Abstract. We use hypercovers to study the homotopy theory of simplicial presheaves. The main result says that model structures for simplicial presheaves involving local weak equivalences can be constructed by localizing at the hypercovers. One consequence is that the fibrant objects can be explicitly described in terms of a hypercover descent condition, and the fibrations can be described by a relative descent condition. We give a few applications for this new description of the homotopy theory of simplicial presheaves.

An object in motivic homotopy theory is called cellular if it can be built out of motivic spheres using homotopy colimit constructions. We explore some examples and consequences of cellularity. We explain why the algebraic K -theory and algebraic cobordism spectra are both cellular, and prove some Künneth theorems for cellular objects.

We present some data on the cohomology of the motivic Steenrod algebra over an algebraically closed field of characteristic 0. Our results are based on computer calculations and a motivic version of the May spectral sequence. We discuss features of the associated Adams spectral sequence and use these tools to give new proofs of some results in classical algebraic topology. We also consider a motivic AdamsNovikov spectral sequence. The investigations reveal the existence of some stable motivic homotopy classes that have no classical analogue. 55T15, 14F42

We show that if U * is a hypercover of a topological space X then the natural map hocolim U * →X is a weak equivalence. This fact is used to construct topological realization functors for the A 1 -homotopy theory of schemes over real and complex fields. In an appendix, we also prove a theorem about computing homotopy colimits of spaces that are not cofibrant.

We present a detailed analysis of 2-complete stable homotopy groups, both in the classical context and in the motivic context over C. We use the motivic May spectral sequence to compute the cohomology of the motivic Steenrod algebra over C through the 70-stem. We then use the motivic Adams spectral sequence to obtain motivic stable homotopy groups through the 59-stem. In addition to finding all Adams differentials in this range, we also resolve all hidden extensions by 2, η, and ν, except for a few carefully enumerated exceptions that remain unknown. The analogous classical stable homotopy groups are easy consequences.We also compute the motivic stable homotopy groups of the cofiber of the motivic element τ . This computation is essential for resolving hidden extensions in the Adams spectral sequence. We show that the homotopy groups of the cofiber of τ are the same as the E 2 -page of the classical Adams-Novikov spectral sequence. This allows us to compute the classical Adams-Novikov spectral sequence, including differentials and hidden extensions, in a larger range than was previously known. ContentsList of Tables vii Chapter 1. Introduction 1.1. The Adams spectral sequence program 1.2. Motivic homotopy theory 1.3. The motivic Steenrod algebra 1.4. Relationship between motivic and classical calculations 1.5. Relationship to the Adams-Novikov spectral sequence 1.6. How to use this manuscript 1.7. Notation 1.8. Acknowledgements Chapter 2. The cohomology of the motivic Steenrod algebra 2.1. The motivic May spectral sequence 2.2. Massey products in the motivic May spectral sequence 2.3. The May differentials 2.4. Hidden May extensions Chapter 3. Differentials in the Adams spectral sequence 3.1. Toda brackets in the motivic Adams spectral sequence 3.2. Adams differentials 3.3. Adams differentials computations Chapter 4. Hidden extensions in the Adams spectral sequence 4.1. Hidden Adams extensions 4.2. Hidden Adams extensions computations Chapter 5. The cofiber of τ 5.1. The Adams E 2 -page for the cofiber of τ 5.2. Adams differentials for the cofiber of τ 5.3. Hidden Adams extensions for the cofiber of τ Chapter 6. Reverse engineering the Adams-Novikov spectral sequence 6.1. The motivic Adams-Novikov spectral sequence 6.2. The motivic Adams-Novikov spectral sequence for the cofiber of τ 6.3. Adams-Novikov calculations Chapter 7. Tables Index Bibliography v 45 Hidden Adams-Novikov η extensions 46 Hidden Adams-Novikov ν extensions 47 Correspondence between classical Adams and Adams-Novikov E ∞ 48 Classical Adams-Novikov boundaries 49 Classical Adams-Novikov non-permanent classes