  The goal of this paper is to illustrate different approaches to understand
Euler characteristics in the setting of totally real commutative and
non-commutative Iwasawa theory. In addition to this, and in the spirit of
Hesselholt and Mitchell, we show how these objects interact with homotopy
theoretic invariants such as ($K(1)$-local) $K$-theory and Topological Cyclic
homology.
