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JOURNALS // Zapiski Nauchnykh Seminarov POMI // Archive

Zap. Nauchn. Sem. POMI, 2013 Volume 415, Pages 109–136 (Mi znsl5691)

On homotopy invariants of finite degree

S. S. Podkorytov

St. Petersburg Department of Steklov Institute of Mathematics of the Russian Academy of Sciences, St. Petersburg, Russia

Abstract: Let $X$ and $Y$ be pointed topological spaces and let $V$ be an abelian group. By definition, a homotopy invariant $f\colon[X,Y]\to V$ has degree at most $r$ if there exists a homomorphism $l\colon\mathrm{Hom}(C_0(X^r),C_0(Y^r))\to V$ such that $f([a])=l(C_0(a^r))$ for all maps $a\colon X\to Y$. Here $C_0(a^r)\colon C_0(X^r)\to C_0(Y^r)$ is the homomorphism of the groups of unreduced zero-dimensional singular chains induced by the $r$th Cartesian power of $a$. Suppose that $X$ is a connected compact CW-complex and $Y$ is a nilpotent connected CW-complex with finitely generated homotopy groups. Then finite-degree homotopy invariants taking values in cyclic groups of prime orders distinguish homotopy classes of maps $X\to Y$. Several similar statements are shown to be false.

Key words and phrases: Shipley's convergence theorem.

UDC: 515.143.3

Received: 27.11.2012


 English version:
Journal of Mathematical Sciences (New York), 2016, 212:5, 587–604

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