Robot motion planning, weights of cohomology classes, and cohomology operations

Abstract

The complexity of algorithms solving the motion planning problem is measured by a homotopy invariant <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper T normal upper C left-parenthesis upper X right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">T</mml:mi> <mml:mi mathvariant="normal">C</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathrm {TC}(X)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of the configuration space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of the system. Previously known lower bounds for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper T normal upper C left-parenthesis upper X right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">T</mml:mi> <mml:mi mathvariant="normal">C</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathrm {TC}(X)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> use the structure of the cohomology algebra of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In this paper we show how cohomology operations can be used to sharpen these lower bounds for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper T normal upper C left-parenthesis upper X right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">T</mml:mi> <mml:mi mathvariant="normal">C</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathrm {TC}(X)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. As an application of this technique we calculate explicitly the topological complexity of various lens spaces. The results of the paper were inspired by the work of E. Fadell and S. Husseini on weights of cohomology classes appearing in the classical lower bounds for the Lusternik-Schnirelmann category. In the appendix to this paper we give a very short proof of a generalized version of their result.