Computing with the $\mathsf{TC}$-canonical class

Abstract

The $\mathsf{TC}$-canonical class introduced by Costa and Farber is a very useful tool for studying the maximality of $\mathsf{TC}$, especially in the context of closed manifolds. We survey some calculations where this class has been used to prove the maximality of $\mathsf{TC}$. We also give some non-maximality results which have been established for manifolds with abelian fundamental group.