On the sequential topological complexity of group homomorphisms

Nursultan Kuanyshov

paper

On the sequential topological complexity of group homomorphisms

arXiv:2402.13389

Abstract

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 $T C (f)$ interacts with $T C (X)$ and $T C (Y) .$ Furthermore, we apply $T C_{r} (f)$ to studying group homomorphisms $I ¨ : I^ \to I^.$ 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.

19 pages