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Topological complexity is a fibrewise L-S category. (English) Zbl 1192.55002

Topology Appl. 157, No. 1, 10-21 (2010); erratum ibid. 159, No. 10–11, 2810–2813 (2012).
Let \(p: X\to B\) be a map that admits a section. I. M. James and J. R. Morris [Proc. R. Soc. Edinb., Sect. A 119, No. 1–2, 177–190 (1991; Zbl 0738.55005)] defined a fibrewise generalization of the usual Lusternik-Schnirelmann category. They considered both unpointed fibrewise category \(\text{cat}_ B (X)\) and pointed (i.e. with a fixed section) fibrewise category \(\text{cat}_ B ^ B(X)\). For example, the unpointed category is defined to be the least number of fibrewise categorical open sets required to cover \(X\); here a subset \(U\) of \(X\) is fibrewise categorical if the inclusion \(U\to X\) is fibrewise homotopic to the composite \(s p\) of \(p\) with a section \(s\).
The authors define another version \(\text{cat} _ B ^ * X\) of the unpointed fibrewise category. Fix a section \(s\) of \(p\), and define \(\text{cat} _ B ^ * X\) as the minimal number \(m\) such that there exists a cover of \(X\) by \(m+1\) of open subsets \(U_i\) each of which is fibrewise compressible into \(s\) in \(X\) by a fibrewise homotopy.
Theorem 1.13. For every fibrewise well-pointed space \(X\) over \(B\), we have \(\text{cat}_ B ^ B (X)=\text{cat}_ B ^ *(X)\).
M. Farber [NATO Science Series II: Mathematics, Physics and Chemistry 217, 185–230 (2006; Zbl 1089.68131)] introduced the topological complexity of a robot motion planning in order to measure the (dis)continuity of a motion planning algorithm. More precisely, the topological complexity \(TC(B)\) is defined as the Schwartz genus of \(\pi:P(B) \to B \times B\) where \(P(B)=B^ I\) is the space of all paths in \(B\) and \(\pi(\alpha)=(\alpha(0), \alpha(1))\).
The authors define the “monoidal complexity” \(TC^ M(B)\) as the minimal number \(m\) such that there exists a cover of \(B\times B\) by \(m\) of open subsets \(U_i\supset \Delta(B)\) each of which admits a continuous section \(s_ i: U_ i\to P(B)\) satisfying \(s_ i(b,b)=c_ b\) for all \(b\in B\), where \(c_ b\) is the constant path. Given a space \(B\), define a fibrewise pointed space \(d(b)=B \times B\) where \(p_ d: B \times B \to B\) is the prjection on 2nd factor and \(s: B \to B \times B\) is the diagonal.
Theorem 1.7. For a space \(B\) we have the following inequalities: (1) \(TC(B)=\text{cat}_ B ^ *(d(B))+1\), (2) \(TC^ M(B)=\text{cat}_ B ^ B(d(B))+1\).
In particular, in view of 1.13, we have \(TC(B)=TC^ M (B)\) for every locally finite simplicial complex.
Editorial remark: The authors point out that the proofs of the theorems 1.12 and 1.13 are incorrect in the erratum [Topology Appl. 159, No. 10–11, 2810–2813 (2012; Zbl 1243.55002)].

MSC:

55M30 Lyusternik-Shnirel’man category of a space, topological complexity à la Farber, topological robotics (topological aspects)
55Q25 Hopf invariants
55R70 Fibrewise topology
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References:

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