This work is motivated by the question of whether there are spaces X for which the Farber–Grant symmetric topological complexity TCS(X) differs from the Basabe–González–Rudyak–Tamaki symmetric topological complexity TCΣ(X). For a projective space ${\open R}\hbox{P}^m$, it is known that $\hbox{TC}^S ({\open R}\hbox{P}^{m})$ captures, with a few potential exceptional cases, the Euclidean embedding dimension of ${\open R}\hbox{P}^{m}$. We now show that, for all m≥1, $\hbox{TC}^{\Sigma}({\open R}\hbox{P}^{m})$ is characterized as the smallest positive integer n for which there is a symmetric ${\open Z}_{2}$-biequivariant map Sm×Sm→Sn with a ‘monoidal’ behaviour on the diagonal. This result thus lies at the core of the efforts in the 1970s to characterize the embedding dimension of real projective spaces in terms of the existence of symmetric axial maps. Together with Nakaoka's description of the cohomology ring of symmetric squares, this allows us to compute both TC numbers in the case of ${\open R}\hbox{P}^{2^{e}}$ for e≥1. In particular, this leaves the torus S1×S1 as the only closed surface whose symmetric (symmetrized) TCS (TCΣ) invariant is currently unknown.

Let $X$ be a $\text{CW}$ complex with a continuous action of a topological group $G$. We show that if $X$ is equivariantly formal for singular cohomology with coefficients in some field $\Bbbk $, then so are all symmetric products of $X$ and in fact all its $\Gamma $-products. In particular, symmetric products of quasi-projective $\text{M}$-varieties are again $\text{M}$-varieties. This generalizes a result by Biswas and D’Mello about symmetric products of $\text{M}$-curves. We also discuss several related questions.

We generalize the notions of the orbifold Euler characteristic and of the higher-order orbifold Euler characteristics to spaces with actions of a compact Lie group using integration with respect to the Euler characteristic instead of the summation over finite sets. We show that the equation for the generating series of the kth-order orbifold Euler characteristics of the Cartesian products of the space with the wreath products actions proved by Tamanoi for finite group actions and by Farsi and Seaton for compact Lie group actions with finite isotropy subgroups holds in this case as well.