= Connective generalized homology detects simply connected equivalences
For a <connective spectrum> with $\pi_0E\cong\mathbb Z$, 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.
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