# Example of a $G$-sphere that is not a $G$-representation sphere

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.

• Maybe working too hard, but: it would suffice to find a $G$-action that doesn't fix a smooth structure. I don't know if this is easy or hard. – Qiaochu Yuan May 27 '15 at 7:22
• You can take the classical example of a Z/2Z-action on $S^4$ with non-manifold fixed set, e.g. Corollary 4.10 of math.rutgers.edu/~sferry/ps/geotop.pdf. – skupers May 27 '15 at 8:35
• @QiaochuYuan, can a finite group not fix a smooth structure? – Mariano Suárez-Álvarez May 27 '15 at 12:10
• @Zev: there are also examples of finite groups acting freely on spheres but the actions are not conjugate to linear actions. Do a Google search for "fake projective space" and you will find several papers. Generally people use this language in the manifold world rather than talking about $G$-spheres. – Ryan Budney May 27 '15 at 15:31