Let $G$ be a finite group with the discrete topology. To set terminology:


* a **$G$-sphere** is a sphere equipped with a continuous $G$-action
* a **$G$-representation sphere** is a $G$-sphere obtained from an orthogonal $G$-representation $V$ by taking its unit sphere $S(V)\subset V$


From what I've heard, there exist $G$-spheres that are not homeomorphic (as $G$-spaces) to any $G$-representation sphere. However, I haven't been able to construct one, or find any reference for this; any help would be appreciated.



Example of a $G$-sphere that is not a $G$-representation sphere
There are involutions on some Brieskorn spheres which are true spheres whose fixed point sets are only Z/2-homology spheres. (But fixed sets of linear actions on spheres are spheres.)


Here it is easy to write down the Brieskorn sphere and the involution, it is more work to see that the Brieskorn sphere is topologically a sphere, and it is even more work if one is interested in the smooth structure.


Reference: Bredon, *Exotic Actions on Spheres*, Proceedings of the Conference on Transformation Groups 1968, pp 47-76. [doi:10.1007/978-3-642-46141-5\_2](http://dx.doi.org/10.1007/978-3-642-46141-5_2)


