Batalin–Vilkovisky algebras and the J-homomorphism

Let X be a topological space. The homology of the iterated loop space H∗ΩnX is an algebra over the homology of the framed n-disks operad H∗fDn [E. Getzler, Batalin–Vilkovisky algebras and two-dimensional topological field theories, Comm. Math. Phys. 159 (2) (1994) 265–285; P. Salvatore, N. Wahl, Framed discs operads and Batalin–Vilkovisky algebras, Q. J. Math. 54 (2) (2003) 213–231]. We explicitly determine this H∗fDn-algebra structure on H∗(ΩnX;Q). We show that the action of H∗(SO(n)) on the iterated loop space H∗ΩnX is related to the J-homomorphism and that the BV-operator on H∗(Ω2X) vanishes on spherical classes only in characteristic other than 2.