# Homotopy classes of maps

This is a reference request.

A theorem of Hurewicz (unfortunately published in the hard-to-get journal Proc.Akad. Wetensch. Amsterdam 1936) asserts $\left[X,Y\right]=Hom(\pi_1X,\pi_1Y)/Inn(\pi_1Y)$ for aspherical spaces.

The generalization to arbitrary (pairs of) spaces reads: when $f,g:(X,X_1)\rightarrow (Y,Y_1)$ are such that $f_\ast=g_\ast:\pi_i(X)\rightarrow \pi_i(Y)$ and $f_\ast=g_\ast:\pi_i(X_1)\rightarrow \pi_i(Y_1)$ for all $i$, then $f$ and $g$ are homotopic (as maps of pairs).

I think I know how to prove this for pairs of CW complexes. My question is just for a citable reference either in a paper or a textbook.

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BTW the theorem of Hurewicz you mention is explained nicely in Appendix 4.A of Hatcher's "Algebraic Topology" text. – Mark Grant May 8 '13 at 9:28
Detecting homotopic maps by means of homotopy groups is a very complicated task. There are even conjectures on it, look up 'Freyd generating hypothesis'. – Fernando Muro May 8 '13 at 10:54
At least homotopy classes from X to K(G,1) correspond to homomorphisms of fundamental groups mod inner autos. – ThiKu May 8 '13 at 12:32
Just a note: that journal is hard to find, but if your library subscribes to Elsevier you may have access to it, but under a different name. See plus.google.com/115215145654669548294/posts/jHJTaqaB5e2 – Charles Rezk May 8 '13 at 12:55

I don't think this is true. For a counter-example take $f$ to be any pointed map $K(G,n)\to K(H,m)$ representing a non-trivial cohomology operation, where $G$ and $H$ are abelian groups and $m>n>1$. Then $f$ and the trivial map $g$ induce the same map on homotopy groups, but are not based homotopic.