TY - JOUR

T1 - Order of a function on the Bruschlinsky group of a two-dimensional polyhedron

AU - Podkorytov, S. S.

PY - 2009/7/1

Y1 - 2009/7/1

N2 - 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 canonically isomorphic to H1 (X; ℤ), Let L be an Abelian group, and let f: B(X) → L be a function. One says that the order of f does not exceed r if for each mapping a: X → T the value f([a]) is ℤ-linearly expressed via the characteristic function Ir(a): (X × T)r → ℤ of (Γa)r, where Γa ⊂ X × T is the graph of a. The (algebraic) degree of f is not greater than r if the finite differences of f of order r + 1 vanish. Conjecturally, the order of f is equal to the algebraic degree of f. The conjecture is proved in the case where dim X ≤ 2. Bibliography: 1 title.

AB - 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 canonically isomorphic to H1 (X; ℤ), Let L be an Abelian group, and let f: B(X) → L be a function. One says that the order of f does not exceed r if for each mapping a: X → T the value f([a]) is ℤ-linearly expressed via the characteristic function Ir(a): (X × T)r → ℤ of (Γa)r, where Γa ⊂ X × T is the graph of a. The (algebraic) degree of f is not greater than r if the finite differences of f of order r + 1 vanish. Conjecturally, the order of f is equal to the algebraic degree of f. The conjecture is proved in the case where dim X ≤ 2. Bibliography: 1 title.

UR - http://www.scopus.com/inward/record.url?scp=70350668455&partnerID=8YFLogxK

U2 - 10.1007/s10958-009-9574-0

DO - 10.1007/s10958-009-9574-0

M3 - Article

AN - SCOPUS:70350668455

VL - 161

SP - 454

EP - 459

JO - Journal of Mathematical Sciences

JF - Journal of Mathematical Sciences

SN - 1072-3374

IS - 3

ER -