Let $f:X\to Y$ be a pointed map of pointed connected $n$-dimensional CW complexes. Whitehead's theorem says that if $f_*:\pi_qX\to \pi_qY$ is an isomorphism for $q\le n$ and a surjection for $q=n+1$, then $f$ is a homotopy equivalence (e.g. Theorem (Whitehead) on p.75 of May's "Concise Course in Algebraic Topology").
I am interested in counterexamples to this when you drop the surjectivity condition for $q=n+1$. That is,
Question: What examples are there of a map $f:X\to Y$ of pointed connected $n$-dimensional CW complexes that induces isomorphism on $\pi_q$ for $q\le n$, but is not a homotopy equivalence?
I would also like to know what are the "minimal" examples of this. For example, it seems impossible for $n\le 2$ (the induced map on universal covers is a homology isomorphism by the Hurewicz theorem, and hence is a weak equivalence and thus induces isomorphism on all higher homotopy groups). I also wonder if there is an example with finite complexes.
Best Answer
It seems to me that the stronger statement is true. If $f$ is only an isomorphism on homotopy in degrees $* \leq n$ then the homotopy fibre $F$ is $(n-1)$-connected and its Hurewicz map is an isomorphism in degree $n$. Considering the Serre spectral sequence of the fibration seqeuence $F \to \widetilde{X} \to \widetilde{Y}$, and using the fact that it vanishes above the $n$th column (by the dimension restriction on $Y$) it follows that there is a short exact sequence $$0 \to H_n(F) \to H_n(\widetilde{X}) \to H_n(\widetilde{Y}) \to 0$$ and so the composition $\pi_n(F) \to H_n(F) \to H_n(\widetilde{X})$ is injective. But this factors through the map $\pi_n(F) \to \pi_n(\widetilde{X})$, which is trivial, and hence $\pi_n(F)=0$ too.