I am trying to understand basic notions from Hill-Hopkins-Ravenel paper: <https://arxiv.org/abs/0908.3724>


In the Example 3.10 we are considering equviariant cellular chain complex for $n$-dimensional representation $V$ of a group $G$
$$
\ldots\to C^{cell}\_n(S^V;\underline{\mathbb{Z}})\to C^{cell}\_{n-1}(S^V;\underline{\mathbb{Z}})\to\ldots\to C^{cell}\_0(S^V;\underline{\mathbb{Z}}).
$$
The underlying homology groups are those of the sphere $S^V$ - in particular kernel of the map $C^{cell}\_n(S^V;\underline{\mathbb{Z}})\to C^{cell}\_{n-1}(S^V;\underline{\mathbb{Z}})$ is isomorphic as a $G$-module to nonequivariant homology of $S^V$ (denoted as $H^u\_n(S^V;\mathbb{Z}))$. 


Now if $V$ is oriented (so I suppose this means that group action preserves orientation), then $G$ acts trivially on $C^{cell}\_n(S^V;\underline{\mathbb{Z}})$ (? I am not sure about this statement) and we obtain that $H^G\_\*(S^V;\mathbb{Z})\cong H^u\_n(S^V;\mathbb{Z}).$


**Question:** Why if $V$ is oriented we obtain such result?


I moved this question from MathStack Exchange.



"Oriented representation" sphere
First of all, note that right before example 3.9 they prove that
$$H^G\_\*(S^V;\underline{\mathbb{Z}})=H\_\*(C^{cell}\_\*(S^V)^G)\,,$$
where $C^{cell}\_\*(S^V)$ is the cellular complex for some $G$-CW-structure on $S^V$ (and so levelwise is just a sum of permutation modules). In particular, since $S^V$ is $n$-dimensional
$$H^G\_n(S^V;\underline{\mathbb{Z}})=\ker\left(C^{cell}\_n(S^V)^G\to C^{cell}\_{n-1}(S^V)^G\right)\,.$$
Hence
$$H^G\_n(S^V;\underline{\mathbb{Z}})=\ker\left(C^{cell}\_n(S^V)\to C^{cell}\_{n-1}(S^V)\right)^G=H^u\_n(S^V;\mathbb{Z})^G\,,$$
since taking fixed points commute with taking kernels.


Now, $V$ is orientable iff the action of $G$ preserves the orientation iff $G$ acts trivially on $H^u\_n(S^V;\mathbb{Z})$ (recall that an orientation of $V$ is the same thing as a generator of $H^u\_n(S^V;\mathbb{Z})$). So the restriction map $H^G\_n(S^V;\underline{\mathbb{Z}})\to H^u\_n(S^V;\mathbb{Z})$ is an isomorphism iff $V$ is orientable.


**WARNING:** In general it is not possible to find a $G$-CW structure such that $G$ acts trivially on $C^{cell}\_n(S^V)$, even if $V$ is orientable.


