On numerical invariants of retraction map

We introduce a notion of retraction between continuous maps of topological spaces and study the behavior of several numerical invariants under such retractions. These include (co)homological dimensions, the Lusternik-Schnirelmann category, the topological complexity, and the sequential topological complexity. We prove that, under the retraction map, the corresponding inequalities between invariants hold. Our results also apply to recent invariants are defined by Dranishnikov, Jauhari \cite{DJ} and Knudsen, Weinberger \cite{KW}.