Geodesic
On homotopy invariants of finite degree
Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 12, Tome 415 (2013), pp. 109-136 Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
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     title = {On homotopy invariants of finite degree},
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S. S. Podkorytov. On homotopy invariants of finite degree. Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 12, Tome 415 (2013), pp. 109-136. http://geodesic.mathdoc.fr/item/ZNSL_2013_415_a13/

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