Algebra/Topology seminar

Speaker: Jonathan Pedersen

Title: Tate L-theory and the Kervaire Invariant

Abstract: In this talk, I will introduce the Tate L-theory Lt(Z) of Z which is a particular commutative ring spectrum coming from hermitian K-theory. We completely calculate its homotopy groups, along with its ring structure. I will explain that Lt(Z) is a natural recipient for framed manifold invariants: The unit map will in degrees 4k take a manifold to its signature and in degrees 4k + 2 to its Kervaire invariant. Combined with a 3-step filtration of Tate L-theory by 2-modules this will allow us to reprove Browder’s theorem that elements with non-zero Kervaire invariant are detected on the 2-line of the Adams spectral sequence. This is joint work with Markus Land and Thomas Nikolaus.