Homotopy Theory Topics Seminar

The Steenrod algebra of (stable) primary operations in mod p cohomology has a rich and fruitful history in homotopy theory, notably with the Adams spectral sequence. Secondary cohomology operations detect additional information not seen by primary operations. We will introduce secondary operations and discuss some of their properties. Then we will present sample calculations and applications from classical homotopy theory, such as the Peterson-Stein formulas and some homotopy groups of spheres.

We will use (primary and) secondary cohomology operations to describe the structure of the stable stems in low dimensions and compute a few Toda brackets in the stable stems.

Notes from the talk

The sequel.

Notes from the talk

We will present a theorem due to J. Cohen that the stable homotopy groups of spheres are generated under higher Toda brackets by the classes in Adams filtration one: the Hopf classes as well as the first alpha element (for odd primes).

I will sketch Adams' original proof of the Hopf invariant one problem.

I will talk about the definition of higher Toda brackets in triangulated categories and possibly about how these are determined by the triple Toda brackets.

We will present a theorem due to R.M.F. Moss which says, roughly, that 3-fold Massey products of permanent cycles in the Adams spectral sequence converge to the corresponding 3-fold Toda brackets in stable homotopy.

The $E_2$ term of the Adams spectral sequence is given by Ext groups over the Steenrod algebra, namely the algebra of primary (stable) cohomology operations. In this talk, we will present work of Baues and Jibladze on secondary chain complexes and secondary derived functors, which generalize the usual chain complexes and derived functors in homological algebra. With this machinery, the $E_3$ term can be expressed as a secondary Ext group over the algebra of secondary cohomology operations.

We give an introduction to A-infinity algebras in this talk, which is a generalisation of differential graded algebras. We show that for a graded algebra A, the Ext-algebra has an A-infinity structure that contains sufficient information to recover A. On the other hand, we will present an example where the usual associative algebra structure on the Ext-algebra cannot recover A. We also show that the A-infinity structure is closely related to Massey products.

Notes from the talk

I will discuss universal Toda brackets due to Sagave. They are Mac Lane cohomology classes that determine Toda brackets in certain stable homotopy theories and provide an obstruction theory to the problem of realizing $\pi_* R$-modules as $R$-modules for a ring spectrum $R$.