Equivariant topological complexities (Q6641605)

In the present paper, the author surveys different versions of topological complexity with symmetries. These include the equivariant topological complexity \(\text{TC}_G(X)\) of \textit{H. Colman} and the author [Algebr. Geom. Topol. 12, No. 4, 2299--2316 (2012; Zbl 1260.55007)], the invariant topological complexity \(\text{TC}^G(X)\) of \textit{W. Lubawski} and \textit{W. Marzantowicz} [Bull. Lond. Math. Soc. 47, No. 1, 101--117 (2015; Zbl 1311.55004)], the strongly equivariant topological complexity \(\text{TC}^\ast_G(X)\) of \textit{A. Dranishnikov} [Topology Appl. 179, 74--80 (2015; Zbl 1304.55003)], and the effective topological complexity \(\text{TC}^{G,\infty}(X)\) of \textit{Z. Błaszczyk} and \textit{M. Kaluba} [Publ. Mat., Barc. 62, No. 1, 55--74 (2018; Zbl 1385.55003)].\N\NAdditionally, the author presents a new cohomological lower bound for \(\text{TC}^\ast_G(X)\) in the case when \(G\) is finite. The paper concludes with a list of interesting research problems in the area.\N\NFor the entire collection see [Zbl 1547.55002].