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DefinitionFor any topological space, cat(X ) is the smallest n for which thereis an open covering {U0, . . . ,Un} by (n + 1) open sets, each ofwhich is contractible in X .
☞ It was, as originally defined for the case of X a manifold, thelower bound for the number of critical points that a real-valuedfunction on X could possess.
☞ This should be compared with the result in Morse theory thatshows that the sum of the Betti numbers is a lower bound forthe number of critical points of a Morse function.
☞ It was, as originally defined for the case of X a manifold, thelower bound for the number of critical points that a real-valuedfunction on X could possess.
☞ This should be compared with the result in Morse theory thatshows that the sum of the Betti numbers is a lower bound forthe number of critical points of a Morse function.
☞ The question will be : Does there exist a continuous motionplanning in X ?
☞ Equivalently, is it possible to construct a motion planning in theconfiguration space X so that the continuous path s(A, B) in X,which describes the movement of the system from the initialconfiguration A to the final configuration B, depends continu-ously on the pair of points (A, B) ?
☞ Absence of continuity will result in the instability of behav-ior : there will exist arbitrarily close pairs (A,B) and (A′
,B′) ofinitial-desired configurations such that the corresponding pathss(A,B) and s(A′
Definition (M. Farber 2003)TC(X) is the smallest number n for which there is an open cover{U0, . . . ,Un} of X × X by (n + 1) open sets, for each of whichthere is a local section si : Ui −→ PX .
Homotopy Invariance : M. Farber (2003)TC(X) depends only on the homotopy type of X.
☞ This property of homotopy invariance allows us to simplify theconfiguration space X without changing the topological com-plexity, and hence, we may predict the character of instabilitiesof the behavior of the robot
☞ the homotopy invariant TC(X) is a new interesting tool whichmeasures the “navigational complexity” of X.
Homotopy Invariance : M. Farber (2003)TC(X) depends only on the homotopy type of X.
☞ This property of homotopy invariance allows us to simplify theconfiguration space X without changing the topological com-plexity, and hence, we may predict the character of instabilitiesof the behavior of the robot
☞ the homotopy invariant TC(X) is a new interesting tool whichmeasures the “navigational complexity” of X.
Homotopy Invariance : M. Farber (2003)TC(X) depends only on the homotopy type of X.
☞ This property of homotopy invariance allows us to simplify theconfiguration space X without changing the topological com-plexity, and hence, we may predict the character of instabilitiesof the behavior of the robot
☞ the homotopy invariant TC(X) is a new interesting tool whichmeasures the “navigational complexity” of X.
M. Farber, Invitation to Topological Robotics, EMS (2008).
M. Farber, Topology of robot motion planning , in : Morse TheoreticMethods in Nonlinear Analysis and in Symplectic Topology (P.Biran et al (eds.)) (2006), 185–230.