# Contents

## Idea

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

${J}_{i}:{\pi }_{i}\left(O\left(n\right)\right)\to {\pi }_{n+i}\left({S}^{n}\right)$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

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

## References

Created on May 24, 2012 00:55:37 by Urs Schreiber (89.204.155.65)