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

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.

Choose basepoints and replace the spaces and map by CW complex models. If the hypothesis is stated for unreduced represented homology theory, the basepoint coefficient summands split naturally, so it also gives an isomorphism in reduced represented homology theory. Let $C$ be the homotopy mapping cone of $f$. Its reduced represented homology theory is zero in every degree, by the exact sequence of the corresponding cofiber sequence of spectra.

The mapping cone $C$ is a simply connected space: the mapping cylinder retracts onto $Z$, and coning off the path-connected space $Y$ adds no fundamental-group generators; the van Kampen calculation gives the quotient of ${\pi }_{1}Z$ by the image of ${\pi }_{1}Y$, which is trivial. Suppose some positive-degree integral homology group of $C$ were nonzero, and let $n$ be the least such degree. Then $n\ge 2$. Starting with ${\pi }_{1}C=0$, the Hurewicz theorem shows inductively that ${\pi }_{i}C=0$ for $i<n$: at each first possible degree $i$, its homotopy group is identified with the zero group $H_i(C;\mathbb{Z})$. It then identifiesApplying the result of part (a) to the $(n-1)$-connected based space $C$ gives ${\tilde{E}}_n(C)\cong {\pi }_{n}C\ne 0$, contradicting the vanishing of its represented homology theory.

Thus $C$ is integrally acyclic, and the ordinary homology exact sequence says that $f$ is an integral homology isomorphism. The allowed homological Whitehead theorem for simply connected spaces makes the map of CW complex models a homotopy equivalence. Returning to the original spaces therefore givesThe simply connected hypothesis is used both in the first-nonzero-degree Hurewicz theorem argument and in the final homological Whitehead theorem.