Relation to the classical J-homomorphism

Remark 10.2.14 (Relation to the classical J-homomorphism). The connected component of \(\S \) in \(\Pic (\S )\) is equivalent to \(\bB \GL _1(\S )\), where \(\GL _1(\S ) := \Aut _{\Sp }(\S )\) is the automorphism group of the sphere spectrum. The restriction of \(J\) to the virtual-rank-zero component lands in this connected component. Passing to loops gives a group homomorphism \(O\to \GL _1(\S )\) in \(\An \), whose effect on positive homotopy groups is the classical stable J-homomorphism \(\pi _n(O)\to \pi _n(\S )\). We refer to Ravenel (1986), Chapter 1 for a classical discussion of this map.