The $E_2$-term of the Adams spectral sequence for $\mathbf{Y}$ may be described in terms its cohomology $E^\ast \mathbf{Y}$, together with action primary operations \mathbf{E}$ on it, ring spectra such as $\mathbf{E} = \mathbf{H}\mathbb{F}_p$. We show how higher can similarly order truncated $\mathbf{E}$-mapping algebra $\; - \;$ that is truncations function $\operatorname{Fun}(\mathbf{Y}, \mat...

1. The telescope conjecture and Bousfield localization 4 1.1. Telescopes 4 1.2. Bousfield localization and Bousfield classes 6 1.3. The telescope conjecture 8 1.4. Some other open questions 9 2. Some variants of the Adams spectral sequence 10 2.1. The classical Adams spectral sequence 11 2.2. The Adams-Novikov spectral sequence 12 2.3. The localized Adams spectral sequence 15 2.4. The Thomified...

Using operad methods and functional homology operations, we obtain inductive formulae for the differentials of the Adams spectral sequence of stable homotopy groups of spheres. The Adams spectral sequence was invented by Adams [1] almost fifty years ago for the calculation of stable homotopy groups of topological spaces (in particular, those of spheres). The calculation of the differentials of ...

Let E be a ring spectrum for which the E-Adams spectral sequence converges. We define a variant of Mahowald’s root invariant called the ‘filtered root invariant’ which takes values in the E1 term of the E-Adams spectral sequence. The main theorems of this paper concern when these filtered root invariants detect the actual root invariant, and explain a relationship between filtered root invarian...