Suppose $M$ is a connected closed smooth $d$-dimensional manifold, and suppose $S^1 = SO(2)$ acts smoothly on $M$. Then the fixed point set $Y = M^{S^1}$ will be a submanifold of $M$ of even codimension, and we also have $\chi(Y) = \chi(M)$.

(For instance, there are different linear circle actions on $S^d$ for which the fixed point set will be $S^{d-2j}$; here $j$ can be any number between $0$ and $d/2$.)

My question is rather vague, I simply want to know what else we can say about $Y$. I am also very interested in seeing more examples where $Y$ is an interesting manifold. **In particular I would like to know whether it can happen that $Y$ has components of different dimensions, and whether the number of components of $Y$ can be bounded in some way.** Is there any result that tells us that the homotopy type of $Y$ cannot be much more complicated than that of $M$?