Friday, April 22, 2022 - 2:00pm
For a given ring R let NTC(R) denote the fiber of the map TC(R[t])->TC(R). In other words NTC surves as the obstruction to TC being homotopy invariant. While NTC(R) is almost never zero in its entirety, due to its close connection to algebraic K-theory it does tend to vanish in high degrees when R is regular and has positive characteristic. One might suspect that this behavior will persist in mixed characteristic when working with perfectoid rings. In this talk I will explain why this is not true. I will review the relevant features of TC and perfectoid rings, and discuss how the question of NTC vanishing is related to psuedo-isotopies of negatively curved manifolds.