University of Arkansas Topology Seminar: 9/23/2021

Speaker: Jorge Robinson

Title: Algebraic topology meets geometric group theory through model categories on Grothendieck topologies

The homotopy groups of spheres in modern homotopy theory are usually computed using the Adams or Adams–Novikov spectral sequence. This method has the disadvantage of being computationally demanding. We can also identify the homotopy groups of spheres as framed cobordism groups. Calculating cobordism groups through geometric topology require the introduction of cobordism invariants. One famous cobordism invariant is the T.C. Wall surgery obstruction who lives in the algebraic K–theory of quadratic forms (L–theory) of the fundamental group. A particular case of the surgery obstruction is the signature of manifolds. To calculate the algebraic K–theory of the fundamental group one makes use of the Farrell–Jones conjecture, which let us calculate it using virtually cyclic subgroups of the fundamental group. This conjecture has been proven for wild geometric groups and we can take advantage of these recent theorems. Unfortunately, the fundamental group of the n–spheres is trivial. In parallel to the étale topology and motivic homotopy, we made use of model categories on Grothendieck topologies to modify the fundamental group of Sn to obtain an exotic fundamental group. Thus, we can define the surgery obstruction living in the L–theory of these wild new fundamental groups.



BACK TO SCHEDULE