The stable Hopf map

Example 4.4.26 (The stable Hopf map). The first positive stable stem already contains information that ordinary homology does not detect. Let \[ h\colon S^3 \longrightarrow S^2 \] be the Hopf fibration. Applying the suspension-spectrum functor and shifting by \(-2\) gives the stable Hopf map \[ \eta \colon \S [1] \longrightarrow \S . \] The classical calculation \(\pi _4(S^3)\cong \Z /2\) identifies the suspension of \(h\) with its nonzero element. The Freudenthal suspension theorem then shows that all subsequent suspensions are isomorphisms, and hence \[ \pi _1(\S )\cong \Z /2, \] generated by \(\eta \); see, for example, [Mosher and Tangora (1968); Switzer (1975)].

Regard \(\CP ^2\) as pointed at its \(0\)-cell. Its usual cell structure \(\CP ^2\cong S^2\cup _h e^4\) gives an exact sequence of spectra \[ \S [3]\xrightarrow {\ \Sigma ^{\infty }h\ }\S [2]\longrightarrow \Sigma ^{\infty }\Piinfty {\CP ^2}. \] Thus the attaching map of the top cell remains nonzero after stabilization, even though it induces the zero map on reduced integral homology. This is the simplest example in which the stable category retains information not visible to ordinary homology groups.