U. Haifa Topology & Geometry seminar: Sunday, Dec. 24, 2023. Speaker: Ayelet Lindenstrauss (Indiana). Title: “Loday Constructions on Spheres and Tori”.
Given a commutative ring or ring spectrum, there is a functor from finite sets to commutative rings sending a set to the tensor product of copies of the ring (or ring spectrum) indexed by that set.
The Loday construction extends this to simplicial sets. The simplest nontrivial example is the Loday construction over a circle, which gives Hochschild homology (or, respectively, topological Hochschild homology).
I will discuss Loday constructions on higher spheres and on higher tori. The calculations for spheres are inductive, using the splitting of an n-sphere into two hemispheres, joined along the equator (n-1)-sphere.
The calculations for tori which we know are for cases where the Loday construction on the torus is equivalent to the Loday construction on the wedge of spheres whose suspension is homotopy equivalent
to the suspension of the torus. This kind of stability is unexpected, definitely not generally true, and yet there are surprisingly many cases where it holds.
All joint work with Birgit Richter; some parts are also joint with others.
