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?



Special cell decomposition for spheres with free $\mathbb{Z}/p\mathbb{Z}$-action by orthogonal transformations?
This is a more explicit version of Ian's answer.


From the representation theory of $\mathbb{Z}/p$, we can assume that $V=\mathbb{C}^{m+1}$ with the generator $g$ of $\mathbb{Z}/p$ acting as $g.z=(\omega\_0z\_0,\dotsc,\omega\_mz\_m)$ for some primitive $p$-th roots of unity $\omega\_0,\dotsc,\omega\_m$. Put $I=[0,1]$ and $W=\{re^{i\theta}:0\leq r\leq 1,\;0\leq\theta\leq 2\pi/p\}$. Then put
\begin{align\*}
 e\_{2k} &= \{z\in S^{2m+1}:z\_k\in I,\; z\_j=0\text{ for } j>k\} \\
 e\_{2k+1} &= \{z\in S^{2m+1}:z\_k\in W,\; z\_j=0\text{ for } j>k\}.
\end{align\*}
There is a homeomorphism $f\_{2k}\colon B^{2k}=B(\mathbb{C}^k)\to e\_{2k}$ given by
$$ f\_{2k}(z) = (z\_0,\dotsc,z\_{k-1},\sqrt{1-\|z\|^2},0,\dotsc,0) $$
There are continuous surjections $p\_k\colon B(\mathbb{C}^k)\times[0,1]\to e\_{2k+1}$ and $q\_k\colon B(\mathbb{C}^k)\times[0,1]\to B(\mathbb{C}^k\oplus\mathbb{R})=B^{2k+1}$ given by
\begin{align\*}
 p\_k(z,t) &= (z\_0,\dotsc,z\_{k-1},\sqrt{1-\|z\|^2}\,e^{2\pi it/p},0,\dotsc,0) \\
 q\_k(z,t) &= (z,\sqrt{1-\|z\|^2}\,(2t-1))
\end{align\*}
One checks that
$$ p\_k(z,t)=p\_k(z',t') \Leftrightarrow (z=z' \wedge (t=t' \vee \|z\|=1))
 \Leftrightarrow q\_k(z,t)=q\_k(z',t').
$$
It follows that there is a unique map $f\_{2k+1}\colon B^{2k+1}\to e\_{2k+1}$ with $f\_{2k+1}\circ q\_k=p\_k$, and that this is a homeomorphism.


One can now check that the cells $\{g^ie\_j:0\leq i<p,\;0\leq j\leq 2m+1\}$ give an equivariant cell structure on $S^{2m+1}$. The cellular boundary operator is $\partial(e\_{2k})=\sum\_ig^ie\_{2k-1}$ and $\partial(e\_{2k+1})=g^{u\_k}e\_{2k}-e\_{2k}$, where $u\_k$ is determined by $\omega\_k^{u\_k}=e^{2\pi i/p}$. In particular, in the basic case where $\omega\_k=e^{2\pi i/p}$ for all $k$ we have $\partial(e\_{2k+1})=g.e\_{2k}-e\_{2k}$.


Most of this is in Section V.5 of the 1962 book [Cohomology operations](https://people.math.rochester.edu/faculty/doug/otherpapers/steenrod-epstein.pdf) by Steenrod and Epstein,


