On the higher topological complexity of manifolds with abelian fundamental group

Abstract We study the higher (or sequential) topological complexity $$\operatorname {TC}_s$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mo>TC</mml:mo> <mml:mi>s</mml:mi> </mml:msub> </mml:math> of manifolds with abelian fundamental group. We give sufficient conditions for $$\operatorname {TC}_s$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mo>TC</mml:mo> <mml:mi>s</mml:mi> </mml:msub> </mml:math> to be non-maximal in both the orientable and non-orientable cases. In combination with cohomological lower bounds, we also obtain some exact values for certain families of manifolds.