Adams spectral sequence

Remarks

The construction resolves $Y$ by a tower of spectra built from mod $p$ Eilenberg–MacLane spectra, the Adams resolution, and applies $[X, -]_*$. The filtration degree $s$ counts how many maps in the tower a class passes through.

The main case is $X = Y = S$, the sphere spectrum. Then $E_2^{s,t} = \operatorname{Ext}_{\mathcal{A}}^{s,t}(\F_p, \F_p)$ and the abutment is the $p$-primary part of the stable homotopy groups of spheres $\pi_{t-s}^s$ for $t - s > 0$. For $p = 2$, the group $\operatorname{Ext}^{1,t}$ has a basis of classes $h_j$ with $t = 2^j$, one for each $j \ge 0$, coming from the indecomposable squares $Sq^{2^j}$. The classes $h_0, h_1, h_2, h_3$ detect $2$, $\eta$, $\nu$ and $\sigma$, while $d_2(h_j) = h_0 h_{j-1}^2 \neq 0$ for $j \ge 4$; this is Adams' theorem on Hopf invariant one.

Without the finiteness hypotheses the spectral sequence still exists, but it may converge only to the homotopy of a $p$-completion of $Y$, or not converge at all.

Going deeper

For a student

The $E_2$ term can be computed by machine with minimal resolutions over the Steenrod algebra. In a range of degrees the differentials are known, and this gives the stable stems far beyond what other methods reach. Its algebraic structure, such as Massey products and the May spectral sequence for $E_2$, controls much of the computation. The Adams–Novikov spectral sequence replaces mod $p$ cohomology by complex cobordism and is better suited for periodic phenomena.