About

The Adams operations are ring homomorphisms $\psi^k : K(X) \to K(X)$ for $k \geq 0$ which are natural in $X$ and satisfy $\psi^k(\ell) = \ell^k$ where $\ell$ is the class of a line bundle. They have the following properties:

• $\psi^k\circ\psi^l = \psi^{kl}$
• $\psi^p(E) \equiv E^p \bmod p$ for $p$ a prime (i.e. $\psi^p(E) = E + pF$ for some $F$)

Note that if $\ell_1, \dots, \ell_n$ are classes of line bundles, then

$$\psi^k(\ell_1+\dots +\ell_n) = \psi^k(\ell_1) + \dots + \psi^k(\ell_n) = \ell_1^k + \dots + \ell_n^k = p_k(\ell_1, \dots, \ell_n)$$

where $p_k(x_1, \dots, x_n) = x_1^k + \dots + x_n^k$. As $p_k$ is a symmetric polynomial, it is a polynomial in the elementary symmetric polynomials $e_1, \dots, e_n$, i.e. there is a polynomial $s_k$ such that

$$p_k(x_1, \dots, x_n) = s_k(e_1(x_1, \dots, x_n), \dots, e_k(x_1, \dots, x_n)).$$

Now $e_i(\ell_1, \dots, \ell_n) = \bigwedge^i(\ell_1 + \dots + \ell_n)$. So, by using the splitting principle, one can show that

$$\psi^k(E) = s_k\left(\lambda^1(E), \dots, \lambda^k(E)\right)$$

where if $E = [V]$, $\lambda^j(E) = \left[\bigwedge^jV\right]$. The maps $\lambda^j : K(X) \to K(X)$ define a $\lambda$-ring structure on $K(X)$, see lambda-rings. Similarly, one can define Adams operations for any $\lambda$-ring.

Adams operations were introduced by Adams in his 1962 paper Vector Fields on Spheres where he calculated the maximum number of linearly independent vector fields there are on a sphere. Adams operations can also be used to solve the Hopf invariant one problem, see Hatcher's Vector Bundles and K-Theory Theorem 2.19.