Batalin–Vilkovisky algebras and the J-homomorphism

Let X be a topological space. The homology of the iterated loop space H ∗ Ω n X is an algebra over the homology of the framed n-disks operad H ∗ f D n [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 ∗ f D n -algebra structure on H ∗ ( Ω n X ; Q ) . We show that the action of H ∗ ( SO ( n ) ) on the iterated loop space H ∗ Ω n X is related to the J-homomorphism and that the BV-operator on H ∗ ( Ω 2 X ) vanishes on spherical classes only in characteristic other than 2.