Topological complexity of \(H\)-spaces. (English) Zbl 1263.55002
The notion of topological complexity \(TC(X)\), introduced in [M. Farber, Discrete Comput. Geom. 29, No. 2, 211–221 (2003; Zbl 1038.68130)], is motivated by the study of motion planning algorithms in robotics. Higher versions of the invariant \(TC(X)\) were defined by Y. Rudyak in [Topology Appl. 157, No. 5, 916–920 (2010); erratum ibid. 157, No. 6, 1118 (2010; Zbl 1187.55001)] and are denoted \(TC_n(X)\) with \(TC_2(X)\) being equal to \(TC(X)\). The authors prove that for a connected CW \(H\)-space one has \(TC_{n+1}(X) = cat(X^n)\) and in particular \(TC(X) = cat(X).\) This result was previously known in the special case when \(X\) is a topological group. The authors also give an inequality applicable when a space \(Y\) acts on another space \(X\).
Reviewer: Michael Farber (Zürich)
MSC:
| 55M30 | Lyusternik-Shnirel’man category of a space, topological complexity à la Farber, topological robotics (topological aspects) |
| 55S40 | Sectioning fiber spaces and bundles in algebraic topology |
| 57T99 | Homology and homotopy of topological groups and related structures |
| 70Q05 | Control of mechanical systems |
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