Homotopy similarity of maps. Maps of the circle

We describe the relation of $r$-similarity and finite-order invariants on the homotopy set $[S^1,Y]=π_1(Y)$.

💡 Research Summary

The paper introduces a new equivalence relation called r‑similarity for continuous maps between cellular spaces, and studies its consequences for the fundamental group π₁(Y) when the source space is the circle S¹.

Framework.
Let X and Y be cellular spaces with X compact. Assume X is equipped with a comultiplication µ : X → X∨X and a coinversion ν : X → X. For any map a : X → Y one defines two operations: a∗b = (a∨b)∘µ (a product) and a† = a∘ν (an inverse). If these operations turn the homotopy set