Atiyah–Hirzebruch spectral sequence

Remarks

The spectral sequence comes from the filtration of $X$ by its skeleta $X^{(p)}$. The $E_1$ term is $h^{p+q}(X^{(p)}, X^{(p-1)})$, which is a sum of copies of $h^q(\text{point})$, one for each $p$-cell, and $d_1$ is the cellular coboundary.

For complex K-theory, $E_2^{p,q} = H^p(X; \Z)$ for $q$ even and $0$ for $q$ odd. So $d_2 = 0$, and the first possible nonzero differential $d_3 \colon H^p(X; \Z) \to H^{p+3}(X; \Z)$ is the integral lift of the Steenrod square $Sq^3$. All differentials are torsion-valued, so the spectral sequence collapses after tensoring with $\Q$, which recovers the Chern character isomorphism.

Example: for $X = S^n$, the $E_2$ page has nonzero columns only at $p = 0$ and $p = n$, and for $K$-theory no differential can be nonzero when $n$ is even. This gives $\tilde{K}(S^n) \cong H^n(S^n; \Z) = \Z$ for even $n$.

Going deeper

For a student

The spectral sequence is the main computational tool for K-theory and cobordism of CW complexes. When the cohomology of $X$ is free and concentrated in even degrees, all differentials vanish for K-theory. Then $K^0(X)$ is free of rank equal to the sum of the Betti numbers; this applies to complex projective spaces and Grassmannians. The same method, applied to a Serre fibration and a generalized cohomology theory, gives a version of the Serre spectral sequence for that theory.