TITLE: Fixed point set of smooth circle action
QUESTION [11 upvotes]: 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 $M$?

REPLY [6 votes]: To answer your question: Yes $Y$ can have components in different dimensions: Consider for example the following action on $\mathbb{CP}^2$, where we identified $S^1\subset \mathbb{C}$:
$$
w\cdot [z_0,z_1,z_2]=[w z_0,z_1,z_2].
$$
Then $[1,0,0]$ is an isolated fixed point, and also $[0,z_1,z_2]$ is a fixed $\mathbb{CP}^1$. To get more of such examples you should have a look at Delzant's classification of Hamiltonian Torus actions. Suitably choosing an $S^1$ sitting inside the torus will produce many more examples. 
There is a large literature on $S^1$ actions. Even the case of a discrete fixed point set is very interesting. In the almost complex setting one could have a look at https://arxiv.org/pdf/1411.6458.pdf and references therein.