On homotopy invariants of maps to the circle

Homotopy classes of maps of a space $X$ to the circle $T$ form an Abelian group $B(X)$ (Bruschlinsky group). A map $f\colon B(X)\to C$, where $C$ is an Abelian group, has order at most $r$ if for a continuous map $a\colon X\to T$ the value $f([a])$ can be $\mathbb Z$-linearly expressed in terms of the indicator function $I_r(a)\colon(X\times T)^r\to\mathbb Z$ of the $r$th Cartesian power of the graph of $a$. We prove that the order of $f$ equals the algebraic degree of $f$. (A map between abelian groups has degree at most $r$ if its finite differences of order $r+1$ vanish.) Bibl. – 2 titles.