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In this question all groups are finite, and all spaces are nice (eg, simplicial sets).

Given a $G$ space $X$, which we assume has finitely many nonzero cohomology groups, we can compute the trace of any element $g$ in $G$ on $H^*(X,\mathbb{Q})$. If $X$ has finitely many cells, then we can also compute this as the Euler characteristic of $X^g$, the fixed points of $\langle g \rangle$ on $X$.

This agreement of trace and Euler characteristic of fixed points is false in general though, for instance take $EG$. Both of these quantities are invariant under $G$ weak equivalences, so my question is whether there is a (homotopy) class of $G$ spaces for which this equality holds which is larger than $G$ spaces with finitely many cells (up to homotopy)?

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    $\begingroup$ Probably not: there is a 4-dimensional (infinite) contractible simplicial complexes with a simplicial action of a cyclic group of order $pq$ that has no global fixed point, for any distinct primes $p$, $q$. $\endgroup$
    – IJL
    Commented Nov 7, 2022 at 14:02

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