Much of homotopy theory originates from algebraic topology, the study of topological spaces (and of homotopy types) through algebraic means. Homotopy theory is now a separate field, but it does sometimes go back to its roots, and I sometimes do too.
In On the multiplicativity of the Euler characteristic, John Klein, Cary Malkiewich and I revisit the classical fact that Euler characteristics are multiplicative along fibrations, but with a twist: this is usually stated under the additional assumption that the base is a finite CW-complex (or equivalent to one), and we generalize this to finitely dominated CW-complexes. It seems like this was not known, and we give two proofs, both of which involving non-trivial facts about algebraic K-theory of integral group rings.
The less elementary proof is based on this paper with Shachar Carmeli, Bastiaan Cnossen and Lior Yanovski.