Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 22 / 4 / a / Solution

Take a based CW complex model for $X$. Since it is $(n-1)$-connected and $n\ge 2$, its reduced ordinary homology vanishes below degree $n$, with arbitrary constant coefficients. This also follows from a CW complex model with no non-basepoint cells below degree $n$. The connective spectrum $E$ has ${\pi }_{q}E=0$ for $q<0$.

Filter $X$ by its skeleta. The quotients of consecutive stages are wedges of spheres, so the exact sequences in represented homology theory give the homological Atiyah-Hirzebruch spectral sequenceIt is first quadrant because $E$ is a connective spectrum. In a fixed total degree it converges with a finite filtration: cells of sufficiently high dimension cannot contribute in that degree or affect it through a boundary, by the same connectivity bound.

In total degree $n$, every term with $p<n$ is zero, and terms with $q<0$ are zero. The only possible term isNo differential can leave it: the target has spatial degree $n-r<n$. No differential can enter it: its source would be $(n+r,1-r)$, whose coefficient degree is negative for every $r\ge 2$. Thus it survives unchanged and is the only filtration quotient of ${\tilde{E}}_n(X)$; there is no extension problem. The ordinary Hurewicz theorem now yieldsThe isomorphism uses the stated identification ${\pi }_{0}E\cong \mathbb{Z}$ and is natural in $X$. This is the bottom-degree generalized Hurewicz isomorphism.