On the sequential topological complexity of group homomorphisms

We define and develop a homotopy invariant notion for the sequential topological complexity of a map $f:X\to Y,$ denoted $T{C}_r(f)$, that interacts with $T{C}_r(X)$ and $T{C}_r(Y)$ in the same way Jamie Scott’s topological complexity map $TC(f)$ interacts with $TC(X)$ and $TC(Y)\mathrm{.}$ Furthermore, we apply $T{C}_r(f)$ to studying group homomorphisms $\phi :\mathrm{\Gamma }\to \mathrm{\Lambda }\mathrm{.}$ In addition, we prove that the sequential topological complexity of any nonzero homomorphism of a torsion group cannot be finite. Also, we give the characterisation of cohomological dimension of group homomorphisms.