On sequential versions of distributional topological complexity

We define a (non-decreasing) sequence {dTC(m)(X)}(m >= 2) of sequential versions of distributional topological complexity (dTC) of a space X introduced by Dranishnikov and Jauhari [5]. This sequence generalizes dTC(X) in the sense that dTC(2)(X)=dTC(X), and is a direct analog to the well-known sequence {TCm(X)}(m >= 2). We show that like TCm and dTC, the sequential versions dTC(m) are also homotopy invariants. Furthermore, dTCm(X) relates with the distributional LS-category (dcat) of products of X in the same way as TCm(X) relates with the classical LS-category (cat) of products of X. On one hand, we show that in general, dTC(m) is a different concept than TCm for each m >= 2. On the other hand, by finding sharp cohomological lower bounds to dTC(m)(X), we provide various examples of closed manifolds X for which the sequences {TCm(X)}(m >= 2) and {dTC(m)(X)}(m >= 2) coincide. (c) 2025 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.