Stable J-homomorphism

Construction 10.2.13 (Stable J-homomorphism). Applying reduced suspension spectra to Proposition 10.2.11 gives a symmetric monoidal functor \[ \Vect _{\R }(\pt )^{\simeq } \xrightarrow {\,S^{(-)}\,} \An _* \xrightarrow {\Sigma ^\infty } \Sp . \] Its image lies in \(\Pic (\S )\), since \(\Sigma ^\infty S^V\cong \S [\dim (V)]\) is invertible. Since tensor product of spectra turns \(\Pic (\S )\) into a commutative group in animae, the resulting functor to \(\Pic (\S )\) extends uniquely over group completion, giving a symmetric monoidal functor \[ J\colon \BOP \longrightarrow \Pic (\S ). \] We call \(J\) the stable J-homomorphism. We use the same notation for its restriction to any component of \(\BOP \), in particular for \[ J\colon \bB O \simeq \{0\}\times \bB O \longrightarrow \Pic (\S ). \]