Intertwining category and complexity

We develop the theory of the intertwining distributional versions of the LS-category and the sequential topological complexities of a space X, denoted by icat(X) and iTCₘ(X), respectively. We prove that they satisfy most of the nice properties as their respective distributional counterparts dcat(X) and dTCₘ(X), and their classical counterparts cat(X) and TCₘ(X), such as homotopy invariance and special behavior on topological groups. We show that the notions of iTCₘ and dTCₘ are different for each m ≥ 2 by proving that iTCₘ(H)=1 for all m ≥ 2 for Higman's group H. Using cohomological lower bounds, we also provide various examples of locally finite CW complexes X for which icat(X) > 1, iTCₘ(X) > 1, icat(X) = dcat(X) = cat(X), and iTC(X) = dTC(X) = TC(X).