We show that the normalized topological complexity of the Klein bottle is equal to 4. For any non-orientable surface N g of genus g ! 2, we also show that TCðN g Þ ¼ 4. This completes recent work of Dranishnikov on the topological complexity of non-orientable surfaces.

Based on a Whitehead-type characterization of the sectional category, we develop the notion of weak sectional category. This is a new lower bound of the sectional category, which is inspired by the notion of weak category in the sense of Berstein-Hilton. We establish several properties and inequalities, including the fact that the weak sectional category is a better lower bound for the sectional category than the classical one given by the nilpotency of the kernel of the induced map in cohomology. Finally, we apply our results in the study of the topological complexity in the sense of Farber.

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In this paper we analyze some relationships between the topological complexity of a space X and the category of C∆ X , the homotopy cofibre of the diagonal map ∆X : X → X × X. We establish the equality of the two invariants for several classes of spaces including the spheres, the H-spaces, the real and complex projective spaces and almost all the (standard) lens spaces.

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