Topological complexity is a fibrewise L-S category

Norio Iwase, Michihiro Sakai

    Research output: Contribution to journalArticlepeer-review

    24 Citations (Scopus)


    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) = catB* (d (B)) + 1 and TCM (B) = catBB (d (B)) + 1, where d (B) = B × B is a fibrewise pointed space over B whose projection and section are given by pd (B) = pr2 : B × B → B the canonical projection to the second factor and sd (B) = ΔB : B → B × B the diagonal. In addition, our method in studying fibrewise L-S category is able to treat a fibrewise space with singular fibres.

    Original languageEnglish
    Pages (from-to)10-21
    Number of pages12
    JournalTopology and its Applications
    Issue number1
    Publication statusPublished - Jan 1 2010

    All Science Journal Classification (ASJC) codes

    • Geometry and Topology


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