Consider the unit sphere $S^d$ in $\mathbb{R}^{d+1}$ with the antipodal action $\nu \colon x\mapsto -x$. This turns $S^d$ into a free $\mathbb{Z}/2\mathbb{Z}$-space.

Construct a CW-complex structure for $S^d$ with 2 cells in each dimension (which we think of as hemispheres) as follows: start with two vertices/$0$-cells. Attach to segments to get a circle, glue in two disks to get a $2$-sphere etc.

$\nu$ induces an action on chains. Write $\mathrm{id}$ for the identity map on chains and let $\theta = \mathrm{id}+\nu$. The cell structure described above gives rise to an interesting family of chains $h_i\in C_i(S^d;\mathbb{Z}/2\mathbb{Z})$, given by one of the two hemispheres, such that $h_0$ is an elementary $0$-chain, $\theta h_d$ is the fundamental cycle of $S^d$ and $\partial h_i = \theta h_{i-1}$ for all $i\geq 1$.

Some handwavy argument tells me that this should generalize to free $\mathbb{Z}/p\mathbb{Z}$-actions on $S^d$ as follows: Assume the cyclic group $G=\mathbb{Z}/p\mathbb{Z}$ of prime order $p$ acts freely on $S^d$ by linear orthogonal transformations (so we are considering the unit sphere of a $(d+1)$-dimensional linear orthogonal representation $V$ of $G$ such that the induced action on that sphere is free). Let $\nu\colon S^d\rightarrow S^d$ be a generator for this action. Consider the two special elements $s= \mathrm{id}-\nu$ and $t=\mathrm{id}+\nu+\dots+\nu^{p-1}$ acting on chains of $S^d$. Can we find chains $h_i\in C_i(S^d;\mathbb{Z}/p\mathbb{Z})$ such that $h_0$ is an elementary $0$-chain, $t h_d$ is the fundamental cycle of $S^d$ and $\partial h_i=s h_{i-1}$ if $i$ is odd and $\partial h_i=t h_{i-1}$ if $i$ is even? Is there an analogous CW-complex structure like the hemispheres in the case of the antipodal action on $S^d$ from which we can read of the chains $h_i$? How to make this precise?