nLab
J-homomorphism

Idea

The $J$-homomorphism is traditionally a family of group homomorphisms

$$J_i:{\pi }_i(O(n))\to {\pi }_{n+i}(S^n)$$J_i : \pi_i(O(n)) \to \pi_{n+i}(S^n)

from the homotopy groups of the orthogonal group to those of the sphere spectrum.

This refines to a morphism of infinity-groups

$$J:O\to {\mathrm{GL}}_1(𝕊)$$J : O \to GL_1(\mathbb{S})

to the infinity-group of units of the sphere spectrum, regarded as an E-infinity ring spectrum.

Applications

By postcomposition with $BJ:BO\to B{\mathrm{GL}}_1(𝕊)$ the $J$-homomorphism sends real vector bundles to sphere-spectrum bundles ((infinity,1)-vector bundles over the sphere spectrum). See also at Thom space.

Related concepts

• orientation in generalized cohomology

References

• Akhil Mathew, Notes on the J-homomorphism (pdf)