On homotopy invariants of finite degree

S. S. Podkorytov

Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 12, Tome 415 (2013), pp. 109-136

Résumé

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.