Euler-characteristic divisibility from a fixed-point-free cyclic action
= Euler-characteristic divisibility from a fixed-point-free cyclic action
Let $f$ have prime order $p$ on a finite CW complex, and suppose every nonidentity power is fixed-point-free. On rational homology,
$$
P_q=\frac1p\sum_{j=0}^{p-1}(f_*|_{H_q})^j
$$
is the projection onto the invariant subspace. Therefore
$$
\sum_q(-1)^q\dim H_q^{\langle f\rangle}
=\frac1p\sum_{j=0}^{p-1}L(f^j)
=\frac{\chi(X)}p.
$$
The left side is an integer, so $p$ divides $\chi(X)$.