology algebra $H^*(M;\mathbb{R})$ requires at least two generators, then the Betti numbers of $M^{S^1}$ are unbounded. The finiteness assumption implies that $M^{S^1}$ has only finitely many path components, and so at least one of them is not a finite CW-complex.

There are some negative results about. For instance, a theorem of Sullivan and Vigué-Poirrier states that if $M$ is a closed manifold with $\pi_1(M)$ finite, and if the cohomology algebra $H^*(M;\mathbb{R})$ requires at least two generators, then the Betti numbers of $M^{S^1}$ are unbounded. The finiteness assumption implies that $M^{S^1}$ has only finitely many path components, and so at least one of them is not a finite CW-complex.

There are some negative results about. For instance, a theorem of Sullivan and Vigué-Poirrier states that if $M$ is a closed manifold with $\pi_1(M)$ finite, and if the cohomology algebra $H^*(M;\mathbb{R})$ requires at least two generators, then the Betti numbers of $M^{S^1}$ are unbounded. The finiteness assumption implies that $M^{S^1}$ has only finitely many path components, and so at least one of them is not homotopy equivalent to a finite CW-complex.

There are some negative results about. For instance, a theorem of Sullivan and Vigué-Poirrier states that if $M$ is a closed manifold with $\pi_1(M)$ finite, and if the cohomology algebra $H^*(M;\mathbb{R})$ requires at least two generators, then the Betti numbers of $M^{S^1}$ are unbounded. The finiteness assumption implies that $M^{S^1}$ has only finitely many path components, and so at least one of them is not a finite CW-complex.