Bott self-map

Construction 9.4.5 (The Bott self-map). The Bott class is represented by a morphism \(\beta \colon \S [2]\to \ku \), or equivalently by a morphism \(\beta \colon \S \to \ku [-2]\). Multiplication by this class defines a morphism of spectra \[ b\colon \ku \simeq \S \otimes \ku \xrightarrow {\beta \otimes \id } \ku [-2]\otimes \ku \xrightarrow {\mu [-2]}\ku [-2], \] where \(\mu \) denotes the multiplication on \(\ku \). We call \(b\) the Bott map.

By the compatibility of the preceding comparison with reduced external products, the map \[ b_*\colon \pi _n\ku \longrightarrow \pi _{n+2}\ku \] for \(n\geq 0\) is reduced external multiplication with the classical Bott class. More explicitly, if \(q\colon S^n\times \CP ^1\to S^n\wedge \CP ^1\) is the quotient map, then the corresponding reduced class is characterized by the fact that its pullback along \(q\) is the ordinary external product.