"Oriented representation" sphere

Question

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.

Answer 1

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.