Tate L-theory and the Kervaire Invariant

In this talk, I will introduce the Tate L-theory $L^t(Z)$ 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 $L^t(Z)$ 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 $\mathbb{F}_2$-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.