Order of a~function on the Bruschlinsky group of a~two-dimensional polyhedron

S. S. Podkorytov

Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 10, Tome 353 (2008), pp. 181-190

Résumé

Homotopy classes of mappings of a compact polyhedron $X$ to the circle $T$ form an Abelian group $B(X)$, which is called the Bruschlinsky group and is isomorphic to $H^1(X;\mathbb Z)$. A function $f\colon B(X)\to L$, where $L$ is an Abelian group, has order at most $r$ if for each mapping $a\colon X\to T$ the value $f([a])$ is $\mathbb Z$-linearly expressed via the characteristic function $I_r(a)\colon(X\times T)^r\to\mathbb Z$ of $(\Gamma_a)^r$, where $\Gamma_a\subset X\times T$ is the graph of $a$. The function $f$ has degree at most $r$ if the finite differences of $f$ of order $r+1$ vanish. Conjecturally, the order of $f$ equals the algebraic degree of $f$. The conjecture is proved in the case where $\dim X\le2$. Bibl. – 1 title.