Connective spectrum

A connective spectrum has ${\pi }_{q}E=0$ for $q<0$. This makes the homological Atiyah-Hirzebruch spectral sequence first quadrant for a based space, and prevents negative coefficient degrees from affecting its first nonzero represented homology theory.

If $E$ is connective spectrum in the sense of a connective spectrum, ${\pi }_{0}E\cong \mathbb{Z}$, and $X$ is $(n-1)$-connected with $n\ge 2$, then ${\tilde{E}}_n(X)\cong {\pi }_n(X)$. In the homological Atiyah-Hirzebruch spectral sequence, the only nonzero term in total degree $n$ is ${\tilde{H}}_n(X;{\pi }_{0}E)$. Incoming differentials have negative coefficient degree, and outgoing differentials have spatial degree below $n$. The Hurewicz theorem identifies this term with ${\pi }_{n}X$.

For a connective spectrum with ${\pi }_{0}E\cong \mathbb{Z}$, an $E$-homology isomorphism of simply connected spaces is a weak homotopy equivalence. Its simply connected homotopy mapping cone is $E$-acyclic. If it had a first nonzero integral homology group, the Hurewicz theorem and the bottom-degree generalized Hurewicz isomorphism would produce a nonzero $E$-homology group. Thus the cone is integrally acyclic and the homological Whitehead theorem applies.