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- W2014773175 abstract "Topological complexity $TC{B}$ of a space $B$ is introduced by M. Farber to measure how much complex the space is, which is first considered on a configuration space of a motion planning of a robot arm. We also consider a stronger version $TCM{B}$ of topological complexity with an additional condition: in a robot motion planning, a motion must be stasis if the initial and the terminal states are the same. Our main goal is to show the equalities $TC{B} = catBb{double{B}}+1$ and $TCM{B} = catBB{double{B}}+1$, where $double{B}=B{times}B$ is a fibrewise pointed space over $B$ whose projection and section are given by $p_{double{B}}=proj_{2} : B{times}B to B$ the canonical projection to the second factor and $s_{double{B}}=Delta_{B} : B to B{times}B the diagonal. In addition, our method in studying fibrewise L-S category is able to treat a fibrewise space with singular fibres. Recently, we found a problem with the proof of Theorem 1.13 which states that for a fibrewise well-pointed space $X$ over $B$, we have $catBB{X}$ = catBb{X}$ and that for a locally finite simplicial complex $B$, we have $TC{B} = TCM{B}$. While we still conjecture that Theorem 1.13 is true, this problem means that, at present, no proof is given to exist. Alternatively, we show the difference between two invariants $catBb{X}$ and $catBB{X}$ is at most 1 and the conjecture is true for some cases. We give further corrections mainly in the proof of Theorem 1.12." @default.
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- W2014773175 date "2010-01-01" @default.
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- W2014773175 title "Topological complexity is a fibrewise L–S category" @default.
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- W2014773175 doi "https://doi.org/10.1016/j.topol.2009.04.056" @default.
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