TY - JOUR

T1 - Homotopy invariants of mappings to the circle

AU - Podkorytov, S. S.

PY - 2011/6/1

Y1 - 2011/6/1

N2 - Homotopy classes of mappings of a space X to the circle T form an Abelian group B(X) (the Bruschlinsky group). If a: X → T is a continuous mapping, then [a] denotes the homotopy class of a, and Ir(a): (X × T)r→ → ℤ is the indicator function of the rth Cartesian power of the graph of a. Let C be an Abelian group and let f: B(X) → C be a mapping. By definition, f has order not greater than r if the correspondence Ir(a) → f([a]) extends to a (partly defined) homomorphism from the Abelian group of ℤ-valued functions on (X × T)r to C. It is proved that the order of f equals the algebraic degree of f. (A mapping between Abelian groups has degree at most r if all of its finite differences of order r +1 vanish.) Bibliography: 2 titles.

AB - Homotopy classes of mappings of a space X to the circle T form an Abelian group B(X) (the Bruschlinsky group). If a: X → T is a continuous mapping, then [a] denotes the homotopy class of a, and Ir(a): (X × T)r→ → ℤ is the indicator function of the rth Cartesian power of the graph of a. Let C be an Abelian group and let f: B(X) → C be a mapping. By definition, f has order not greater than r if the correspondence Ir(a) → f([a]) extends to a (partly defined) homomorphism from the Abelian group of ℤ-valued functions on (X × T)r to C. It is proved that the order of f equals the algebraic degree of f. (A mapping between Abelian groups has degree at most r if all of its finite differences of order r +1 vanish.) Bibliography: 2 titles.

UR - http://www.scopus.com/inward/record.url?scp=79958026397&partnerID=8YFLogxK

U2 - 10.1007/s10958-011-0376-9

DO - 10.1007/s10958-011-0376-9

M3 - Article

AN - SCOPUS:79958026397

VL - 175

SP - 609

EP - 619

JO - Journal of Mathematical Sciences

JF - Journal of Mathematical Sciences

SN - 1072-3374

IS - 5

ER -