### Abstract

When studying classification problems in the theory of transformation groups one usually focuses on smooth actions of compact Lie groups $G$ on specific manifolds $M$, such as Euclidean spaces, disks, spheres, and complex projective spaces. We discuss results related to the realisation problem for actions on Euclidean spaces, disks, and spheres, and then to describe new results for actions on complex projective spaces obtained by the first author in his PhD Thesis.

Publication

*The Topology and the Algebraic Structures of Transformation Groups*, Proceedings of RIMS – Kokyuroku, Volume **1922**, (2014), p. 147-153

Let $G$ be a compact Lie group. For specific manifolds $M$, such as Euclidean spaces, disks, spheres, and complex projective spaces consider the following realisation problem.

- Which manifolds $F$ are diffeomorphic to the corresponding fixed points sets $M^G$ for some action of $G$ on $M$?

Our goal is to discuss results related to the first question obtained so far for actions on Euclidean spaces, disks, and spheres, and then to describe new results for actions on complex projective spaces obtained by the first author in his PhD Thesis. Hence, every manifold $F$ which occurs as the fixed point set is a second-countable space, i.e., $F$ is paracompact and $F$ has countably many connected components, possibly not of the same dimension.