On the sectional category of subgroup inclusions and Adamson cohomology theory. (English) Zbl 1485.55004
In the paper under review the authors study the sectional category of subgroup inclusions. Given any group \(G\) and a subgroup \(H\), they define the sectional category of the inclusion, \(\mbox{secat}(H\hookrightarrow G),\) as the ordinary sectional category of the map \(K(H,1)\rightarrow K(G,1)\) between the corresponding Eilenberg-MacLane spaces. This definition generalizes the already established notion of topological complexity of a group \(\pi \), originally defined as \(\mbox{TC}(\pi ):=\mbox{TC}(K(\pi,1))\). In this sense, \(\mbox{TC}(\pi )\) can be also seen as the sectional category of \(\Delta _{\pi}\hookrightarrow \pi \times \pi \) where \(\Delta _{\pi}\) denotes the diagonal subgroup of \(\pi \times \pi .\) Among their results we can mention an extension of a certain characterization of topological complexity of a group, by M. Farber et al. [Algebr. Geom. Topol. 19, No. 4, 2023–2059 (2019; Zbl 1471.55003)], to this new setting of sectional category of group inclusions. The authors also introduce Adamson cohomology theory into their study.
Reviewer: José Calcines (La Laguna)
MSC:
| 55M30 | Lyusternik-Shnirel’man category of a space, topological complexity à la Farber, topological robotics (topological aspects) |
| 68T40 | Artificial intelligence for robotics |
Citations:
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